Testing fundamental physics with distant star clusters: theoretical models for pressure-supported stellar systems
Hosein Haghi, Holger Baumgardt, Pavel Kroupa, Eva K. Grebel, Michael Hilker, Katrin Jordi
aa r X i v : . [ a s t r o - ph . GA ] F e b Mon. Not. R. Astron. Soc. , 1– ?? (2008) Printed 3 November 2018 (MN L A TEX style file v2.2)
Testing fundamental physics with distant star clusters:theoretical models for pressure-supported stellar systems
Hosein Haghi , ⋆ , Holger Baumgardt , Pavel Kroupa , Eva K. Grebel ,Michael Hilker and Katrin Jordi Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P. O. Box 45195-1159, Zanjan, 45195, Iran Argelander Institute for Astronomy (AIfA), Auf dem H¨ugel 71, D-53121 Bonn, Germany Astronomisches Rechen-Institut, Zentrum fuer Astronomie, Universitaet Heidelberg/Germany ESO, Garching/Germany
Accepted . . . . Received . . . ; in original form . . .
ABSTRACT
We investigate the mean velocity dispersion and the velocity dispersion profile of stel-lar systems in MOND, using the N-body code N-MODY, which is a particle-meshbased code with a numerical MOND potential solver developed by Ciotti, Londrilloand Nipoti (2006). We have calculated mean velocity dispersions for stellar systemsfollowing Plummer density distributions with masses in the range of 10 M ⊙ to 10 M ⊙ and which are either isolated or immersed in an external field. Our integrations repro-duce previous analytic estimates for stellar velocities in systems in the deep MONDregime ( a i , a e ≪ a ), where the motion of stars is either dominated by internal ac-celerations ( a i ≫ a e ) or constant external accelerations ( a e ≫ a i ). In addition, wederive for the first time analytic formulae for the line-of-sight velocity dispersion inthe intermediate regime ( a i ∼ a e ∼ a ). This allows for a much improved compari-son of MOND with observed velocity dispersions of stellar systems. We finally derivethe velocity dispersion of the globular cluster Pal 14 as one of the outer Milky Wayhalo globular clusters that have recently been proposed as a differentiator betweenNewtonian and MONDian dynamics. Key words: galaxies: clusters: general- galaxies: dwarf - gravitation - methods:analytical – methods: N-body simulations
The flattening of rotation curves of disk galaxies at largeradial distances, i.e. the apparently non-Newtonian mo-tion, is usually explained by invoking the otherwise unde-tected, so called Cold Dark Matter (CDM) (Bosma 1981;Rubin & Burstein 1985). This hypothesis has successfullyexplained the internal dynamics of galaxy clusters, grav-itational lensing and the standard model of cosmologywithin the framework of general relativity (GR). Despitethe fact that the dark matter model has been notably suc-cessful on large scales (Spergel 2003), dark matter parti-cles has not been detected after much experimental ef-forts and the results of high resolution N-body simula-tions do not seem to be compatible with observationson galactic scales (Klypin et al. 1999; Moore et al. 1999; ⋆ E-mail: [email protected] (HH); [email protected](HB); [email protected] (PK);[email protected] (EKG); [email protected](MH);[email protected] (KJ)
Metz et al. 2008). Another approach to explain galaxy ro-tation curves would be an alternative theory of gravity. Onepromising alternative theory is Modified Gravity (MOG)which has recently been successfully applied for dwarf satel-lite galaxies (Moffat & Toth 2007a) and distant globularclusters (Moffat & Toth 2007b). One of the most famous al-ternative theories is the so-called modified newtonian dy-namics (MOND) theory, which was introduced by Milgrom(1983). According to MOND, the flat rotation curves of spi-ral galaxies at large distances can be explained by a modifi-cation of Newton’s second law of acceleration below a char-acteristic scale of a ≃ − ms − without invoking darkmatter(Bekenstein & Milgrom 1984).It has been shown that on galactic scales MONDcan explain many phenomena at least as well as CDM(Sanders & McGaugh 2002). For example, Sanchez-Salcedoand Hernandez (2007) studied the tidal radii of dis-tant globular clusters and dwarf spheroidal satellite galax-ies in MONDian dynamics. The most serious chal-lenges for MOND come from clusters of galaxies, whereMOND cannot completely explain the galaxy velocities c (cid:13) Haghi et al. (Sanders & McGaugh 2002), and the merging of galaxyclusters, where the baryonic mass is clearly separatedfrom the gravitational mass, as indicated by gravitationallensing (Clowe et al. 2006). Both phenomena can be ex-plained in MOND if some kind of hot dark matter is as-sumed, perhaps in the form of a massive ( ∼ ρ is given by(Bekenstein & Milgrom 1984): ∇ · ( µ ( aa ) a ) = 4 πGρ = ∇ · a N , (1)where a N is the Newtonian acceleration vector, a is theMONDian acceleration vector, a = | a | is the absolute valueof MONDian acceleration and µ is an interpolating func-tion which runs smoothly from µ ( x ) = x at x ≪ µ ( x ) = 1 at x ≫
1. The standard interpolating functionis µ ( x ) = x √ x , but Famaey & Binney (2005) suggestedanother function µ ( x ) = x x , which provides a better fitto the rotation curve of the Milky Way. Equation (1) can betransformed into ∇· ( µ ( aa ) a − a N ) = 0, where the expressionin parentheses is thus a curl field, and we may write µ ( aa ) a = a N + ∇ × H . (2)The value of the curl field H depends on the boundary con-ditions and the mass distribution, but vanishes for some spe-cial symmetries. In realistic geometries, the curl field is non-zero and leads to difficulties for standard N-body codes. Inother words, the non-linearity of the MOND field equationmakes the use of the usual Newtonian N-body simulationcodes impossible in the MOND regime.Many stellar systems (e.g. globular clusters) have tidalradii much larger than their sizes, therefore the external fieldis approximately constant over the cluster area and the mo-tions of stars are not influenced by tidal effects.In Newtonian dynamics, a stellar system evolving un-der the influence of a uniform external acceleration, will, inthe frame of the system, have the same internal dynamicsas an isolated system. In MOND, due to the non-linearity ofPoisson’s equation, the strong equivalence principle (SEP)is violated (Bekenstein & Milgrom 1984), and consequentlythe internal properties and the morphology of a stellar sys-tem are affected both by the internal and external field. Thisso-called external field effect (EFE) significantly affects non-isolated systems and can provide a strict test for MOND.The EFE postulation originated from observations of openclusters in the solar neighborhood, which do not show massdiscrepancies even if the internal accelerations are below a (Milgrom 1983). The EFE has several consequences, for ex-ample it allows high velocity stars to escape from the poten-tial of the Milky Way (Famaey et al. 2007; Wu et al. 2007),and it decreases the velocity of satellite galaxies at very largeradii, which is in conflict with the asymptotically flatteningof rotation curves (Gentile et al. 2007; Wu et al. 2008). TheEFE implies that rotation curves of spiral galaxies shouldfall where the internal acceleration becomes equal to theexternal acceleration. In addition, if the EFE is taken into account, internal properties of Galaxies such as the Tully-Fisher relation should be changed (Wu et al. 2007).Milgrom derived the mean velocity dispersion of stellarsystems for two special cases of internal or external fielddominated systemsanalytically, assuming that the systems are everywherein the deep-MOND regime ( a e , a i ≪ a ). If the externalacceleration a e is much larger than the internal one a i , thesystem of mass M is in the quasi Newtonian regime but witha normalized gravitational constant larger than the standardNewtonian one by a factor a a e , and therefore the line-of-sightvelocity dispersion is (Milgrom 1986) σ LOS,M = σ LOS,N r a a e , (3)where σ LOS,N is the Newtonian velocity dispersion. If a e ≪ a i ≪ a , the cluster is isolated and the line-of-sight velocitydispersion is given by σ LOS,M = 0 . GMa ) . (4)Many systems which can be used to test MOND are notcompletely internally or externally dominated, for exampleglobular clusters or dwarf galaxies of the Milky Way haveinternal and external accelerations which are of the same or-der (Baumgardt et al. 2005). Since Milgrom’s relations arevalid only for systems that have either a i ≫ a e or a i ≪ a e and are in the deep-MONDian regime, one has to deter-mine the velocity dispersions numerically for intermediatecases. Milgrom found that for isolated systems (internal ac-celeration dominated), the mass M of a system is nearlyproportional to the forth power of the line of sight velocitydispersion σ los and the ratio σ los /GM must be somewherebetween a and a . But how does the velocity dispersionchange while the system transits from the Newtonian to theMONDian regime? In an attempt to answer this question,we have performed N-body simulations and present analyt-ical formulae for the velocity dispersion of stellar systemsin the intermediate MOND regime. We have calculated thevelocity dispersion for a number of isolated systems in whichthe internal accelerations a i are in the range from a i ≪ a to a i ≫ a . We also give formulae for systems with differ-ent strengths of external fields. It should be noted that theisolated systems are in equilibrium only in the Newtoniancase, and reach a MONDian equilibrium state after collapse.For non-isolated systems we start from the MONDian equi-librium state which is created as described in section 4.3.These results could be useful for comparison with observa-tional data of several GCs and dSph galaxies that are faraway from the host galaxy, so that the external accelerationdue to the host galaxy is small ( a e < . a ) these objectsshould therefore provide straightforward possibilities to testMOND. Since the external field affects the velocity disper-sion by both tidal effects and EFE, and in order to see thepure MONDian effects, we concentrate on systems in whichthe tidal radius is much larger than the virial radius andtherefore tidal effects are unimportant.This is the first of a series of papers that deals withthe numerical calculations for stellar systems. In the forth-coming papers, the observational constraints on mass andvelocity dispersion of Pal 14 will be studied by Hilker et al.(2008) and Jordi et al. (2008). c (cid:13) , 1– ?? elocity dispersion in MOND The paper is organized as follows: In Section 2 we in-troduce theoretical predictions for the velocity dispersionin different regimes. In Section 3, we give a brief review ofthe N-MODY code which we use for our modelling. Thenumerical results for isolated and non-isolated systems andcomparison with observational data are discussed in Section4. We present our conclusion in Section 5.
In Newtonian gravity, the mean-square velocity, σ , of a stel-lar system of mass M is given by the following equation(Equation (4-80a) of Binney and Tremaine (1987)): σ = GMr g , (5)where r g is the gravitational radius defined as (Equation(2-132) of Binney and Tremaine (1987)): r g = GM | W | . (6)Here W is the total potential energy. In the case of a Plum-mer model (Plummer 1911), and if we assume an isotropicvelocity distribution, the line-of-sight velocity dispersion be-comes σ LOS,N = 0 . r GMR h , (7)where R h is the half-mass radius. If we define the half-mass-radius acceleration as a h = GM R h , we can re-write the aboverelation as σ LOS,N GM = 0 . a h . (8) In the case of MOND, and in the presence of an externalfield, the total acceleration, which is the sum of the inter-nal a i and external a e acceleration, satisfies the modifiedPoisson equation (Bekenstein & Milgrom 1984), ∇ . [ µ ( a e + a i a )( a i + a e )] ≃ πGρ, (9)where a e is approximately constant, a i = ∇ φ is the non-external part of the potential and ρ is the density of the starcluster. The boundary condition is ∇ φ = a e ˆ x for r → ∞ .Equation 9 was postulated by Milgrom (1983) to explain thedynamical properties of nearby open clusters in the MilkyWay and is an outcome of the MOND phenomenology. As anapproximation for a spherical system one can write equation9 as: a i µ ( | a e + a i | a ) = a N . (10)Note however that Eq.10 is only an approximate andeffective way to take into account the external field effect(EFE), in order to avoid solving the modified Poisson equa-tion with an external source term ρ ext on the right-handside. The EFE is indeed a phenomenological requirement of MOND, which has important consequences for non-isolated systems. For example, if a e ≪ a i ≪ a , thenthe dynamics is in the MOND regime, and the externalfield can be neglected. When a i ≪ a e ≪ a , µ tendsto its asymptotic value µ ( a e /a ) = a e /a ( saturation ofthe µ function ), and the gravitational potential is thusNewtonian with a renormalized gravitational constant to( G eff = G/µ ( a e /a ) ≈ Ga /a e [(Milgrom 1986)]).Recently, several papers were published usingthis formulation to take into account the EFE(Gentile et al. 2007; Wu et al. 2007; Wu et al. 2008;Angus 2008; Famaey et al. 2007; Klypin & Prada 2008).For example, in order to estimate the order of magnitudeof the EFE, Famaey et al. (2007) and Gentile et al. (2007),pointed out a a ( a + a e ) / ( a + a + a e ) = a N using a simple µ -function. Other authors replaced | a i + a e | = p a i + a e (Angus 2008; Klypin & Prada 2008). A more rigoroustreatment of EFE on galactic rotation curves was made byWu et al. (2007, 2008). In the work by Wu et al. (2007), forthe mass density of the internal system, the MOND Poissonequation was solved as if the system was isolated, but theboundary condition on the last grid point was changed tobe nonzero.As a first approximation, we considered that a cluster isin a non-inertial frame, which free-falls with a uniformsystematic acceleration. Since the calculation of φ is donefor an isolated cluster, we did not change the boundarycondition and at each step of potential solving, we addedthe constant external field with a i inside the µ function.This method might be only an approximation but as it isclear from the figures 1,3 and 4, the transition region fromNewtonian to MONDian case is reproduced reasonably wellby our method.Analytical solutions exist only for some special casesthat can be subdivided as follows:1 - If a i ≫ a or a e ≫ a then the value of the interpolatingfunction is equal to one and the system is in the Newtonianregime and the velocity dispersion is given by equation (8).2 - If a e ≪ a i ≪ a , the system is in the deep MOND regimeand the external field can be neglected (isolated system).In this case, the line-of-sight velocity dispersion is given byEquation (4), which can be re-written as σ LOS,M GM = 0 . a . (11)3 - If a i ≪ a e ≪ a , the system is externally field domi-nated. µ ( x ) becomes in this case µ ( a e /a ) = a e /a = const (saturation of the µ function) and the system goes to aquasi-Newtonian regime but with an effective gravitationalconstant G e = Gµ − ( a e /a ) ≃ Ga /a e that is larger thanthe standard Newtonian one. Using equation (3), the line-of-sight velocity dispersion is therefore equal to, σ LOS,M GM = 0 . a a e ) a h . (12)Comparing equations (8), (11) and (12) with each other sug-gests that the line-of-sight velocity dispersion in the generalcase should be given by σ LOS GM = f ( a h ) , (13)where f ( a h ) ∼ a h if a h ≫ a and f ( a h ) ∼ a if a h ≪ a . c (cid:13) , 1– ?? Haghi et al.
In this paper, we attempt to investigate the universalfunctional form for f ( a h ) for systems with a wide range ofinternal and external accelerations. In order to numerically solve the non-linear MOND fieldequations, recently two N-body codes have been developed(Ciotti et al. 2006; Tiret & Combes 2007). In the presentwork we apply the N-MODY code developed by thefirst group, which can be used to do numerical exper-iments in either MONDian or Newtonian dynamics. N-MODY is a parallel, three-dimensional particle-mesh codefor the time-integration of collision-less N-body systems(Londrillo & Nipoti 2008). The potential solver of N-MODYis based on a grid in spherical coordinates and is best suitedfor modeling isolated systems. N-MODY uses the leap-frogmethod to advance the particles. The code and the potentialsolver have been presented and tested by Ciotti et al. (2006)and Nipoti et al. (2007).In the present study we used a spherical grid ( r, θ, ϕ )made of N r × N θ × N ϕ = 64 × ×
128 grid cells for the inte-gration. We use twice as many cells in the ϕ direction since ϕ runs from 0 < ϕ < π while θ runs only from 0 < θ < π . Thetotal number of particles was in the range N p = 10 − .The details of the scaling of the numerical MOND modelsand code units are discussed in Nipoti, Londrillo & Ciotti(2007). In order to include the EFE for non-isolated systems,we changed the N-MODY code and put a constant exter-nal field in the MONDian potential solver. We also chose a = a i + a e within the interpolating function as the totalacceleration of particles.In the present work, the Plummer model (Plummer1911) was used as the initial cluster model. It has a den-sity distribution ρ ( r ) = 3 M πr Pl (cid:18) r r Pl (cid:19) − / (14)where M is the total mass and r Pl is the ’scale radius’. Thehalf-mass radius of a Plummer model is R h ≃ . r Pl andthe virial radius is R v = π r Pl . The total potential energy, | W | = π GM r Pl , is used in equation (6) to calculate r g . In this Section, we present N-MODY solutions for stellarsystems that are both isolated and non-isolated, allowingfor different values of the external field.
We have performed a large set of dissipationless N-MODYcomputations for isolated systems. Since the modeled sys-tems are in equilibrium in the Newtonian case, in the MON-Dian case, they initially collapse. In order to have a MON-Dian equilibrium initial system, we rescaled the velocitiesby an amount given by our fitting formulae ( Section 4.2Equations 15 ) to prevent collapse. In order to create theinitial condition, there is another substantial method devel-oped by Nipoti et al. (2007b, 2008), in which the distribution -3 -2 -1 0 1 2 3
Log (a h /a ) -2-1.6-1.2-0.8-0.400.40.8 L og ( σ / G M a ) Figure 1.
Line-of-sight (LOS) velocity dispersion profiles for iso-lated stellar systems as calculated by N-MODY. In order to havedifferent internal half-mass accelerations a h , several cases withdifferent half-mass radii were calculated. As expected from equa-tion (13), all curves follow the same functional form. The dashedline shows the asymptotic behavior in the Newtonian (equation8) regime. The solid line shows the asymptotic behavior in deepMONDian (equation (11)) regime. For high internal acceleration( a h ≫ a ), the models are consistent with the Newtonian resultand for low acceleration ( a h ≪ a ), they are consistent with thedeep MONDian prediction. ( Log ≡ log ). -2 -1 0 1 Log (a h /a ) -0.0500.05 S i m u l a ti on -f it f un c ti on -3 -2 -1 0 1 2 Log (a h /a ) -2-1.5-1-0.500.51 L og ( σ ( G M a ) - ) fit functionsimulation Figure 2.
Fit of our best fitting curve (equation (15)) to the nu-merical solution for an isolated stellar system. The difference ofeach point from the fit function is presented in the inset. The av-erage residual of this function from our numerical solution, whichis defined as ∆ = | f theory − f fit | , is less then 10 − . ( Log ≡ log ).c (cid:13) , 1– ?? elocity dispersion in MOND -3 -2 -1 0 1 2 3 Log (a h /a ) -2-1.6-1.2-0.8-0.400.40.8 L og ( σ / G M a ) µ = x (1+x ) -0.5 µ = x (1+x) -1 Figure 3.
Effect of different choice of interpolation function onthe line-of-sight velocity dispersion for systems with different in-ternal accelerations. Both functions have the same value in theNewtonian and the deep MONDian regime. In the transition zone,the simple function, µ , produces a larger velocity dispersion.This means that if the observed velocity dispersion of a stellarsystem shows a value smaller than the MONDian prediction withthe standard interpolation function, µ , the simple function µ would not help to decrease this discrepancy. The largest differencebetween both functions is of order 20% and occurs at a h = a .( Log ≡ log ). function is obtained numerically with an Eddington inver-sion with the far field logarithmic behavior of the MONDpotential. Here we have used our method to set up MON-Dian initial condition. This method could generalis to nonisolated systems easily (see section 4.2).As discussed in Nipoti et al. (2007a), all simulationswith different masses but with the same value of a h areidentical, in the sense that they can be simply rescaled todifferent masses, provided that M/r h = 2 a h /G remains con-stant. As a consequence, systems of any mass with a h in theexplored range follow the same functional form. Therefore,we consider only one simulation for given a h . In order to pro-duce different internal acceleration regimes, we changed thehalf-mass radii of the system from 1 pc to 1 kpc. The modelsare evolved for several crossing times to reach the equilib-rium state, which is identified by stationary Lagrange radii(e.g. Fig 6).The resulting global velocity dispersions as a functionof internal acceleration of the stellar systems are plottedin Fig. 1. As expected from equation (13), all of them fol-low the same functional form. The dashed line shows theNewtonian prediction for the velocity dispersion (equation(8)). The asymptotic behavior of the models in the Newto-nian regime are compatible with this analytical prediction.The solid line shows the analytical velocity dispersion in thedeep MONDian regime (equation (11)). In the low accelera-tion region, the numerical solutions are compatible with theanalytical formula. At a h = a , the difference between the -6 -4 -2 0 2 Log (a h /a ) -4-3-2-1012 L og ( σ / G M a ) Simulation resultNewtonian resultM2M1
Figure 4.
The line-of-sight velocity dispersions for stellar sys-tems with an external field of a e = 0 . a with different internalhalf-mass-radii accelerations as calculated by N-MODY (blackline with open squares). M M µ function in the external field dominated regime,the velocity dispersion curve (open squares) starts to fall in aquasi Newtonian way (blue line with open circles) with decreas-ing a h and deviates from the prediction of MOND for isolatedsystems, M
1, (green solid line with closed diamond). Moving to-wards decreasing a h , the first transition occurs near a h ≈ a ,when the system enters into the MONDian regime and the veloc-ity dispersion deviates from the Newtonian prediction (red dashedline with filled circles). The horizontal axis gives the Newtonianhalf-mass-radius acceleration. Since the Newtonian internal ac-celeration of a system is the square of the MONDian acceleration( a M = √ a N a ), the point log ( a h,N a ) = −
4, corresponds to a h,M = 0 . a in MOND. The velocity dispersion remains on thehorizontal line which corresponds to the isolated system until theinternal acceleration reaches a i ≈ a e ≈ . a . ( Log ≡ log ). numerical model and the MONDian prediction is about 0 . log , which means that σ ( a h ) ≃ . × σ LOS,M .We now try to find an expression for a function f ( x )where x = a h a which fits the numerical results. In the New-tonian regime ( x ≫ f has to approach f ( x ) = x + const , while in the deep MONDian regime( x ≪ f has to be constant. We therefore make an ansatz, f ( x ) = a ln(exp( xa ) + b ) + c, (15)for the function f . The best-fitting coefficients are then de-termined by a least-squares fit to the data and are foundto be a = 0 . b = 1 .
78, and c = − .
48. This functionis shown in Fig. 2 as a solid line. The average residual ofthis function from our numerical results, which is defined as∆ = | f theory − f fit | , is less then 10 − . Therefore, for any iso-lated system, if the internal half-mass-radius acceleration, a h , can be measured, it is possible to find out the MON-Dian prediction by this function. This is especially useful c (cid:13) , 1– ?? Haghi et al. -6 -4 -2 0 2 4
Log (a h /a ) -4-3-2-1012 L og ( σ / G M a ) a e = 0a e = 0.01 a a e = 0.03 a a e = 0.1 a a e = a a e = 10 a Figure 5.
External field effect on predicted line-of-sight velocitydispersions for stellar systems with different internal accelerationsas calculated by N-MODY. The x-axis gives the Newtonian in-ternal acceleration of the system. In order to see the transitionregime we assume several values of a e . When the internal ac-celeration of the system decreases, there are two transitions inthe velocity treatment. The first transition occurs near a h ≈ a from Newtonian into MONDian regime and the second transi-tion from the MONDian to quasi Newtonian regime occurs whenthe internal acceleration becomes equal to the external accelera-tion. The functional form of each fit curve is given in Table (1).( Log ≡ log ). for the intermediate case which has no analytical predictionin MOND. The corresponding formula for the velocity dis-persion islog ( σ LOS ) = 0 . { .
331 ln[exp (cid:18) . GMa R h (cid:19) + 1 . − .
48 + log ( GMa ) } . A simple relation exists between the three-dimensional half-mass radius and easier to observe two-dimensional, pro-jected half-mass radius R hp : R hp = γR h with γ ≈ . µ function on the results. In Fig. 3 we plot the velocity dis-persion for an isolated system using the simple interpolationfunction µ , and compare it with the results obtained for µ .Since the simple function has a stronger MONDian effect,the velocity dispersion is higher than that of the standardfunction. The difference of the velocity dispersion betweenboth functions at a h = a is about 20%, so in order to deter-mine the µ function from observations, one needs to measurethe mass and overall structure of a stellar system very ac-curately. As expected, in the extreme limit of a h ≪ a or a h ≫ a , both functions predict the same value for the ve-locity dispersion. Systems relevant for testing MOND (e.g. globular clusters ordwarf galaxies) usually move through the gravitational fieldof a host galaxy. Therefore, the internal dynamics is ofteninfluenced by the host galaxy due to the EFE of MOND.We assume that coriolis forces that arise in the rotatingreference frame of the cluster and that tidal forces arisingfrom a gradient of the external field can be neglected. Webelieve this to be a a good first approximation that allowsus to focuss on the effects of the (constant) external field,therewith allowing us to for the first time venture into theintermediate MOND regime in order to study the externalfield effect numerically.As an example which shows the EFE on the predictedLOS velocity dispersion for non-isolated stellar systems withdifferent internal accelerations, we choose an external accel-eration of a e = 0 . a . This corresponds to a cluster ordwarf galaxy being at a distance of about 1 Mpc from theGalactic center for an enclosed Milky Way mass of 10 M ⊙ .The resulting velocity dispersion as calculated by N-MODYis shown in Fig. 4. The first transition occurs near a h ≈ a ,when the systems enter the MONDian regime and the veloc-ity dispersion deviates from the Newtonian prediction. Thevelocity dispersion remains close to the MOND predictionfor the isolated case until the internal acceleration reaches a i ≈ a e ≈ . a at which point a second transition occurs.Due to the saturation of the µ function in the external fielddominated regime, the velocity dispersion curve falls in aquasi Newtonian way if a h < a e and therefore deviates fromthe prediction of MOND for isolated systems.Note that the internal acceleration shown in Fig. 4 is theNewtonian internal acceleration, a h = GM r h , of the system. Inthe deep MOND regime, since the Newtonian accelerationis the square of the MONDian acceleration ( a M = √ a N a ),the point a h,N = 0 . a corresponds to a h,M = 0 . a .In order to see the effect of different external acceler-ations, the MONDian velocity dispersion as a function ofinternal acceleration is plotted in Fig. 5 from a weak to astrong external field. Note that the transition point is de-termined by the strength of the external field. For a smallerexternal field, the transition point occurs at a smaller accel-eration. As predicted by theory, for a strong external field( a e ≫ a ) the system is completely in the Newtonian regime,even for a low internal acceleration.In order to find out the best functional form for thevelocity dispersion as a function of the strength of the ex-ternal field, we use the same procedure as in the isolatedcase, but take into account the different asymptotic behav-ior in Fig. 5. In the Newtonian regime, ( x ≫ f ( x ) stillchanges as f ( x ) = x + const . In the deep MONDian regime( x ≪ f ( x ) is again linear with the same slope as in the Newtonianregime. In the intermediate regime, in which the system isinternal-acceleration dominated, σ los /GM has to be con-stant. A general function satisfying all these constraints istherefore given by f ( x ) = f ( x ) − a ln(exp( − xa ) + b ) + c. (17)The coefficients a , b and c depend on the external accel-eration. Table 1 gives their values for several values of theexternal acceleration a e . Since the asymptotic value of dif- c (cid:13) , 1– ?? elocity dispersion in MOND -1 -0.5 0 0.5 1 Log (radius) [arcmin] -1-0.500.511.52 L og ( Σ )[ a r c m i n - ]
25 50 75 100
R (pc) -5 -4 -3 -2 -1 ρ ( M p c - ) M=1000 M M=4000 M M=10000 M M=40000 M M=100000 M Figure 7.
Upper panel: Density profile of Pal 14 for differentcluster masses obtained by N-MODY computations. The shapesof the profiles are the same for all masses. Lower panel: Surfacedensity profile of Pal 14 for masses as in the upper panel scaled tothe level of data. The surface density profile shapes compare wellwith the observed density profile of Pal 14 (blue dots) as tracedby giant stars (Hilker 2006). The meaning of the different lines isas in the upper panel. (
Log ≡ log ). ferent interpolating functions is the same, the choice of the µ -function does not affect systems which are in the low accel-eration regime. However for the higher acceleration systems( a h ∼ a ), the µ -function plays a more important role. Equa-tion (17) will for example allow it to test MOND against theobserved global velocity dispersions of dwarf galaxies. Therelevant external acceleration can be determined by interpo-lating between the points computed with N-MODY (Table1). log (Mass) [solar mass] V e l o c it yd i s p e r s i on [ k m / s ] Newtonian resultsNumerical solutionsM1M2
Figure 8.
Line-of-sight velocity dispersion for Pal 14 for vari-ous masses as found by N-MODY. In order to compare with thereal cluster (observational velocity dispersion of Pal 14), the halfmass radii of all models are fixed at 33 pc. The analytical predic-tions for different limiting cases are also plotted for comparison. M M M M In order to decide whether MOND or dark matter is the righttheory to explain the dynamics of the universe, it is desir-able to study MOND for objects in which no dark matter issupposed to exist and where the characteristic accelerationof the stars is less than the MOND critical acceleration pa-rameter a . GCs are a perfect candidate since they are thelargest virialized structure that does not contain dark matter(Moore 1996), and their internal accelerations can be lowerthan a . Hence, GCs may provide a good laboratory to testthe law of gravity (Baumgardt et al. 2005).We choose the globular cluster Pal 14, for which thereis a current observational effort to determine its velocity dis-persion (Jordi et al. 2008). We initially choose a Newtonianequilibrium Plummer model initially. While the half-massradius of Pal 14 is about 33 pc (Hilker 2006), the mass isnot actually known, but an observing campaign is underwayto constrain it (Hilker et al. 2008). We change the mass inthe wide range from [10 − ] M ⊙ and consider the half-mass radius to be constant. We perform numerical modelingto obtain the mean velocity dispersion as well as the densityprofile and velocity dispersion profile.Since the modeled systems are in equilibrium in theNewtonian case, in the MONDian case, they initially col-lapse and R h is decreased before the systems virialize again.In order to have a MONDian equilibrium initial system,we increased the velocity of the Newtonian system by an c (cid:13) , 1–, 1–
Line-of-sight velocity dispersion for Pal 14 for vari-ous masses as found by N-MODY. In order to compare with thereal cluster (observational velocity dispersion of Pal 14), the halfmass radii of all models are fixed at 33 pc. The analytical predic-tions for different limiting cases are also plotted for comparison. M M M M In order to decide whether MOND or dark matter is the righttheory to explain the dynamics of the universe, it is desir-able to study MOND for objects in which no dark matter issupposed to exist and where the characteristic accelerationof the stars is less than the MOND critical acceleration pa-rameter a . GCs are a perfect candidate since they are thelargest virialized structure that does not contain dark matter(Moore 1996), and their internal accelerations can be lowerthan a . Hence, GCs may provide a good laboratory to testthe law of gravity (Baumgardt et al. 2005).We choose the globular cluster Pal 14, for which thereis a current observational effort to determine its velocity dis-persion (Jordi et al. 2008). We initially choose a Newtonianequilibrium Plummer model initially. While the half-massradius of Pal 14 is about 33 pc (Hilker 2006), the mass isnot actually known, but an observing campaign is underwayto constrain it (Hilker et al. 2008). We change the mass inthe wide range from [10 − ] M ⊙ and consider the half-mass radius to be constant. We perform numerical modelingto obtain the mean velocity dispersion as well as the densityprofile and velocity dispersion profile.Since the modeled systems are in equilibrium in theNewtonian case, in the MONDian case, they initially col-lapse and R h is decreased before the systems virialize again.In order to have a MONDian equilibrium initial system,we increased the velocity of the Newtonian system by an c (cid:13) , 1–, 1– ?? Haghi et al. t [Myr] L a g . r a d i u s [ p c ] LR10LR20LR30LR40LR50
M=10 M , a h =0.05 a , Newtonian velocities t [Myr] L a g . r a d i u s [ p c ] M=10 M , a h =0.5 a , Newtonian velocities t [Myr] L a g . r a d i u s [ p c ] M=10 M , a h =0.05 a , adjusted velocities t [Myr] L a g . r a d i u s [ p c ] M=10 M , a h =0.5 a , adjusted velocities Figure 6.
Upper left: Evolution of Lagrangian radii for a low mass cluster with stellar velocities corresponding to Newtonian virialequilibrium but in the deep-MOND limit. After rapid collapse, the system reaches an equilibrium state. Upper right: By increasingthe initial velocities of particles by a factor of 2.8, calculated from equation (15), the system reaches equilibrium in MOND withoutcollapsing. Lower panel: Evolution of the Lagrangian radius for a high mass cluster in the intermediate MOND regime. In the left panel,the velocities are not adjusted and the system still collapses. After adjusting the velocities by a factor of 1.38, calculated from equation(15), there is no collapse (right panel). amount given by our fitting formulae (Equations (15) and(17)) to avoid a collapse. For low mass systems that arein the deep-MOND regime ( a i ≪ a ) the increase is largerthan for massive systems that are in the intermediate regime( a i ∼ a ). In Fig. 6 we plot the evolution of the Lagrangianradii for two clusters with the same half-mass radius anddifferent mass in the deep-MOND and intermediate MONDregime. In deep-MOND (low mass cluster), after a rapid col-lapse, the system reaches an equilibrium state. By increasingthe initial velocity of the particles, the collapse can be pre-vented.In Figs. 7 and 8 we show the numerical solution forPal 14. We plot the density profile of Pal 14 for different masses and compare it with the observed profile in Fig. 7(observational data from Hilker (2006)). The shape of thedensity profiles is the same for all masses, but the centraldensity differs significantly. All calculated surface densityprofiles compare well with the observed density profile. Itshould be mentioned that the full observed density profileis not known for Pal 14 and that the surface density profileshown in Fig. 7 is based only on giant stars.Fig. 8 shows the numerical solutions for the line-of-sightvelocity dispersion and compares it with Milgrom’s analyti-cal predictions for the extreme limits (to see how analyticalpredictions differ from the numerical solution). M M c (cid:13) , 1– ?? elocity dispersion in MOND External acceleration a b ca e = 0 . a a e = 0 . a a e = 0 . a a e = 0 . a a e = 1 . a a e = 10 . a Table 1.
Best fitting coefficients for the velocity dispersion pre-dicted by N-MODY simulation for various values of the exter-nal acceleration. The general form is given by equation (17) ,where f ( x ) is due to an isolated cluster (equation (15)). In case a e = 10 a , the function is nearly linear and the best fit functioncan also be obtained by f ( x ) = x − . to the quasi-Newtonian case which is for the external fielddominated case (equation (12)). As expected, the analyti-cal estimates are consistent with the numerical solution ei-ther in the external field dominated (small mass) or internalfield dominated case, but have significant deviations in theintermediate regime. For Pal 14 the external accelerationof the Galaxy is about a e ∼ . a . For low cluster masseswhich means low internal accelerations, the prediction of M M R g .However for larger systems such as galaxies, the anisotropycould be observable for a case in which a i ≪ a e ≪ a . In this work we have calculated global line-of-sight velocitydispersions of stellar systems in MOND for both isolatedand non-isolated stellar systems. The velocity dispersion ofstellar systems in MOND was so far only known in the caseof the deep MONDian limit where all accelerations are muchsmaller than the critical acceleration, a , and even in thiscase only if either the internal acceleration is much largerthan the external acceleration or the internal acceleration ismuch lower than the external acceleration. We used the N-MODY code to calculate for the first time the line-of-sightvelocity dispersions of stellar systems also for the interme-diate regime.We have obtained a large set of dissipationless N- MODY numerical solutions for isolated systems with massesin the range 10 M ⊙ to 10 M ⊙ and with the Plummer modelas the initial condition. In order to produce different inter-nal acceleration regimes, for each mass, we changed the half-mass radius of the system. We deduce the analytical formu-lae for the velocity dispersion of a stellar system as a func-tion of its half-mass-radius-internal-acceleration, a h (equa-tion (13)), and investigate the universal functional form forthe velocity dispersion of isolated systems (equation (15)).We have also studied the effect of a different choice ofthe interpolation function on the line-of-sight velocity dis-persion for systems with different internal accelerations. Wefound that the simple function suggested by Famaey andBinney (2005) produces a larger velocity dispersion thanthe prediction with the standard interpolation function sug-gested by Milgrom (1984), with the maximum difference oc-curring at a h ∼ a and being of order 20%.Since most stellar systems (e.g. globular clusters ordwarf galaxies) are not isolated and usually move throughthe gravitational field of a host galaxy, the internal dynam-ics is often influenced by the host galaxy due to the exter-nal field effect (EFE) of MOND. Therefore, we have inves-tigated non-isolated systems, adding the external field toN-MODY. Our simulations reproduce previous analytic es-timates for stellar velocities in systems in the deep MONDregime ( a i , a e ≪ a ), where the motion of the stars is eitherdominated by internal accelerations ( a i ≫ a e ) or externalaccelerations ( a e ≫ a i ). In addition, we calculate the line-of-sight velocity dispersion for intermediate cases and derivefor the first time analytic formulae for the line-of-sight veloc-ity dispersion in the intermediate regime ( a i ∼ a e ∼ a ) andfound a smooth functional form for the velocity dispersionof stellar systems under the EFE. These formulae will allowto test MOND more thoroughly than was hitherto possible.We finally calculated the velocity dispersion of the glob-ular cluster Pal 14, and will compare it with observationaldata in a forthcoming paper (Jordi et al. 2008). An addi-tional observational study in order to constrain the mass ofPal 14 is also underway by our team (Hilker et al. 2008).In a future contribution we will also discuss the fascinatingpossibility of ”freezing” a cluster on a highly eccentric or-bit: as a cluster moves from the Newtonian regime (small σ N ) to the MONDian regime on a time scale comparableto or faster than the internal crossing time it will retain aNewtonian velocity dispersion (Haghi et al. 2008).Additional observational efforts to determine the veloc-ity dispersion of stellar systems such as GCs or dSph satel-lites would be highly important as such data also providea strict test of MOND. On the other hand, if we believein MOND, these observations could be used to constrainthe external field and consequently to put constrains on thepotential in which the systems are embedded. Moreover itwould be worthwhile to observe the stellar system in theintermediate regime to constrain the µ -function and a . ACKNOWLEDGEMENTS
We would like to thank C. Nipoti for providing us with theN-MODY code and his help in using it. H.H thanks the Ira-nian Cosmology and Particle Physics Center of Excellence,at the physics department of Sharif University of Technol- c (cid:13) , 1– ?? Haghi et al.
R (pc) σ ( k m / s )
20 40 60 8010000.20.40.60.81
M= 4000 M R (pc) σ ( k m / s )
20 40 60 8010000.511.5
M= 10000 M R (pc) σ ( k m / s )
20 40 60 8010000.511.522.53
M= 40000 M R (pc) σ ( k m / s )
20 40 60 8010001234
M= 100000 M Figure 9.
Line-of-sight velocity dispersion profiles of Pal 14 for different cluster masses obtained by N-MODY computation. Thehorizontal blue (dashed) line indicates the mean global velocity dispersion and the vertical red (dashed) line indicates the half-mass-radius of Pal 14. The velocity dispersion changes only slightly (about 10%) from the center to the half-mass-radius. ogy and the Argelander Institute for Astronomy for provid-ing fellowships in support of this research. K.J and E.K.Ggratefully acknowledge support by the Swiss National Sci-ence Foundation.
REFERENCES
Angus, G.W., Famaey, B. & Zhao, H.S., 2006 MNRAS,371, 138Angus, MNRAS 387, Issue 4, pp. 1481-1488Baumgardt H., Grebel E.K. and Kroupa P. 2005, MNRAS,359, L1 Bekenstein J.D., Milgrom M., 1984, ApJ, 286, 7Bekenstein J., 2004, PRD, 70, 083509Binney S., Tremaine S. 1987, Galactic Dynamics, PrincetonUniv. Press, Princeton, NJ.Bosma, A. 1981, AJ, 86, 18251846Clowe, D., et al., 2006 ApJ, 648, L109.Ciotti L., Londrillo P. & Nipoti C., 2006, ApJ, 640, 741.Famaey B., Binney J., 2005, MNRAS, 361 633Famaey B., Bruneton J., & Zhao H.S., 2007, MNRAS, MN-RAS, 377L, 79Gentile G., Fammaey B., Combes F., Kroupa P., Zhao H.S.,& Tiret O., 2007, A&A, 472, L25.Haghi H., et al., 2008, under preparation. c (cid:13) , 1– ?? elocity dispersion in MOND Hilker M., 2006, A&A, 448, 171.Hilker M., et al., 2008, under preparationJordi K., et al., 2008, under preparationKlypin A. et al., 1999, ApJ, 522, 82Anatoly Klypin and Francisco Prada (2007), as-troph/0706.3554Londrillo & Nipoti 2008,(arXiv:0803.4456v1, SAIt-Suppl.2008,in press)Metz M., Kroupa P. & Libeskind N., 2008, accepted in ApJ,(arXiv:0802.3899v1)Milgrom M., 1983, ApJ, 270, 365Milgrom M. 1986, ApJ, 302, 617Milgrom M. 1994, ApJ, 429, 540Milgrom M. 1995, ApJ, 455, 439Moffat, J. W. and Toth 2007,(arXiv:0708.1264v2 ).Moffat, J. W. and Toth 2007, accepted for publication inApJ (arXiv:0708.1935v3).Moore B., et al., 1999, ApJ, 524, L19.Moore B., 1996, ApJ, 461, L13.Nipoti C., Londrillo P., Ciotti L., 2007a, ApJ, 660, 256Nipoti C., Londrillo P., Ciotti L., 2007b, MNRAS, 381,L104Nipoti C., Ciotti L., Binney J., Londrillo P., 2008, MNRAS,386, 2194Plummer H.C., 1911, MNRAS, 71, 460.Rubin, V. C., and Burstein, D., 1985, ApJ, 297, 423435.Sanchez-Salcedo, F. J., & Hernandez, X., 2007, ApJ, 667,878890Sanders R. H., McGaugh S., 2002, Ann. Rev. Astron. As-trophys., 40, 263.Scarpa R., Marconi G., Gilmozzi R. & Carraro G., 2007,A&A, 462, L9.Spergel D.N., et al., 2003, ApJS, 148, 175Tiret O. & Combes F., 2007, A&A, 464, 517-528.Wu, X., et al., 2007, ApJ, 665, L.101, (arXiv:0706.3703v2)Wu, X., et al., 2008, accepted for publication in MNRAS,(arXiv:0803.0977v1).Zhao H.S, Famaey B, 2006, ApJ,638, L9-L12(astro-ph/0512425) c (cid:13) , 1–, 1–