Galactic orbital motions of star clusters: static versus semicosmological time-dependent Galactic potentials
aa r X i v : . [ a s t r o - ph . GA ] M a y Mon. Not. R. Astron. Soc. , 1–12 (2013) Printed 16 September 2018 (MN L A TEX style file v2.2)
Galactic orbital motions of star clusters: static versussemicosmological time-dependent Galactic potentials
Hosein Haghi ⋆ , Akram Hasani Zonoozi , Saeed Taghavi , Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran Department of Mathematics, Mazandaran University of Science and Technology, Behshahr 48518-78413, Iran
Accepted ....; Received ...
ABSTRACT
In order to understand the orbital history of Galactic halo objects, such as globularclusters, authors usually assume a static potential for our Galaxy with parameters thatappear at the present-day. According to the standard paradigm of galaxy formation,galaxies grow through a continuous accretion of fresh gas and a hierarchical mergingwith smaller galaxies from high redshift to the present day. This implies that the massand size of disc, bulge, and halo change with time. We investigate the effect of assuminga live Galactic potential on the orbital history of halo objects and its consequences ontheir internal evolution. We numerically integrate backwards the equations of motionof different test objects located in different Galactocentric distances in both staticand time-dependent Galactic potentials in order to see if it is possible to discriminatebetween them. We show that in a live potential, the birth of the objects, 13 Gyrago, would have occurred at significantly larger Galactocentric distances, comparedto the objects orbiting in a static potential. Based on the direct N -body calculationsof star clusters carried out with collisional N -body code, nbody6 , we also discussthe consequences of the time-dependence of a Galactic potential on the early- andlong-term evolution of star clusters in a simple way, by comparing the evolution oftwo star clusters embedded in galactic models, which represent the galaxy at presentand 12 Gyr ago, respectively. We show that assuming a static potential over a Hubbletime for our Galaxy as it is often done, leads to an enhancement of mass-loss, anoverestimation of the dissolution rates of globular clusters, an underestimation of thefinal size of star clusters, and a shallower stellar mass function. Key words: galaxies: star clusters: general, galaxies: evolution methods: numerical
Tens of satellite galaxies and about 160 globular clus-ters (GCs; Harris 1996, 2010), have been identified in theMilky Way (MW), that are distributed out to more than200 kpc, orbiting around the centre of our Galaxy. Ob-servations show that nearly all galaxies host these sys-tems, with giant ellipticals having almost the largest pop-ulation (Brodie & Strader 2006). Stellar population stud-ies have revealed that GCs have ages up to 13 Gyr (e.g.,Chaboyer & Krauss 2002; Hansen et al. 2002), and there-fore they represent fossil records of the earliest epoch ofgalaxy formation. As such, they are potentially powerful ⋆ E-mail: [email protected] (HH) [email protected] (AHZ);[email protected] (ST) probes of physical conditions in the high redshift Universe(Brodie & Strader 2006).All star clusters lose mass over time and this dependson a number of internal and external processes, as e.g.,mass-loss due to stellar evolution, mass segregation, andcore collapse due to two-body relaxation, and the externaltidal field of the parent galaxy within which the cluster or-bits (Vesperini & Heggie 1997; Baumgardt & Makino 2003;Heggie & Hut 2003; Gieles, Heggie, & Zhao 2011). In addi-tion, the evolution of star clusters depends crucially on theinitial conditions (e.g., the initial mass profile of the starcluster, the initial mass function (IMF) of the stars, andthe initial binary fraction) and the orbital parameters of thecluster (Madrid, Hurley, & Sippel 2012; Webb et al. 2013,2014; Haghi et al. 2014).Based on N -body simulations of GC systems, it is c (cid:13) Haghi et al. well accepted that the GC populations we observe to-day are only the very remnants of much richer systems(e.g., Bonaca, Geha, & Kallivayalil 2012; Grillmair et al.2013; Brockamp et al. 2014; Koposov et al. 2014). The rateof GC erosion strongly depends on the details of the gravi-tational potential of the host galaxy as well as on the inter-nal properties of the GCs (Brockamp et al. 2014). Therefore,the present-day distribution of GC systems around the MW,and their properties may be valuable probes of the Galaxypotential.The survival or dissolution of star clusters in the galac-tic tides within which they orbit also depends crucially ontheir orbital history: star clusters with large radii spend-ing the major part of their lifetime in the innermost re-gions of our Galaxy are more susceptible to tidally in-duced mass-loss, whereas the outer halo objects can sur-vive for a Hubble time (e.g., Giersz & Heggie 1997; Hurley2007; Heggie & Giersz 2008; Gieles, Heggie, & Zhao 2011;Brockamp et al. 2014; Haghi et al. 2014). Also, calculationof orbital motions backward in time is necessary to modelthe formation of the stellar and gaseous streams emergingfrom GCs or accreting satellite galaxies.Therefore, a detailed understanding of the orbital his-tory which requires a better understanding of the evolutionof the Galactic potential since its formation, is an essentialissue in investigating the tidal erosion of GCs (via mass-loss) and accreting disruption of satellite galaxies. Manyauthors usually use the static potential, i.e., assume thatit remains unchanged during the orbital integration. But,observations have revealed that the size and the mass con-tent of galaxies change significantly with redshift such thatthe sizes of the galaxies at high redshifts are smaller in com-parison with galaxies of similar mass in the local universe(e.g.,Franx et al. 2008; Williams et al. 2010; Mosleh et al.2011; Law et al. 2012; Mosleh, Williams,& Franx 2013).There are many proposed scenarios to explain thephysical processes of galaxy assembly that well reproducethe observable properties like, e.g., the stellar mass andsize of galaxies at different redshifts. Among them are thegalaxy minor or major mergers (e.g., Khochfar& Silk 2006;Khochfar & Silk 2009; Naab, Johansson, & Ostriker 2009),and the accretion of fresh gas in outer regions activating newstar formation (e.g., Elmegreen, Bournaud, & Elmegreen2008; Dekel, Sari, & Ceverino 2009). Indeed, in the stan-dard picture of galaxy formation, galaxies are em-bedded in massive virialized haloes of dark matter(Springel, Frenk, & White 2006). These dark matter haloesaccumulate over time hierarchically, continuously growingvia accretion of dark matter and merging with other haloesfrom high redshift to the present day. The fraction of GCsthat have survived to the present day has evolved in a time-dependent potential of the host galaxy.Understanding the influence of the time-dependence ofthe Galactic potential on the orbital history of the halo ob-jects (e.g., GCs) rotating around the MW at different Galac-tocentric distances, and its consequence on their early- andlong-term evolution over a Hubble time is the main motiva-tion for this paper.Indeed, because of the wide range of time-scales andsize scales, from the two-body strong encounters of starsto the galactic scales, the N -body simulations of starclusters in a live galactic potential are challenged. How- ever, attempts have been made to overcome this difficulty.A pioneering study of this subject was carried out byRenaud, Gieles, & Boily (2011) who investigated the evo-lution of star clusters including a time-dependent poten-tial. They proposed a novel approach to extract the tidalinformation as tables of tensors from a galaxy or cosmol-ogy simulation along one orbit. Their method has been ap-plied to a large number of star clusters in a galaxy ma-jor merger (Renaud & Gieles 2013) emulating the Antennagalaxies (NBC 4038/39). A more recent improvement of thismethod using any definition of the external potential as afunction of space and time can be found in Renaud & Gieles(2015). Similar work was done by Rieder et al. (2013) whotracked the tidal history of clusters in a cosmological con-text by inserting clusters into a dark matter only cold darkmatter simulation. They also found that mergers tend toincrease the mass-loss rates of clusters.In this work, we will estimate how the mass, character-istic radii, and the mass function (MF) slope of a star clusterwould change if it is evolved in the time-dependent potentialby comparing the evolution of two star clusters embeddedin two galactic models, which represent our Galaxy with thepresent-day parameters and with parameter values at 12 Gyrago, respectively.As a concrete example of motion within the Galactichalo, we will also use the backward motion of the LargeMagellanic Cloud (LMC) located at about 50 kpc from thecentre of the MW in both static and live Galactic poten-tials to see if, at least in principle, our approach is able todiscriminate between them.In Section 2, we describe the characteristics of the time-dependent galactic potential we have used in this paper.We compare the orbital motions of different test particlesaround the centre of the Galaxy in Section 3. In Section 4,we present results of N -body calculations for the dynamicalevolution of star clusters moving through an external galaxywith different background potential parameters. The simu-lations were carried out with the collisional N-body code nbody6 on desktop workstations with Nvidia 690 GraphicsProcessing Units at the Institute for Advanced Studies inBasic Sciences (IASBS). Finally, in Section 5, we summa-rize our results. In the first part, for the Galaxy potential we, like manyother authors, assumed that the galaxy potential is staticand consists of three idealized components,Φ tot = Φ d + Φ b + Φ h , (1)including a Miamoto–Nagai disc potential(Miyamoto & Nagai 1975) given byΦ d ( x, y, z ) = − GM d q x + y + (cid:0) a + √ z + b (cid:1) , (2)a central bulge, the Hernquist (1990) model, given byΦ b ( x, y, z ) = − GM b r + r c , (3) c (cid:13)000
Tens of satellite galaxies and about 160 globular clus-ters (GCs; Harris 1996, 2010), have been identified in theMilky Way (MW), that are distributed out to more than200 kpc, orbiting around the centre of our Galaxy. Ob-servations show that nearly all galaxies host these sys-tems, with giant ellipticals having almost the largest pop-ulation (Brodie & Strader 2006). Stellar population stud-ies have revealed that GCs have ages up to 13 Gyr (e.g.,Chaboyer & Krauss 2002; Hansen et al. 2002), and there-fore they represent fossil records of the earliest epoch ofgalaxy formation. As such, they are potentially powerful ⋆ E-mail: [email protected] (HH) [email protected] (AHZ);[email protected] (ST) probes of physical conditions in the high redshift Universe(Brodie & Strader 2006).All star clusters lose mass over time and this dependson a number of internal and external processes, as e.g.,mass-loss due to stellar evolution, mass segregation, andcore collapse due to two-body relaxation, and the externaltidal field of the parent galaxy within which the cluster or-bits (Vesperini & Heggie 1997; Baumgardt & Makino 2003;Heggie & Hut 2003; Gieles, Heggie, & Zhao 2011). In addi-tion, the evolution of star clusters depends crucially on theinitial conditions (e.g., the initial mass profile of the starcluster, the initial mass function (IMF) of the stars, andthe initial binary fraction) and the orbital parameters of thecluster (Madrid, Hurley, & Sippel 2012; Webb et al. 2013,2014; Haghi et al. 2014).Based on N -body simulations of GC systems, it is c (cid:13) Haghi et al. well accepted that the GC populations we observe to-day are only the very remnants of much richer systems(e.g., Bonaca, Geha, & Kallivayalil 2012; Grillmair et al.2013; Brockamp et al. 2014; Koposov et al. 2014). The rateof GC erosion strongly depends on the details of the gravi-tational potential of the host galaxy as well as on the inter-nal properties of the GCs (Brockamp et al. 2014). Therefore,the present-day distribution of GC systems around the MW,and their properties may be valuable probes of the Galaxypotential.The survival or dissolution of star clusters in the galac-tic tides within which they orbit also depends crucially ontheir orbital history: star clusters with large radii spend-ing the major part of their lifetime in the innermost re-gions of our Galaxy are more susceptible to tidally in-duced mass-loss, whereas the outer halo objects can sur-vive for a Hubble time (e.g., Giersz & Heggie 1997; Hurley2007; Heggie & Giersz 2008; Gieles, Heggie, & Zhao 2011;Brockamp et al. 2014; Haghi et al. 2014). Also, calculationof orbital motions backward in time is necessary to modelthe formation of the stellar and gaseous streams emergingfrom GCs or accreting satellite galaxies.Therefore, a detailed understanding of the orbital his-tory which requires a better understanding of the evolutionof the Galactic potential since its formation, is an essentialissue in investigating the tidal erosion of GCs (via mass-loss) and accreting disruption of satellite galaxies. Manyauthors usually use the static potential, i.e., assume thatit remains unchanged during the orbital integration. But,observations have revealed that the size and the mass con-tent of galaxies change significantly with redshift such thatthe sizes of the galaxies at high redshifts are smaller in com-parison with galaxies of similar mass in the local universe(e.g.,Franx et al. 2008; Williams et al. 2010; Mosleh et al.2011; Law et al. 2012; Mosleh, Williams,& Franx 2013).There are many proposed scenarios to explain thephysical processes of galaxy assembly that well reproducethe observable properties like, e.g., the stellar mass andsize of galaxies at different redshifts. Among them are thegalaxy minor or major mergers (e.g., Khochfar& Silk 2006;Khochfar & Silk 2009; Naab, Johansson, & Ostriker 2009),and the accretion of fresh gas in outer regions activating newstar formation (e.g., Elmegreen, Bournaud, & Elmegreen2008; Dekel, Sari, & Ceverino 2009). Indeed, in the stan-dard picture of galaxy formation, galaxies are em-bedded in massive virialized haloes of dark matter(Springel, Frenk, & White 2006). These dark matter haloesaccumulate over time hierarchically, continuously growingvia accretion of dark matter and merging with other haloesfrom high redshift to the present day. The fraction of GCsthat have survived to the present day has evolved in a time-dependent potential of the host galaxy.Understanding the influence of the time-dependence ofthe Galactic potential on the orbital history of the halo ob-jects (e.g., GCs) rotating around the MW at different Galac-tocentric distances, and its consequence on their early- andlong-term evolution over a Hubble time is the main motiva-tion for this paper.Indeed, because of the wide range of time-scales andsize scales, from the two-body strong encounters of starsto the galactic scales, the N -body simulations of starclusters in a live galactic potential are challenged. How- ever, attempts have been made to overcome this difficulty.A pioneering study of this subject was carried out byRenaud, Gieles, & Boily (2011) who investigated the evo-lution of star clusters including a time-dependent poten-tial. They proposed a novel approach to extract the tidalinformation as tables of tensors from a galaxy or cosmol-ogy simulation along one orbit. Their method has been ap-plied to a large number of star clusters in a galaxy ma-jor merger (Renaud & Gieles 2013) emulating the Antennagalaxies (NBC 4038/39). A more recent improvement of thismethod using any definition of the external potential as afunction of space and time can be found in Renaud & Gieles(2015). Similar work was done by Rieder et al. (2013) whotracked the tidal history of clusters in a cosmological con-text by inserting clusters into a dark matter only cold darkmatter simulation. They also found that mergers tend toincrease the mass-loss rates of clusters.In this work, we will estimate how the mass, character-istic radii, and the mass function (MF) slope of a star clusterwould change if it is evolved in the time-dependent potentialby comparing the evolution of two star clusters embeddedin two galactic models, which represent our Galaxy with thepresent-day parameters and with parameter values at 12 Gyrago, respectively.As a concrete example of motion within the Galactichalo, we will also use the backward motion of the LargeMagellanic Cloud (LMC) located at about 50 kpc from thecentre of the MW in both static and live Galactic poten-tials to see if, at least in principle, our approach is able todiscriminate between them.In Section 2, we describe the characteristics of the time-dependent galactic potential we have used in this paper.We compare the orbital motions of different test particlesaround the centre of the Galaxy in Section 3. In Section 4,we present results of N -body calculations for the dynamicalevolution of star clusters moving through an external galaxywith different background potential parameters. The simu-lations were carried out with the collisional N-body code nbody6 on desktop workstations with Nvidia 690 GraphicsProcessing Units at the Institute for Advanced Studies inBasic Sciences (IASBS). Finally, in Section 5, we summa-rize our results. In the first part, for the Galaxy potential we, like manyother authors, assumed that the galaxy potential is staticand consists of three idealized components,Φ tot = Φ d + Φ b + Φ h , (1)including a Miamoto–Nagai disc potential(Miyamoto & Nagai 1975) given byΦ d ( x, y, z ) = − GM d q x + y + (cid:0) a + √ z + b (cid:1) , (2)a central bulge, the Hernquist (1990) model, given byΦ b ( x, y, z ) = − GM b r + r c , (3) c (cid:13)000 , 1–12 he time-dependent galactic potential Figure 1.
Left-hand panel: logarithmic plot of the size scale of the MW’s mass components as a function of cosmological time. The virialradius of the dark matter halo r vir (dashed green), scale radius of the disc a (dotted blue), the scale-height of the disc b (dash-dotteddark blue), and the scale radius of the bulge r c (solid red) are shown in the figure. Right-hand Panel: the evolution of the mass scale ofthe MW’s mass components as a function of cosmological time. The virial mass of the dark matter halo M vir (dashed green), the scalemass of the disc M d (dotted blue), and of the bulge M b (solid red) are shown in the figure. All components show fast growth in the pasttime. Disc Bulge Halo M d = 7 . × M b = 2 . × M vir = 9 × a = 5 . b = 0 . r c = 0 . r vir = 250 c = 13 . Table 1.
The present-day parameters of the mass componentsof the MW-like potential used in our calculations. Masses anddistances are in M ⊙ and in kpc, respectively.Disc Bulge Halo M d = 6 . × M b = 2 . × M vir = 7 . × a = 0 . b = 0 . r c = 0 . r vir = 29 . c = 2 . Table 2.
The parameters of the mass components of the MW-likepotential at T = −
12 Gyr. Masses and distances are in M ⊙ andin kpc, respectively. Eqs. 5 - 9 are used to calculate these valuesat z = 3 . T = −
12 Gyr). and a NFW dark matter halo (Navarro, Frenk, & White1997) with a potential of the formΦ
NF W = − GM vir r [log(1 + c ) − c c ] log (cid:16) c rr vir (cid:17) . (4)Here, r = p x + y + z is the distance from the galacticcentre at any given time; M b , M d and M vir are the char-acteristic mass of the bulge, disc, and halo, respectively; r c and r vir are the characteristic radius of the bulge and halo,respectively; and a is the scale radius and b the scale heightthat adjust the shape of the disc. The present-day numericalvalues of these parameters at redshift z = 0 are given in theTable 1. To model the evolution of the Galactic potential, we as-sume a semicosmological time-dependent gravitational po-tential in which the characteristic parameters vary in time.The evolution of the mass and the virial concentration ofthe galaxy’s halo as a function of redshift are given by(Wechsler et al. 2002; Zhao et al. 2003; G´omez et al. 2010) M vir ( z ) = M vir (0) exp( − a c z ) , (5)where the formation epoch is set to be a c = 0 .
34, and c ( z ) = c (0)1 + z . (6)For the disc and bulge we follow the recipe given byBullock & Johnston (2005), as for masses we have M d,b ( z ) = M vir ( z ) M d,b (0) M vir (0) , (7)and for scale-lengths the evolution can be expressed as { a, b, r c } ( z ) = r vir ( z ) { a, b, r c } (0) r vir (0) . (8)Here r vir is the virial radius of the dark matter halo, varyingas r vir ( z ) = (cid:18) M vir ( z )4 π ∆ vir ( z ) ρ c ( z ) (cid:19) / , (9)where ∆ vir ( z ) denotes the virial overdensity ,∆ vir ( z ) = 18 π + 82[Ω( z ) − − z ) − (10)with Ω( z ) the mass density of the universe,Ω( z ) = Ω m, (1 + z ) Ω m, (1 + z ) + Ω Λ , , (11)and ρ c ( z ) is the critical density of the universe at a givenredshift, ρ c ( z ) = 3 H ( z )8 πG (12)with c (cid:13) , 1–12 Haghi et al.
Figure 2.
Comparison of the circular velocity curve as a functionof Galactocentric distance at z = 0 (black solid line) with rotationcurve at z = 3 . − after about 12 Gyr of evolution. H ( z ) = H p Ω Λ , + Ω m, (1 + z ) . (13)We also adopted a flat cosmology defined byΩ m, = 0 . Λ , = 0 . H ( z = 0) = H = 70kms − Mpc − .In cosmology one can label the time t since the big bangin terms of the redshift of light emitted at t . By integratingthe Friedmann equation, the behaviour of the cosmologicalredshift in terms of time for a flat universe can be found as z = Ω m, sinh ( H t p Ω Λ , )Ω Λ , ! − / − , (14)In Fig. 1, we display how the characteristic parametersof the MW (i.e., all Galactic mass components and scale-lengths) vary as a function of cosmological time. The pa-rameters of the mass components of the MW-like potentialat T = −
12 Gyr ( z = 3 .
5) are given in Table 2.The Galactic rotation curves deduced from the three-component mass model, described in Eqs. 2-13, at z = 0 and z = 3 . − at 8.5 kpc from the galactic centre, whileit is about 117 kms − at z = 3 .
5. The smaller value ofasymptotic rotational velocity at R G = 100 kpc occurs inthe time-dependent gravitational potential, while the staticpotential yields the larger one, differing by about 75 kms − after 12 Gyr of evolution. In this section, we describe the results from the numerical in-tegration of the equation of motion to find the trajectory of atest object located in different Galactocentric distances mov-ing under the static Galactic potential and compare themwith the same calculations in a time-dependent Galactic po-tential by using Eqs. 5 - 13 and the present-day parametersgiven in Table 1.Using the gravitational potential components described
Figure 3.
Upper panel: planar shape of the orbits of a test par-ticle w.r.t. the MW, starting from R G = 4 kpc (the first line ofTable 3) . The red dashed line shows the trajectory of test objectwithin the time-dependent potential while the blue line is for thestatic potential. Lower panel: the time evolution (for the past 13Gyr) of the radial distance to the Galactic centre of a test par-ticle, starting from R G = 4 kpc in the time-dependent potential(red dashed line) and the static potential (blue line). above, we will numerically integrate the three scalar differen-tial equations, written in Cartesian coordinates, correspond-ing to the vector differential equation,¨ r = −∇ Φ tot . (15)Here we use the fourth-order Runge–Kutta method to cal-culate the equation of motion of test objects and then toextract the trajectory of object by backtracking orbit fromits current position and velocity. Using the initial conditionsof Table 3, the equations of motion are then numerically in-tegrated backward in time for 13 Gyr in both live and staticgravitational potentials. In this manner, we therefore obtainsets of initial positions and velocities required for forwardintegration in time and can be applied in e.g., N -body sim-ulations of realistic GCs or satellite galaxies.We calculate the orbit for three test objects: an innertest object currently located at R G = 4 kpc, a solar-distanceobject located at R G =8.5 kpc, and an outer object with c (cid:13) , 1–12 he time-dependent galactic potential Figure 4.
The same as Fig. 3 but for a test object that locatedinitially at R G = 8 . the present-day Galactocentric distance of R G = 100 kpc.For all objects, we calculate the trajectory within two dif-ferent Galactic models: the static Galactic potential andtime-dependent Galactic potential. The orbital parametersof these test objects are summarized in Table 3.The initial velocities are extracted from the present-dayrotation curve. That is, the orbits are circular in the staticpotential, while in the live potential the Galactocentric dis-tances increase looking backwards in time. Note that, thetest objects evolve on circular orbits at different galactocen-tric distances in the disc plane (i.e., the inclination angleof the orbits w.r.t the galactic disc is zero). In Figs. 3 - 5,we plot the orbital sections in the Galactocentric coordi-nate planes of the numerically integrated trajectories of thetest objects from now to 13 Gyr ago. The time evolution ofdistances from the centre of the MW for all test objects areshown in the lower panel of Figs. 3 - 5. As can be seen in Fig.6, the Galactocentric distances of these objects at t = − t = 0 by a factor ofabout 2 (for test object initially located at R G = 100 kpc),and 2.5 (for test objects initially located at 4 and 8.5 kpc).It should be noted such a large difference in determining thebirth place of realistic objects (like e.g., GCs in the MW) Figure 5.
The same as Fig. 3, but for a test object initiallylocated at R G =100 kpc as shown in the third line of Table 3. Figure 6.
The evolution of the relative difference of the Galacto-centric distance in the static and live Galactic potential in timethat are plotted for individual test objects moving in differentinitial distances from the centre of Galaxy as given in Table 3.The differences are small at the beginning of orbital backwardmotion and starts to increase at about t = − R G = 100 kpc, while it increasesby a factor of about 2.5 for other two test objects at R G = 4 and8.5 kpc. may pose problems concerning the dynamical evolution. Iwill be back to this important issue in more details in Sec.4. As a concrete example of motion within the Galactichalo, let us consider the orbital motion of the LMC c (cid:13) , 1–12 Haghi et al.
Table 3.
Initial coordinates, in kpc, and initial velocities, in kms − , in Galactocentric rest frame adopted for different testobjects used in this paper. We have used the present-day rota-tion curve of our Galaxy for initial circular velocities. r ( x, y, z ) 3D v ( x, y, z )model1 (inner-part object) (4,0,0) (0,215,0)model2 (solar-distance) (8.5,0,0) (0,221,0)model3 (outer-part object) (100,0,0) (0,195,0) Table 4.
The coordinate, in kpc, and velocity component, in kms − , of LMC used in this work in a Galactocentric rest frame withthe z -axis pointing towards the North Galactic Pole (NGP), the x -axis pointing in the direction from the Sun to the Galactic centre,and the positive y -axis is directed towards the Sun’s Galacticrotation (Mastropietro et al. 2005). r ( x, y, z ) 3D v ( x, y, z )(0,–43.9,–25.04) (–4.3,–182.45,169.8) currently located at about 50 kpc from the centre of theMW. Many authors have shown that the position of theMagellanic Stream (MS) follows the orbits of MCs ( seee.g. Moore & Davis 1994; Connors, Kawata, & Gibson2006; Haghi, Rahvar, & Zonoozi 2006; Besla et al.2007; Haghi, Zonoozi, & Rahvar 2009; Haghi & Rahvar2010). Moreover, the shape and the kinematics of theMS is strongly influenced by the overall propertiesof the underlying potential (Murai & Fujimoto 1980;Lin & Lynden-Bell 1982; Heller & Rohlfs 1994; Sofue 1994;Gardiner & Noguchi 1996). Therefore, it is most striking tocompare the trajectory of the LMC with orbits predicted inboth static and live Galactic potentials. Table 4 summarizesthe present-day Galactocentric Cartesian coordinates andvelocities of the LMC (Mastropietro et al. 2005).Applying the same method for the LMC to numericallyintegrate the equation of motion, we extract its trajectoryby backtracking orbit from its current position and velocityfor 13 Gyr. Fig. 7 depicts the planar shape of the orbits ofthe LMC in Y − Z plane (upper panel). We also show theevolution of the distance to the MW of the LMC in bothstatic and live Galactic potentials in the lower panel of Fig.7. Another issue which should, in principle, take into ac-count is the number of disc (i.e., the MW’s disc) passage inboth Galactic potentials. In fact, crossing the disc would im-ply a strong orbital perturbation of the LMC and perhapsa gas shock which can lead to the formation of stars. Wefound that the number of disc passage with Galactocentricdistances smaller than 100 kpc, in the static potential is five,while it is three in the live potential. A narrow band of neutral hydrogen clouds lies along a greatcircle from ( l = 91 ◦ ; b = − ◦ ) to ( l = 299 ◦ ; b = − ◦ ), startedfrom the MCs and oriented towards the South Galactic Pole. Figure 7.
Top: section in the Z − Y plane of the integrated tra-jectories of the LMC for the static potential (blue line) and thelive potential (red dashed line). The time span of the integra-tion is −
13 Gyr t
0. The initial conditions for position andvelocity are taken from (Mastropietro et al. 2005). Bottom: theevolution of the Galactocentric distance of LMC as a functionof time. The red dashed line: time-dependent Galactic potential.Blue line: static potential. The galactocentric distance of LMC,13 Gyr ago, would have been at R G = 60 kpc for Galactic modelwith static potential, while in the time-dependent Galactic poten-tial it would have been at larger Galactocentric distance of about R G = 230 kpc. Our Galaxy hosts around 160 GCs. It is well understoodthat the gravitational potential of host galaxy has a directinfluence on the survival and the evolution of GCs: they losestars through tidal stripping and disc shocking. As alreadyhave shown by Praagman, Hurley, & Power (2010), varyingthe mass and concentration of the halo affects the rate atwhich the star cluster loses mass. Using N -body models oflow-number star clusters, they found that increasing the halomass and concentration drives enhanced mass-loss rates and,in principle, implies shorter dissolution time-scales.Several studies have been addressed the evolutionof clusters in time-dependent galactic potential by ar-bitrary switching tidal effects to mimic the accretionof a dwarf satellite on to a massive host galaxy(Miholics, Webb, & Sills 2014; Bianchini et al. 2015), or c (cid:13)000
0. The initial conditions for position andvelocity are taken from (Mastropietro et al. 2005). Bottom: theevolution of the Galactocentric distance of LMC as a functionof time. The red dashed line: time-dependent Galactic potential.Blue line: static potential. The galactocentric distance of LMC,13 Gyr ago, would have been at R G = 60 kpc for Galactic modelwith static potential, while in the time-dependent Galactic poten-tial it would have been at larger Galactocentric distance of about R G = 230 kpc. Our Galaxy hosts around 160 GCs. It is well understoodthat the gravitational potential of host galaxy has a directinfluence on the survival and the evolution of GCs: they losestars through tidal stripping and disc shocking. As alreadyhave shown by Praagman, Hurley, & Power (2010), varyingthe mass and concentration of the halo affects the rate atwhich the star cluster loses mass. Using N -body models oflow-number star clusters, they found that increasing the halomass and concentration drives enhanced mass-loss rates and,in principle, implies shorter dissolution time-scales.Several studies have been addressed the evolutionof clusters in time-dependent galactic potential by ar-bitrary switching tidal effects to mimic the accretionof a dwarf satellite on to a massive host galaxy(Miholics, Webb, & Sills 2014; Bianchini et al. 2015), or c (cid:13)000 , 1–12 he time-dependent galactic potential Figure 8.
The evolution of total mass (left-hand panel) and half-mass radius (right-hand panel) with time for two simulated starclusters orbiting within different Galactic potentials with parameters listed in Tables 1 and 2. Cluster moving in a circular orbit with aGalactocentric radius of R G = 15 . z = 3 . T = −
12 Gyr) is shown by a black dashed line, and with Galactocentric radius of R G = 8 . z = 0 (corresponding to the present day) is shown by a red solid line. For the cluster close to thegalactic centre, expansion is limited by the strong tidal field and dissolves before a Hubble time. by rapidly varying evolutions (e.g., galaxy interac-tions and mergers) combining the galaxy simulations tostar cluster simulations (Renaud, Gieles, & Boily 2011;Renaud & Gieles 2013, 2015), or by evolving star clustersin a cosmological environment (Rieder et al. 2013).This issue has been simplified by several authors. Forexample, Madrid, Hurley, & Martig (2014) have studied theimpact of the host disc mass and geometry on the survival ofstar clusters by means of N -body simulations. They showedthat a more massive disc enhances the mass-loss rate of anorbiting star cluster owing to a stronger tidal field such thatdoubling the mass of the disc halves the dissolution time of astar cluster located at R G = 6 kpc from the centre of Galaxy.They placed several of these simulations together, each timeincreasing the galaxy’s mass, to represent a realistic massgrowth history of the MW driven by mergers of satellitegalaxies.As we have shown by backward tracking of test ob-jects, in the frame work of time-varying Galactic potential,the Galactocentric distance of a test object (like e.g., a starcluster in the MW) at t = −
12 Gyr is quite a bit larger thaninitial values at t = 0, by a factor of about 2. In other words,in order to be a star cluster in its current position and veloc-ity, it should be located at a larger Galactocentric distancein the past, i.e., 12 Gyr ago, when the Galaxy was also muchlighter than its present-day mass. This implies that the clus-ter presumably was tidally underfilling at the beginning ofits evolution. Therefore, the slower mass-loss rate of clustersinitially lying inside their tidal radii, takes a longer time tolose a given amount of mass in comparison to tidally fillingclusters (see e.g., Baumgardt, De Marchi, & Kroupa 2008;Marks, Kroupa, & Baumgardt 2008). Indeed, in tidally lim-ited clusters, the early evolution of massive stars leads to arapid expansion, and hence a larger flow of mass over the with r h /r t values smaller than 0.05, where r h and r t are thecluster half-mass and tidal radii, respectively. tidal boundary. It may therefore help to dissolve them morerapidly. Tidally underfilling clusters, however, can survivethis early expansion.We therefore expect that different scenarios for Galac-tic potential (i.e., the time-varying versus the static invari-ant Galactic potentials), in principle, can lead to differentevolution and survival of star clusters and consequently dif-ferent depletion rates of satellite star clusters. Here, in thissection we assess this difference by direct N -body simula-tions of GCs in a realistic MW-like potential using the code nbody6 .The current version of nbody6 does not allow for atreatment of tides with an explicit time-dependent back-ground potential. Within the current framework of nbody6 and in order to estimate the fate of a star cluster which itshost galaxy grows with time we have calculated two inde-pendent models of star clusters with different masses andsizes for host galaxy. • First we assume a star cluster in a circular orbit witha Galactocentric distance of R G = 8 . ”heavy” galaxy with a mass and geometrical parameters ofthe present-day MW-like potential listed in Table 1. • We then simulate the same star cluster in a circular or-bit with Galactocentric distance of R G = 15 . ”light” host galaxy with parameters of the MW-like poten-tial at T = −
12 Gyr given in Table 2. The reason for thischoice of Galactocentric distance is that a cluster which iscurrently located at 8.5 kpc from the centre of the Galaxyhad previously started to evolve (12 Gyr ago) at a Galacto-centric distance of R G = 15 . SSE/BSE routines and analytical fit-ting functions developed by Hurley, Pols & Tout (2000) andHurley, Tout & Pols (2002), the two-body relaxation, and arealistic treatment of the external tidal field. Both star clus-ters are evolving with initial particle number of N = 50000 c (cid:13) , 1–12 Haghi et al.
Figure 9.
The evolution of the core (lower curves) and tidal ra-dius (upper curves) is plotted as a function of time in logarithmicscale. Models are the same as Fig. 8. The tidal radius of clusterwith orbit at R G = 8 . T ≃ ”light” galaxy on a circular orbit with radius of R G = 15 . (corresponding to M ≈ ⊙ ) that were distributed asa Plummer density profile (Plummer 1911) with the sameinitial half-mass radius of r h, = 3 pc. The models startedwith a Kroupa stellar IMF (Kroupa 2001; Kroupa et al.2013), which consists of two power laws with slope α = 1 . . ⊙ and slope α = 2 . ⊙ . The simulated clus-ters evolve on circular orbits at different galactocentric dis-tances in the disc plane (i.e., the inclination angle of theorbits with respect to the galactic disc is zero). Our mainfocus is to study how quantities which can be checked ob-servationally, as e.g., the slope of the MF or the size scaleof the cluster change with time. The evolution of the total mass of the simulated clusterswith time for two different Galactic parameters as given inTables 1 and 2 are plotted in Fig. 8. In fact, the long-termmass-loss for these clusters can be regarded as a runawayoverflow over the tidal boundary. It can be seen, a star clus-ter that evolves in a ”light” galaxy on a circular orbit withradius of R G = 15 . M ⊙ ,i.e., 30% of its initial mass after a Hubble time of evolu-tion, while a star cluster evolving in a ”heavy” galaxy with R G = 8 . heavy ” Galaxy(with the present-day values of parameters of MW listedin Table 1) at 8.5 kpc from the galactic centre, reaches a Figure 10.
Shown are the evolution of Lagrange radii and coreradius of a cluster that evolves in a ”heavy” galaxy with orbitat R G = 8 . T ≃ T ≃ maximum value, which appears to be clearly linked to itsgalactocentric distance, before it decreases again until thecluster dissolves after 11 Gyr of evolution. As shown in Fig.8, the evolution of a star cluster orbiting at R G =15.5 kpcunder the tidal field of a ” light ” Galaxy with the parametersof the MW-like potential at T = −
12 Gyr (Table 1) appearsto have not reached its tidal limit yet and keeps expandingtill the end of the simulation. This is easy to understand:a smaller orbital radius leads to a faster disruption and asmaller half-mass radius after 13 Gyr evolution owing tothe enhanced mass-loss driven by the galactic tide, and thestronger cut-off it inflicts on the clusters.Another useful diagnostic of the difference betweenmodels at different galactocentric distances is provided bythe core and tidal radii. Their time evolutions are illus-trated in Fig. 9. Accentuated mass-loss at R G = 8 . ∼ R G = 8 . heavy ” galaxymodel which we identify as the moment of the initial core- In our simulations, the core radius is a density weighted aver-age distance of each star to the point of highest stellar densitywithin the cluster (Casertano & Hut 1985; Aarseth 2003), whilein observational studies the core radius is generally defined as theradius where the surface brightness falls to half its central value(King 1962). c (cid:13) , 1–12 he time-dependent galactic potential collapse phase ends (Fig. 9). According to Fig. 10, the coreradius expands in the beginning (within the first 2 Gyr) dueto the weak tidal field, but eventually at T ≃ T ≃ light ” Galaxy) doesnot reach the end of the core-collapse phase in 13 Gyr.Therefore, we conclude that the core radius is affectedby changes in tidal forces on the cluster, in agreementwith Madrid, Hurley, & Sippel (2012) who have found thatthe galactocentric distance of a star cluster has an im-pact on its core radius and the onset of core collapse.This is in contradiction with the conclusion reached byMiholics, Webb, & Sills (2014) who showed that the coreradius of a cluster will depend on the initial structural con-ditions of the cluster and will not be affected by its tidalhistory. It can be seen that the tidal radius of the clusterin the heavier (present day) galaxy is smaller (tidally lim-ited) than the cluster in the lighter (12 Gyr ago) galaxy. Wetherefore conclude that clusters survive longer in an evolvinggalaxy than in a galaxy which is kept static.As expected, galaxy with masses between these two ex-tremes (i.e., a time-dependent galaxy mass model) define in-termediate regime of mass-loss. One can therefore concludethat after a Hubble time of evolution, a star cluster has lessmass in a static potential than a simulated cluster evolvingin a live potential. This is because, in the static potentialthe galaxy mass on average (over time) is larger than thelive potential. As already have shown by several authors,increasing the mass of host galaxy accelerate the destruc-tion time of star clusters (Praagman, Hurley, & Power 2010;Madrid, Hurley, & Martig 2014).However, the evolution of a star cluster strongly de-pends on its filling factor ( r h /r t ). The present-day under-filling clusters (i.e., r h /r t < .
05) would remain underfillingin an evolving-potential over the whole 12 Gyr of evolution.This is because of the larger R G and lighter M G comparedto a static potential (see Fig. 9). So one would expect thecluster’s final size to be nearly similar to that of the staticcase. But, a present-day tidally limited cluster would be un-derfilling for a while in an evolving galactic potential. The stellar MF is one of the most important observableparameters that changes through the dynamical evolutionof star clusters. It is evident that the two-body relaxationdriven evaporation through the tidal boundary of a star clus-ter gives rise to a significant correlation between the MF-slope and the strength of tidal field of host galaxy. Becauseof the dynamical mass segregation occurs due to two-bodyrelaxation, the evaporation rate is larger for low-mass stars
Figure 11.
The evolution of the global stellar MF-slope for low-mass ( m . M ⊙ , top panel) and high-mass stars ( m > . M ⊙ ,bottom panel) within the half-mass radius is plotted as a functionof time. Models are the same as Fig. 8. A flatter MF is the directresult of increased tidal stripping of outer region stars that arepreferentially low in mass due to dynamical mass segregation. than it is for high-mass ones (Giersz & Heggie 1997, Baum-gardt & Makino 2003). Thus, the preferential escape of low-mass stars leads to the flattening of the MF as the dynamicalevolution of a star cluster proceeds. It is shown that the evo-lution of MF-slope is faster for clusters at smaller galacto-centric distances (Vesperini & Heggie 1997), i.e., experiencea stronger tidal field (Webb et al. 2013).The canonical IMF as observed in young star clusters inthe MW is often expressed as a two-part power-law function( dNdm ∝ m − α ) with near Salpeter-like slope above 0 . ⊙ (i.e., α = 2 .
3; Salpeter 1955), and a shallower slope of α =1 . . − . ⊙ (Kroupa 2001;Kroupa et al. 2013).Fig. 11 depicts the evolution of the MF-slope at the low-mass end as a function of time. The slope was determinedfrom a fit to the distribution of stars with masses m .
5. Inboth cases the MF flattens as the cluster loses stars. There-fore, as the galactocentric distance of a GC decreases, thestrength of the stellar mass-loss driven by two-body relax-ation increases, and hence the amount of the flattening of theMF enhances. Fig. 11 confirms that the slope of the MF ina cluster orbiting at R G = 8 . ”heavy” Galaxy c (cid:13) , 1–12 Haghi et al. changes significantly as compared to a cluster evolving in acircular orbit with a Galactocentric radius of R G = 15 . ”light” galaxy model.Recent observational work on a number of MW GCshave shown that the global MF-slope in the low-mass rangeis significantly shallower than a canonical MF-slope of about2.3 (Kroupa 2001, see e.g. De Marchi et al. 2007; Jordi et al.2009; Paust et al. 2010; Frank et al. 2012; Hamren et al.2013). However, the preferential loss of low-mass stars dueto two-body relaxation would be a natural explanation forthe observed MF depletion (Baumgardt & Makino 2003),for diffuse outer halo clusters such as Pal 4 and Pal 14(i.e., a low mass together with a large half-mass radius),the present-day two-body relaxation time is of the order ofa Hubble time. Therefore, relaxation should be inefficient inthese clusters and the observations should be an indicationfor primordial mass segregation (Zonoozi et al. 2011, 2014).Our findings in this section show that considering a live po-tential for our Galaxy makes it even more difficult to explainthe observed MF flattening. This is because, the changes inthe MF due to the tidal stripping is less in a live potentialcompared to the static potential model. Many authors usually use the static potential for our Galaxy,as the common assumption is that it remains unchangedduring the orbital integration. In this paper, we have inves-tigated the influence of the time-dependence of the Galacticpotential on the orbital history of the halo objects and itsconsequences on their internal evolution. First, we numer-ically integrated backwards the orbits of different test ob-jects over a Hubble time, located in different Galactocentricdistances within both static and live (cosmologically moti-vated) Galactic potentials to assess the possible differences.It turns out that, the static and live potential do yielddifferent trajectories for our test objects orbiting in differ-ent Galactocentric distances. We have shown that the spa-tial extinction of the orbit’s section in the coordinate planesis larger for live potential w.r.t static potential, such thatin a live potential, the birth of the objects, 13 Gyr ago,would have occurred at significantly larger Galactocentricdistances, compared to the objects orbiting in a static po-tential.As a concrete example of motion within the Galactichalo, we also used the backward motion of the LMC in bothstatic and live Galactic potentials. In addition to the differ-ent trajectories of LMC we uncovered here, we found thatthe orbital period of the LMC around the MW is about 2 . N -body code, nbody6 . Sincethe current version of Nbody6 does not allow for a treat-ment of live potential of host galaxy with an explicit time- dependence, we calculated two models of star cluster withdifferent masses and sizes for host galaxy (which representsthe galaxy at present and 12 Gyr ago), to roughly estimatehow a cluster’s half-mass radius, the total mass, and theMF-slope develops over the time.We followed the evolution of clusters at different Galac-tocentric distances with different Galaxy mass models, andfound that the weaker mass-loss of clusters evolving in aweaker tidal field (i.e., at a larger Galactocentric distanceand within a light-mass Galaxy) leads to a significantlylarger final size.Our computations demonstrate that for two star clus-ters moving in circular orbits with different Galactocen-tric distances and within different Galactic models, onewith R G = 8 . R G = 15 . • a stronger tidal truncation and a smaller size • an enhanced mass-loss rate and a shorter dissolutiontime • a flatter MFSince a galaxy with parameters between these two ex-tremes defines intermediate regime of mass-loss, one cantherefore conclude that over a Hubble time of evolution ina semicosmological time-dependent Galaxy model, the starcluster has more mass than a simulated cluster evolving in astatic invariant potential; this is because in the static poten-tial the galaxy mass on average (over time) is larger than thelive potential. This implies that assuming a static potentialfor our Galaxy (as it is often done) leads to an enhance-ment of mass-loss rate, an overestimation of the dissolutionrates of GCs, and an underestimation of the final size of starclusters.Consequently, after a Hubble time of evolution in theframework of a live Galactic potential, we expect to seethe more survival of star clusters as compared to simu-lated star clusters, which evolve in a galaxy with a con-stant mass components. Clearly, we do not claim thatthe exercise above represents a realistic effect of time-dependence of Galactic potential on the evolution of starclusters. Investigating the fate of a star cluster within agalaxy which grows with time and comparing with the dif-ferent N -body methods (e.g. amuse ; Rieder et al. 2013 and nbody6tt ; Renaud, Gieles, & Boily 2011) is our upcomingproject (Zonoozi et al., in preparation). ACKNOWLEDGEMENTS
This paper is dedicated to Professor Yousef Sobouti, thefounder of IASBS, for his tireless and distinguished effortsin promoting the scientific research in the theoretical as-trophysics in Iran. We would like to thank the referee forconstructive comments and suggestions. HH would like tothank Andreas K¨upper and Holger Baumgardt for helpfulcomments. This work was made possible by the facilities ofGraphics Processing Units at the IASBS. c (cid:13) , 1–12 he time-dependent galactic potential REFERENCES
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This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000