Sextans' cold substructures as a dynamical judge: Core, Cusp or MOND?
aa r X i v : . [ a s t r o - ph . C O ] S e p Draft version October 21, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
SEXTANS’ COLD SUBSTRUCTURES AS A DYNAMICAL JUDGE: CORE, CUSP OR MOND?
V. Lora , E. K. Grebel , F. J. S´anchez-Salcedo and A. Just Draft version October 21, 2018
ABSTRACTThe cold dark matter model predicts cuspy dark matter halos. However, it has been found that, insome low-mass galaxies, cored dark halos provide a better description of their internal dynamics. Herewe give constraints on the dark halo profile in the Sextans dwarf spheroidal galaxy by studying thelongevity of two cold kinematic substructures detected in this galaxy. We perform N -body simulationsof a stellar clump in the Sextans dwarf galaxy, including a live dark matter halo and the main stellarcomponent. We find that, if the dark halo is cuspy, stellar clumps orbiting with semi-major axis ≈ ∼ r c = 5 pc. Stellar clusters in an initial orbit with semi-major axis ≤
250 pc may survive to dissolutionbut their orbits decay towards the center by dynamical friction. In contrast, the stellar clumps canpersist for a Hubble time within a cored dark matter halo, even if the initial clump’s radius is asextended as r c = 80 pc. We also study the evolution of the clump in the MONDian context. In thisscenario, we find that even an extended stellar clump with radius r c = 80 pc survives for a Hubbletime, but an unrealistic value for the stellar mass-to-light ratio of 9 . Subject headings: cosmology: dark matter – galaxies: dwarf – halos – kinematics and dynamics –methods: numerical INTRODUCTION
The Λ cold dark matter (ΛCDM) model has provedto be successful in reproducing structure formation atlarge scales, but it faces some difficulties at galacticscales. For example, cosmological N -body simulationspredict halos with a central cusp (Navarro et al. 1997;Moore et al. 1999; Jing & Suto 2000), whereas observa-tions of the rotation curves of dwarf and low surfacebrightness galaxies indicate that a cored dark halo ispreferred (van den Bosch et al. 2000; de Blok & Bosma2002; Kuzio de Naray et al. 2008; de Blok et al. 2008;Donato et al. 2009).Dwarf spheroidal (dSph) galaxies are the natu-ral targets to study the properties of dark matter(DM) halos at very small masses. The analysis ofthe cusp-core controversy in dSph galaxies has mo-tivated much work (Kleyna et al. 2003; Goerdt et al.2006; S´anchez-Salcedo et al. 2006; Gilmore et al.2007; Battaglia et al. 2008; Walker & Penarrubia2011; Jardel & Gebhardt 2012; Salucci et al. 2012;Agnello & Evans 2012; Amorisco et al. 2013;Breddels & Helmi 2013). Dynamically cold stellarsubstructures as those observed in some dSph galaxiesare sensitive probes of the gravitational potential.Kleyna et al. (2003) presented evidence that the stellarsubstructure in the Ursa Minor (UMi) dwarf spheroidalis incompatible with a cusped DM halo. They arguedthat the second peak located on the north-eastern side ofthe major axis of UMi is a disrupted stellar cluster thathas survived in phase-space because the underlying grav-itational potential is close to harmonic (Kleyna et al. [email protected] Astronomisches Rechen-Institut, Zentrum f¨ur Astronomieder Universit¨at Heidelberg,M¨onchhofstr. 12-14, 69120 Heidelberg, Germany Instituto de Astronom´ıa, Universidad Nacional Aut´onomade M´exico, AP 70-264, 04510 D.F., M´exico N -bodysimulations of the Sextans dSph galaxy. We model itsstellar components (the main stellar component + stel-lar clump) and the DM halo. We explore different profilesfor the DM halo and different sizes for the starting stel-lar clump. We also study the survivability of stellar sub-structures in MOdified Newtonian Dynamics (MOND).This article is organized as follows. In Section 2 wedescribe some properties of Sextans and their stellarclumps. The initial conditions for the N -body simula-tions are given in Section 3. The evolution of the sub-structures is described in Section 4. Finally, we discuss Lora et al.the implications and give our conclusions in Section 5. SEXTANS AND ITS KINEMATIC SUBSTRUCTURES
Sextans is a dSph galaxy satellite of the Milky Way. Itis located at a Galoctocentric distance of R GC = 86 kpc(Mateo 1998) and it has a luminosity of L V = (4 . ± . × L ⊙ ( Lokas 2009). It has a core radius of R core = 16 . ∼ . R tidal = 160 arcmin ( ∼ . ± . × M ⊙ by integrating the light from a single 12 Gyr old stel-lar population. This mass is consistent with the value(8 . × M ⊙ ) obtained by Woo et al. (2008). The cor-responding B-band stellar mass-to-light ratio is Υ ⋆ ≈ × and 1 . × M ⊙ . Adopting aNFW profile for the dark halo, Walker et al. (2007) ob-tained a dynamical mass of 2 . × M ⊙ within a ra-dius of 600 pc, whereas Strigari et al. (2007) estimateda total mass of 0 . × M ⊙ , also within the central 600pc. More recently, Lokas (2009) reported a total mass of(4 . ± . × M ⊙ assuming an NFW density profile,which implies a M/L value of ∼
100 (M/L) ⊙ . All thesedynamical studies show that Sextans is a DM dominateddSph galaxy.Here, we are interested in the existence of putativestellar substructures in Sextans. Kleyna et al. (2004) re-ported some evidence for a kinematically and photomet-rically distinct population at the Sextans center. Theseauthors found that the dispersion at the center of Sex-tans was close to zero, and that such a change in thedispersion profile coincides with a change in the ratioof red horizontal branch stars to blue horizontal branchstars, i.e. in the stellar populations. They suggested thatthis is caused by the sinking and gradual dissolution ofa stellar cluster at the center of Sextans.In a later work, Walker et al. (2006) presented radialvelocities of 294 possible Sextans members. Their largerdata set did not confirm Kleyna et al.’s (2004) report ofa kinematically distinct stellar population at the centerof Sextans but they did obtain similar evidence whenthey restricted their analysis to a similar (small) numberof stars as used by Kleyna et al. (2004). When consider-ing their full radial velocity sample instead, Walker et al.(2006) detected a region near Sextans’ core radius thatis kinematically colder than the overall Sextans sample,with 95% confidence. They estimated a substructure lu-minosity of 3 × L ⊙ . We will refer to it as substructureA .Recently, Battaglia et al. (2011) reported nine oldstars that share very similar spatial location, kinematicsand metallicities. The average metallicity of their 9-stargroup is low ([Fe/H]= − . .
15 dex scatter),consistent with being the remnant of a old stellar clus-ter. This group of stars was taken from the six innermostmetal-poor stars, which show a cold velocity dispersion of1 . − and an average velocity of 72 . ± . − . Battaglia et al. (2011) suggested that the number of starsin this substructure (nine stars) is significant with respectto the total number of Sextans members for which spec-troscopic measurements exist (174 stars). This substruc-ture would account for 5% of Sextans’ stellar population,which corresponds to a luminosity of 2 . × L ⊙ . Wewill refer to it as substructure B .The present spatial extent of the substructures is veryuncertain. The contours of statistical significance for re-gions of cold kinematics in Walker et al. (2006) showthat substructure A is centered on a location 15 arcminnorth of the Sextans center and has a radial size of 4arcmin ( ∼
100 pc). On the other hand, the nine inner-most metal-poor stars that constitute the substructureB are found at
R < ◦ .
22 (Battaglia et al. 2011), i.e. at ≈
330 pc from the center of Sextans if we assume a dis-tance to Sextans of 86 kpc. In Battaglia et al.’s (2011)data, there are no metal-poor stars at
R < ◦ .
1. Thissuggests that the substructure B extends in projectedgalactocentric radius from 0 ◦ . ◦ .
22 (i.e. between150 and 330 pc), indicating that, in projection, its centeris at ∼
240 pc from the Sextans center, and its radius is ∼
90 pc at most. THE N-BODY MODEL
Sextans’ DM component
For our simulations, we constructed a live DM halowith a mass density profile given by ρ ( r ) = ρ ( r/r s ) γ [1 + ( r/r s ) α ] ( β − γ/α ) , (1)where r s is the scale radius and α , β and γ define the DMhalo’s slope. This general density profile equation is veryuseful to define different density profiles. For example,a pseudo-isothermal sphere is obtained for ( α, β, γ ) =(2 , ,
0) and an NFW profile is obtained for ( α, β, γ ) =(1 , , r s = 3 kpc, and a mass of 4 × M ⊙ within the lastmeasured point ( ∼ . . . × M ⊙ . For the NFW DMhalo, they derived a concentration c = 10 and a virialmass M V = 2 . × M ⊙ , resulting a DM mass within0 . . × M ⊙ .To generate the initial conditions of the DM particles,we used the distribution function proposed by Widrow(2000), assuming an isotropic velocity dispersion tensor. Sextans’ main stellar component
The main stellar component in Sextans was modeledusing the density profile ρ ∗ ( r ) = (3 − γ ) M ∗ π ar γ ( r + a ) β − γ , (2)where M ∗ is the total stellar mass and a is the scaleradius. We set M ⋆ ≈ × M ⊙ , assuming a typicalextans: core, cusp or MOND? 3value of the mass-to-light ratio Υ ⋆ = 2. We took β = 4,which corresponds to the Dehnen models (Dehnen 1993;Tremaine et al. 1994). In these models, the density de-clines as r − at large radii and diverges in the center as r − γ . We used γ = 3 /
2, because it most closely resem-bles the de Vaucouleurs model in surface density. Wetook the scale radius for the main stellar componentin the Sextans galaxy to be a ≈ . . N -body simulation with the DM halo(cored and NFW) and the main stellar component to-gether. Each component was found to be stationary fora Hubble time (i.e., the density profile of the stellar com-ponent and its velocity dispersion stayed approximatelyconstant for a Hubble time). Sextans’ stellar clumps
Our starting hypothesis is that the cold substructuresin Sextans were initially stellar clusters that are now inthe process of very slow dissolution. The substructuresare the gravitationally unbound remnants of the stellarclusters. Adopting Υ ⋆ = 2, typical for an old stellarpopulation, the mass in the substructures lies between M = 4 . × M ⊙ for substructure B to ≃ × M ⊙ forsubstructure A, which are reasonable for stellar clusters.In all our runs, we assumed, for simplicity, a fixed massof 4 . × M ⊙ for the initial stellar cluster.For the initial density profile of the stellar clumps, weused a Plummer model, where the mass density profileis given by the following equation: ρ c ( r ) = ρ (cid:18) r r c (cid:19) − / , (3)(Plummer 1911). We explored different values for theinitial core radius r c between 5 pc, which correspondsto the size of a typical stellar cluster, and 80 pc, whichis of the order of the present size of the observed sub-structures. Simulations with initial core radius of r c = 5pc are aimed to represent a scenario where the stellarcluster has been caught in the last stage of tidal dis-ruption. This scenario may present a timing problembecause this stage is expected to proceed on a time-scaleof one crossing time of the system and, thus, it would bevery unlikely to observe them during this phase. Simu-lations with an initial radius of r c = 80 pc correspondto a situation where the stellar cluster became unboundimmediately after formation due to supernova ejection ofgas (Goodwin 1997).Without the loss of generality, we set the clumps withan orbit in the ( x, y ) plane. Since we do not know theorbital parameters of the substructures, we explored dif-ferent orbits for the clumps around the Sextans center.We only know lower limits for the semimajor axes ofthe substructures (it is &
400 pc for substructure A and &
200 pc for substructure B). Because projection effectslead to an underestimation of the galactocentric distanceof the objects, there is a probability of 20% that the sub-structure B is at a deprojected distance of ≥
400 pc to Sextans’ center. Therefore, since either substructure Aor B or both may be on an orbit with a characteristic ra-dius of 400 pc, we considered orbits with this size. Notethat a distance of ∼
400 pc corresponds to the core radiusof the main stellar component in Sextans (see Section 2).We also considered the limiting case where the galacto-centric distance of the clump is 250 pc in order to studya wider range of possible orbits of the clump; this case isrelevant for substructure B. Since the substructures arenot necessarily on circular orbits; it is also worthwhile toconsider eccentric orbits.
The code
Since the internal two-body relaxation timescales forthe three components (clump, main stellar componentand halo) are much larger than a Hubble time, Sextanscan be represented as collisionless (Binney & Tremaine2008). We simulated the evolution of the Sextans dwarfgalaxy (stellar clump, main stellar component and DMhalo) using the N -body code SUPERBOX (Fellhauer et al.2000; Bien et al. 2013).
SUPERBOX is a highly efficientparticle-mesh, collisionless-dynamics code with high res-olution sub-grids.In our case,
SUPERBOX uses three nested grids centeredon the center of density of the Sextans dSph galaxy. Weused 128 cubic cells for each of the grids. The inner gridis meant to resolve the inner region of Sextans and theouter grid (with radii of 100 kpc for all cases) resolves thestars that are stripped away from Sextans’ potential. Thetidal field created by the Milky Way was not included.The spatial resolution is determined by the number ofgrid cells per dimension ( N c ) and the grid radius ( r grid ).Then the side length of one grid cell is defined as l = r grid N c − . For N c = 128, the resolution is of the order of thetypical distance between the particles in the simulation. SUPERBOX integrates the equations of motion with aleap-frog algorithm, and a constant time step dt . Weselected a time step of dt = 0 . RESULTS
Our N -body simulations were carried out from an in-tegration time t = 0 to t = 10 Gyr. We used 1 × particles to model each of the DM halos, 1 × parti-cles to model the main stellar component, and 1 × particles to model the stellar clump. The parametersof our 14 models, from M M
14, are summarized inTable 1.
The cored DM halo case
We first consider the evolution of the clump when itis embedded in a pseudo-isothermal DM halo, having acore radius r s = 3 kpc and a mass within 0 . . × M ⊙ (see Section 3.1). In order to visualize theevolution of the clump in Sextans, we built a map ofthe surface density (in units of M ⊙ pc − ) of the stellarclump in the ( x, y )-plane. Figure 1 shows the temporalevolution of the mass surface density of the clump in themodels M r c = 12 pc) and M r c = 80 pc) for the cored DM halo. The whitecircle shows the initial orbit of the stellar clump, and thewhite cross shows the center of Sextans. Lora et al.Model M . M
1, the stellar cluster remainsessentially intact over the duration of the run, 10 Gyr(see Figure 1), because the tidal heating by the parentgalaxy’s halo is very low.Unbound clumps as extended as the observed substruc-tures (our model M
2) also remain essentially unalteredfor 10 Gyr (see the bottom panels of Figure 1). Thephysical reason is that the underlying potential withinthe DM core is harmonic and, therefore, the substruc-ture is long-lived even if it is a gravitationally unboundsystem (Kleyna et al. 2003; Lora et al. 2009). We alsochecked that the substructure can persist for a long timeif it is dropped on a circular orbit with a radius of 250 pc(simulation M & The NFW DM halo case
The top panels of Figure 2 show the evolution of themass surface density of the stellar clump with initial ra-dius r c = 12 pc (model M c = 10 and a virial mass of M V = 2 . × M ⊙ (see Section 3.1). For the first 3 Gyr, the stellar clumpappears almost unperturbed (see panel ( c ) of Figure 2),but after 5 Gyr the orbital phase mixing dissolves it com-pletely and so only a tidal debris can be seen in panel ( d )of Figure 2.The lower panels of Figure 2 show four snapshots of thesubstructure when its initial radius is r c = 80 pc (model M ∼ . ∼
10 Gyr), the substructureis short lived.At any given time t in the simulation we sample thetwo-dimensional map searching for the 10 ×
10 pc sizeparcel that contains the highest mass (number of clumpparticles). We define the destruction time as the time atwhich the parcel with the highest mass surface densityhas reached a value of ∼ ⊙ pc − . When such value isreached, the column density of the clump is so low thatit would be indistinguishable from Sextans’ main stellarcomponent, and would thus be undetectable. In Figure 3,we plot the surface density of this parcel as a function oftime. We see that the destruction time is ∼ . M ∼ .
45 Gyr in model M M
6, we reduced the initial size of the clumpto a radius r c = 5 pc (Figure 4). We found that not evenwith such a small clump is able to survive for more than ∼ M M
8, see Table 1) where clump’s orbit is non-circular. Inmodel M
7, we set the 5 pc radius clump in a pure radialorbit with apocenter at 400 pc (see Figure 6). We foundthat the clump loses its identity in 4 Gyr (Figure 6). Inmodel M
8, we set the 5 pc radius cluster at a radialdistance of 400 pc with a tangential velocity twice thecircular velocity at that distance ( v y = 2 × v c ). Thus,the orbit is eccentric and its pericenter is located at 400pc. In this case, the clump is disrupted within ∼ ∼ ∼
200 pc or less. In order to explore if this clump couldsurvive for a significant time, we ran a simulation ( M t ≈ M ∼ M
10, where the orbit of the 5 pc radius clumphas its apocenter at 250 pc and its pericenter at 100 pc.The clump spirals to the center of Sextans and loses mass,until the density of the clump decreases by a factor of ∼ t ≈ ∼
400 pc would not survive in an NFWhalo. Only substructures with initial orbital radii . <
100 pc) because of the or-bital decay due to dynamical friction. Since substructureA is at a projected distance of 400 pc, it is difficult tounderstand how it survived against mixing in an NFWhalo. A more accurate determination of the projecteddistance of substructure B would be very important toconstrain the models further.
The case of MOND
It is interesting to explore if the gravitational poten-tial predicted in MOND could explain the survival ofcold substructures. To do so, we followed a similartreatment as in S´anchez-Salcedo & Lora (2010) for theUMi dSph galaxy. In the MOND framework, the grav-itational potential that describes the force acting on astar in Sextans follows the modified Poisson equation ofBekenstein & Milgrom (1984) ∇ · [ µ ( x ) ∇ Φ] = 4 πGρ , (4)where x = |∇ Φ | /a , a ≃ . × − cm s − is the uni-versal acceleration constant of the MOND theory, and µ ( x ) is the interpolating function, which runs smoothlyfrom µ ( x ) = x at x ≪ µ ( x ) = 1 at x ≫ ∇ φ → − g E , where g E is the ex-ternal gravity acting on Sextans and has a magnitude g E = V /R GC . V is the Galactic rotational veloc-ity at R GC which coincides with the asymptotic rota-tion velocity V ∞ for the Milky Way. We set this valueto V ∞ = 170 km s − which is obtained by adopt-ing a mass model for the Milky Way under MOND(Famaey & Binney 2005; S´anchez-Salcedo & Hernandez2007).extans: core, cusp or MOND? 5A star in the stellar clump of Sextans feels the exter-nal acceleration created by the Milky Way (denoted by g E ), the acceleration generated by Sextans smooth stel-lar component ( g I ), and the acceleration generated by allother stars that form the stellar clump ( g int ), and thusall must be taken in consideration.Since the circular velocity of a test particle at thestellar core radius ( r ∗ ≃ . ∼ . − , then the characteristic internal accelera-tion [ v c ( r ∗ )] /r ∗ ≃ . × − cm s − . This is muchsmaller than MOND’s characteristic acceleration a ≃ . × − cm s − . Moreover, the external acceleration, V ∞ /R GC ≃ . × − cm s − is also much smallerthan a . We can conclude then that the Sextans inter-nal dynamics lies deep in the MOND regime. The ratiobetween the internal acceleration at Sextans’ stellar coreradius and the external acceleration ( g I /g E ≈ .
26) tellsus that the dynamics in Sextans is dominated by theexternal field ( g E ≫ g I ).S´anchez-Salcedo & Hernandez (2007) studied the dSphgalaxies of the Milky Way under MOND and comparedthe results with DM halos. For Sextans, they obtaineda high value of (M/L) of ∼ . −
36 assuming that theexternal field is dominant in this galaxy. Angus (2008)analyzed the line-of-sight velocity dispersion as a func-tion of radius for eight Milky Way dSph galaxies and cal-culated the mass-to-light ratio in the MONDian regimethrough a Jeans analysis. He found that Sextans requiresa rather high mass-to-light ratio of 9 .
2. We adopted thisvalue here.We carried out N -body simulations under the MONDapproximation with the code described in Section 3 start-ing with clumps of different radii ( r c = 12, 35 and 80 pc,which correspond to models M M
12 and M
13, re-spectively). The parameters of the different models aresummarized in Tables 1 and 2. The clump survives formore than 10 Gyr in the three cases (see Figure 3). It isinteresting to note that when the initial stellar clump isextended ( r c = 80 pc), it spirals to the center of Sextans;at t = 10 Gyr it orbits within ∼ . M
14) of the 80 pc radiusclump in a circular orbit with a galactocentric distance of250 pc. Also in this case the clump remains undisturbedfor 10 Gyr. In this case, the clump also spirals to thecenter of Sextans, and at t = 10 Gyr it orbits within ∼ . CONCLUDING REMARKS
Using N -body simulations, we studied the survival ofcold kinematic substructures in the DM halo of the Sex-tans dwarf galaxy against phase mixing. We comparedthe evolution of substructures when the dark halo has acore and when the dark halo follows the NFW profile, those having the parameters derived by Battaglia et al.(2011) to explain the projected velocity dispersion pro-file. We found that the core in the pseudoisothermalmodel is large enough to make the potential almostharmonic and, thus, to guarantee the survival of sub-structures. Even if the clump is initially very extended( r c = 80 pc), it easily survives for 10 Gyr. We concludethat the stellar clump in Sextans is in agreement with acored DM halo.On the contrary, stellar clumps orbiting with semi-major axes of ∼
400 pc and initial Plummer radii be-tween 12 and 80 pc are destroyed if they are embeddedin the NFW DM halo. Not even a stellar clump witha small Plummer radius of r c = 5 pc (model M
6) cansurvive in such a NFW DM halo. Stellar clumps initiallyorbiting at a radius ≤
250 pc from the Sextans centerspiral to the center due to dynamical friction and, as aconsequence, phase mixing is reduced. Clumps in theseorbits may survive but would merge forming a centralstar cluster at the center of Sextans.It has to be noted that we cannot rule out a scenariowhere the DM profile was initially cuspy and evolvedto a cored halo. For instance, energy feedback fromsupernova explosions and stellar winds, may lead se-vere gravitational potential fluctuations, which may re-duce the central mass density of dwarf galaxies (e.g.,Mashchenko et al. (2008)). Similarly, Pasetto et al.(2010) found that initial cuspy DM profiles flatten withtime as a result of star formation, which would explainthe observations without contradicting a cuspy DM pro-file. On the other hand, Pe˜narrubia et al. (2012) showsome difficulties with the fine-tuning of the scenarios justmentioned. Goerdt et al. (2010) discuss that the trans-fer of energy from sinking massive objects may destroycentral cusps.As a last point, we investigated whether or not theclump in Sextans could survive in the MONDian frame-work. We found that even a stellar clump with r c = 80 pcremains undisturbed for a Hubble time and slowly spiralsto the center of Sextans. However, it has to be noticedthat the adopted MOND value for the stellar mass-to-light ratio of M/L V = 9 .
2, which was derived by Angus(2008) to explain the observed velocity dispersion profileof Sextans, is very high and inconsistent with the prop-erties expected from an old purely stellar population.
ACKNOWLEDGMENTS
We wish to thank the anonymous referee for veryuseful comments that greatly improved the content ofthis paper. V.L. gratefully acknowledges support froman Alexander von Humboldt Foundation fellowship andthe FRONTIER grant. F.J.S.S. acknowledges financialsupport from CONACyT project 165584 and PAPIITproject IN106212.
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Fig. 1.—
Time evolution ( t = 0, 3, 6 and 10 Gyr) of the stellar clump’s mass surface density in the Sextans dSph galaxy. The top panelsshow the evolution of the clump in model M r c = 12 pc), whereas the bottom panels show the evolution of the clump in model M r c = 80 pc). The white circle shows the initial orbit with 0 . Fig. 2.—
The top panels show the mass surface density of Sextans’ stellar clump at four different times ( t = 0, 1 .
5, 3 and 5 Gyr) in model M r c = 12 pc). In the bottom panels, we show the mass surface density (at t = 0, 0 .
5, 2 and 3 . M r c = 80 pc). Thewhite circle shows the initial orbit. The white cross marks the center of Sextans. Lora et al.
Fig. 3.—
Mass surface density of Sextans’ stellar clump mapped in the ( x, y )-plane as a function of time, for the models quoted at theright margin of the Figure (see Table 1).
Fig. 4.—
Mass surface density of Sextans stellar clump in the ( x, y )-plane in model M t = 0, 1 .
5, 3 . . extans: core, cusp or MOND? 9 Fig. 5.—
Galactocentric distance of the clump as a function of time, for models M M
10 (bottom panel).
Fig. 6.—
Mass surface density of Sextans stellar clump in the plane of the orbit, in model M t = 0, 1, 2 . Fig. 7.—
Mass surface density of Sextans stellar clump in the ( x, y )-plane in model M
13 for the integration times t = 0, 3, 6 and 10 Gyr.The outer white circle shows the initial orbit of the stellar clump, with 0 . .
29 kpc.
TABLE 1Parameters of the models
Model Halo r c Surviving time Type of orbitprofile a [pc] [Gyr]M1 ISO 12 >
10 circular orbit of radius of 400 pcM2 ISO 80 >
10 circular orbit of radius of 400 pcM3 ISO 80 >
10 circular orbit of radius of 250 pcM4 NFW 12 ∼ . ∼ .
45 circular orbit of radius of 400 pcM6 NFW 5 ∼ ∼ ∼ v x = 0, v y = 2 v c M9 NFW 5 > b circular orbit of radius of 250 pcM10 NFW 5 > c eccentric, apocenter at 250 pcand pericenter at 100 pcM11 MOND 12 >
10 circular orbit of radius of 400 pcM12 MOND 35 >
10 circular orbit of radius of 400 pcM13 MOND 80 >
10 circular orbit of radius of 400 pcM14 MOND 80 >
10 circular orbit of radius of 250 pc a ISO refers to the pseudo-isothermal profile. b The orbit decays to the Sextans center in ∼ c The orbit decays to the Sextans center in ∼ extans: core, cusp or MOND? 11 TABLE 2Parameters of the Sextans dSph and its stellar clump
Sextans D L V r ∗ M ∗ M ( < r ∗ ) v c ( r ∗ ) g I /g E [kpc] [L ⊙ ] [arcmin] [M ⊙ ] [M ⊙ ] [km s − ] at r ∗
86 4 . × . . × . × . . L V r h M v c ( r h ) g int /g E axis [kpc] [L ⊙ ] [pc] [M ⊙ ] [km s − ] at r h Small clump 0 . . × . . × . . . . × . . × . ..