Implications of Recent Measurements of the Milky Way Rotation for the Orbit of the Large Ma

a r X i v :0808.0104v 1 [a s t r o -p h ] 1 A u g 2008

Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 1August 2008

(MN L A T E X style ?le v2.2)

Implications of Recent Measurements of the Milky Way

Rotation for the Orbit of the Large Magellanic Cloud

Genevieve Shattow and Abraham Loeb

Harvard-Smithsonian Center for Astrophysics,60Garden St,Cambridge,MA 02138

1August 2008

ABSTRACT

We examine the implications of recent measurements of the Milky Way rotation for the trajectory of the Large Magellanic Cloud.The ～14±6%increase in the MW circular velocity relative to the IAU standard of 220km s ?1changes the qualitative nature of the inferred LMC orbit.Instead of the LMC being gravitationally unbound,as has been suggested based on a recent measurement of its proper motion,we ?nd that the past orbit of the LMC is naturally con?ned within the virial boundary of the MW.The orbit is not as tightly bound as in models derived before the LMC proper motion was measured.

Key words:Magellanic Clouds –Local Group –galaxies:kinematics and dynamics

1INTRODUCTION

Recently,Kallivayalil et al.(2006;hereafter K06)mea-sured the proper motion of the the Large Magellanic Cloud (LMC),and pioneered a ?rst assessment of its 3D veloc-ity vector.Based on this measurement,Besla et al.(2007;hereafter B07)concluded that the LMC was unlikely to have passed near the Milky Way (MW)before and was most likely formed outside of the boundaries of the Galaxy in contrast to traditional scenarios (see references in van der Marel et al.2002;hereafter vdM02).This new conclusion is surprising given that there are no other examples of massive gas-rich galaxies like the LMC within the much bigger volume that separates the MW from its neighboring galaxy,Andromeda (M31).

A couple of years after K06published their ?ndings,Piatek et al.(2008;hereafter P08)con?rmed independently their results to within one standard deviation,although at the lower end of the inferred range of values.Also,within the past ?ve years,the circular velocity of the Milky Way and the distance between the Sun and the galactic center have been updated by Reid &Brunthaler (2004;hereafter RB04)and Gillessen et al.(2008;hereafter GGE08),respec-tively.These increased the likely circular velocity of the MW from the IAU standard of V circ =220km s ?1to 251km s ?1,with one standard deviation down and up at 236km s ?1and 265km s ?1,respectively.A value of 220km s ?1now corre-sponds to a reduction of the best-?t value by two standard deviations (and equivalent to moving the Sun a total of 1kpc closer to the center of the Galaxy).

The rotation speed of the MW a?ects the analysis of the past LMC orbit in two ways.First,because the proper motion of the LMC is measured relative to the solar system

which orbits the Galaxy,it is necessary to know the rota-tion velocity of the Sun in order to transform to the Galac-tocentric frame (see,e.g.vdM02).Second,the depth of the gravitational potential well of the MW (involving its esti-mated mass and scale radius)depends on the normalization of its rotation curve.B07adopted the IAU standard in their analysis instead of the modi?ed values for the Milky Way’s rotation.In this Letter ,we examine the implications of the change in the MW parameters for the LMC orbit (also with the updated P08value).In the particular geometry of the LMC orbit,both of the above-mentioned e?ects make the LMC more gravitationally bound to the MW owing to an increase in V circ .Despite the small fractional magnitude of the correction in circular velocity (～14±6%),we ?nd that the qualitative nature of the LMC orbit changes.Instead of the LMC being possibly unbound as B07concluded,we ?nd that the LMC’s past trajectory was probably con?ned within the virial radius of the MW.The apogalacticon dis-tance of the orbit is comparable to the MW virial radius,as expected for a satellite that had formed at the outer edge of the Galactic halo.

Traditional studies of the LMC’s orbit around the MW (e.g.,Murai &Fujimoto 1980;Lin &Lynden-Bell 1982;Gar-diner et al.1994;Lin et al.1995;Gardiner &Noguchi 1996;vdM02;Bekki &Chiba 2005;Mastropietro et al.2005)con-sidered the MW as an isolated galaxy.While this might have been acceptable for studies where the LMC’s orbit was con-?ned well within the MW halo,the high LMC velocity mea-sured by K06and P08implies (B07)that the apogalacticon could extend beyond the edge of the MW’s halo –where the gravitational in?uence of M31is non-negligible (see Table 1in B07).Thus,we also include the tidal e?ect of M31in our calculations.

2Shattow&Loeb

In§2we describe our adopted model for the mass dis-tribution of the MW halo.We then calculate the past LMC orbit(§3)and the e?ect of M31on it(§4).Finally,we discuss the implications of our results in§5.

2METHOD

Following B07,we adopt a Navarro,Frenk,&White(1996; hereafter NFW)mass pro?le for the dark matter distribu-tion in the MW halo,and include dynamical friction(Chan-drasekhar1943)of the LMC through the MW halo.We also add the gravitational potential of M31in the form of an-other NFW pro?le at a present-day Galactocentric distance of780kpc(McConnachie et al.2005;Cox&Loeb2008and references therein).The full gravitational potentialΦtot as a function of radius r in each galaxy(either MW or M31) includes contributions from a disk(Φdisk),a bulge(Φbulge), and a dark matter halo(ΦNF W),

Φtot(r)=Φdisk(r)+Φbulge(r)+ΦNF W(r),(1) where(Xue et al.2008)

Φdisk(r)=?GM disk(1?e?r

r

,(2)

Φbulge(r)=?

GM bulge

c3r ln(1+

cr

3

c3

3

ρcr?mδth r3vir∝V3circ,(6) andρcr=3H20/8πG.

Some of our initial conditions(e.g.,M vir and r vir)de-pend on the distance of the Sun from the center of the MW galaxy,as will be discussed in§3,but we consistently adopt M disk=4×1010M⊙,M bulge=0.8×1010M⊙,a disk scale length b=3.5kpc,a disk scale height c0=0.7kpc,and a halo concentration c=12for the MW;and M disk= 7×1010M⊙,M bulge=1.9×1010M⊙,M vir=1.6×1012M⊙, b=5.7kpc,c0=1.14kpc,r vir=300kpc,and c=12for M31,as suggested by Klypin et al.(2002).Aside from the revised normalization of M vir and r vir based on the mod-i?ed value of V circ,we have used the same mass pro?le as B07.

3CONSTRAINTS ON THE ORBITAL

HISTORY

Previous calculations of the history of the LMC have adopted the IAU standard values of d⊙=8.5kpc and V circ=220km s?1,as derived by Kerr&Lynden-Bell (1986;hereafter KLB86).These values assume the angu-lar velocity of circular rotation for the Sun to beΘ0/d⊙=Table1.Modi?ed Values for V circ,M vir,and r vir pf the MW 25.9km s?1kpc?1.RB04directly measured1this value from the proper motion of Sgr A*to beΘ0/d⊙=29.45±.15km s?1kpc?1,which using d⊙=8±.5kpc(Reid1993), implies V circ=236±15km s?1.GGE08has the solar dis-tance to the MW center at d⊙=8.4±.5kpc,almost the same value as KLB86.This brings the maximum V circ up to 265km s?1,which is～20%higher than the220km s?1on which all prior studies were based.Calculations based solely on KLB86are therefore at the lower end of the allowed val-ues for M vir,V circ,and r vir.The new values are listed in Table1.The inferred masses and radii assume the commonly accepted value of M vir=1×1012M⊙for V circ=220km s?1. The values of v X,v Y,v Z(corresponding to Galactic Co-ordinates X,Y,Z)were all calculated using the standard method of vdM02,changing only the value of V circ.

While all of the recalculated masses are consistent with various recent measurements,the Table2entry of P08(251km s?1)corresponding to M vir=1.485×1012M⊙, gives the minimum mass consistent with the timing argu-ment,which put the mass of the Local Group(LG)between 3.2×1012M⊙and5.5×1012M⊙(Binney&Tremaine1987, hereafter BT87).The MW and M31galaxies dominate the LG and are of comparable sizes;a combined mass of3–4×1012M⊙is on the lower end of the timing argument estimate.Li&White(2008),using the Millennium Simula-tion and data on Leo I,?nd the true halo mass of the Milky Way to be M MW=2.34×1012M⊙with a lower limit of M MW=0.8×1012M⊙,giving a more than adequate range to acommodate our inferred masses.Additional mass esti-mates of the MW halo are found in Table3and discussed further in§5.

The P08(251km s?1)entry in Table2for the LMC velocity of339km s?1is signi?cantly lower(by more than 2–σ)than the K06(220km s?1)value of378±18km s?1, although it is higher than the vdM02(220km s?1)weighted average of previous studies,293±39km s?https://www.360docs.net/doc/e314000607.html,paring the corresponding LMC velocities for V circ=251km s?1,the K06and P08values are much closer(in agreement with their initial transverse velocities being within one standard deviation of each other),and well below the escape velocity of the MW,v esc≈380km s?1(B07).P08,K06,and vdM02 with V circ=251km s?1are all compared in Figure2.

1The supermassive black hole,Sgr A*,appears motionless at the center of the Galaxy,and so all of its motion within the galaxy plane is the re?ex of the sun’s orbit around the galactic center (RB04).

Orbit of the LMC

3

Time Gyr

D i s t a n c e k p c

Figure 1.Distance of the LMC from the center of the MW galaxy as a function of time.T=0is the present day.The yellow trajectory (corresponding to V circ =220km s ?1)is gravitation-ally unbound to the MW.The green trajectory (corresponding to V circ =236km s ?1)is also unbound.The blue orbit (corre-sponding to V circ =251km s ?1)is bound to the MW,with a period of ～6.3Gyr and an apogalacticon

of 347kpc.The value of r vir at this V circ is 295kpc (see Table 1and equation 6).The purple orbit,corresponding to V circ =265km s ?1,is more tightly bound,with a period of ～3.5Gyr and an apogalacticon distance of 214kpc.This distance is well within the virial radius r vir =310kpc at this value of V circ .

Time Gyr

D i s t a n c e k p c

Figure 2.Distance of the LMC from the center of the MW galaxy for orbital velocities published in previous studies (see Table 2),but assuming V circ =251km s ?1for the MW.The blue line is from P08,the gray line is from vdM02,and the black line is from K06/B07.

Table 2.Calculated velocities for the LMC

Y kpc

Z kpc Figure 3.Projection of the path of the LMC onto the Y–Z plane (Galactic coordinates)of the Local Group over the past 8Gyr.The yellow,green,blue,and purple trajectories correspond to V circ =220,236,251,and 265km s ?1,respectively.The red line traces the path of M31(represented by the large red dot)away and back towards the MW.The Galactic (MW)center is located at the black dot and the LMC center is represented by the small blue dot.The virial radii of the MW and M31galaxies are traced out by the dashed lines.

Z kpc

Figure 4.A close-up of Figure 3showing the projection of the LMC trajectory onto the Y–Z plane (in Galactic coordinates).The yellow,green,blue,and purple trajectories refer to V circ =220,236,251,and 265km s ?1,respectively.

4TIDAL EFFECT OF M31

In our simpli?ed analysis of the dynamics of M31,we ignore its transverse motion and consider only the perturbative in-?uence of M31as its radial distance changes relative to the MW (BT87).While the transverse component of M31’s ve-locity might be non-negligible (Loeb et al.2005),recent anal-ysis suggests that it is lower than the radial component (van der Marel &Guhathakurta 2008),and its inclusion would

4Shattow &Loeb

Time Gyr

D i s t a n c e k p c Figure 5.Distance of the LMC from the center of the MW as a function of time,with (darker line)and without (lighter line)the gravitational in?uence of M31,assuming V circ =251km s ?1.

only have a weak e?ect on the results reported here.We also ignore any di?use intergalactic mass in between the MW and M31,although future modeling might take the related un-certainty into consideration (Cox &Loeb 2007).To account for the changing separation between the two galaxies,we used the standard radial dynamics model (BT87).

We ?nd that M31has a small but non-negligible e?ect on the trajectory of the LMC.Based on the current position of the LMC (see Figures 3and 4),the addition of M31pulls the LMC away from the Galactic center and towards the LG center of mass.Figure 5compares the distance of the LMC from the Galactic (MW)center with (blue line)and without (light blue line)the gravitational in?uence of M31,assuming V circ =251km s ?1for the MW.Our ?gures keep the MW at the center of the coordinate system (corresponding to an accelerated frame of reference).Without M31,the LMC is more tightly bound to the MW;in the P08(251km s ?1)case,the orbital period of the LMC is 5Gyr,about 1.5Gyr shorter than if M31is included.

5DISCUSSION

We have found that the ～14±6%increase in the MW circular velocity,relative to the IAU standard of 220km s ?1,allows the LMC to be gravitationally bound to the MW.Despite its relatively high proper motion (K06;P08),the orbit of the LMC remains con?ned within the virial radius of the MW.

Since the MW and M31galaxies account for most of the mass in the LG,the LG itself is not much more spatially extended than the two are individually,having an estimated zero energy surface at a radius of ～0.9Mpc (Karachentsev et al.2008).If the LMC had followed the path dictated by the P08(220km s ?1)parameters in Table 2,it would have originated from a distance of 1.5Mpc away from the MW center (see Figure 3).In comparison,if the LMC and MW admit the P08(251km s ?1)parameters,then the LMC originated roughly at the virial radius of the MW halo and well within the boundaries of the LG.The path suggested by P08(236km s ?1)puts the LMC origin outside the LG but closer than the P08(220km s ?1)case.On the other extreme,if the P08(265km s ?1)model is to be believed,the LMC is on its third pass by the MW center and also originated on the edge of the MW halo.The Galactocentric distance on which this result is based (d ⊙=9.0kpc)is one

Table https://www.360docs.net/doc/e314000607.html,ky Way Halo Mass in Recent Studies

1FRC

denotes a Flat Rotation Curve,as in an isothermal sphere.2NB

denotes an N-Body simulation,such as the Millennium simulation.

standard deviation (1–σ)above the most recent estimate (8.4±0.5kpc;see GGE08),so it is not entirely excluded.

We note that during the long time-scale between perigee passes (～6Gyr,see Fig.1),the MW halo mass could evolve by tens of percent (Diemand,Kuhlen &Madau 2007).Fu-ture studies might use cosmological simulations to incorpo-rate the evolution of both the host (MW-like)galaxy and its most massive (LMC-like)satellite to get a statistical under-standing of their likely past interaction.Also,all our quoted uncertainty for the LMC orbit stem from the uncertainty in d ⊙(and therefore V circ ).Errors in the proper motion were not taken into account.The K06measurements,for exam-ple,are slightly over 1–σo?from the P08measurements of the LMC proper motion.The di?erence between these two cases (as seen in Figure 2)is drastic –in one case the LMC is clearly bound to the MW (P08),and in the other it is not (K06).A 1–σshift in the other direction,however,would alter the path of the LMC into an even tighter orbit,similar to the P08(265km s ?1)model.

With the value of the distance between the Sun and the Galactic center at its IAU value (GGE08,KLB86),～8.5±.5kpc,the total mass of the MW Galaxy increases by a factor of 1.23–1.75relative to the values inferred for the lower distance of 8±.5kpc (Reid 1993).B07correctly rules out a larger mass for the MW but only considers the low mass (1×1012M ⊙)and the high mass (2×1012M ⊙)models of Klypin et al.(2002)which bracket our preferred range.Xue et al.(2008)describe the uncertainties of the prior assumption that the galactic V circ is 220km s ?1.Ta-ble 3provides the corresponding range of masses for the MW halo,all calculated from observational data,?t either to a ?at rotation curve or an NFW pro?le,such as the one we have adopted in this work.

The previous suggestion (B07)of possibly disqualify-ing the LMC and SMC as MW satellites has undesirable implications for the overall number of satellites in the MW halo.M31has 18satellites,5of which are gas rich,whereas the Milky Way,if the LMC and SMC are no longer bound to it,has only 12bound satellites,2of which are gas rich (Karachentsev 2005).M31’s satellites range in mass from 0.58to 500×108M ⊙,whereas the remaining MW satellites are in the range of 0.1–1×108M ⊙(Mateo 1998).There is no reason for the two comparably sized galaxies to have such a large disparity in the number of massive gas-rich satel-lites.Moreover,the chance of ?nding massive galaxies like the LMC and the SMC so close to the MW center requires

Orbit of the LMC5

a special coincidence if they are unbound,since these galax-ies should have spent most of their orbital time far away from the MW.Yet,no similar galaxies are known to exist in the much larger volume between M31and the MW.These arguments support indirectly the higher updated values of V circ(RB04)and d⊙(GGE08)for the MW. Acknowledgments.We thank Gurtina Besla and Mark Reid for useful discussions.This work is supported by in part by NASA grant NNX08AL43G,by FQXi,and by Har-vard University and Smithsonian Astrophysical Observatory funds.

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