The Magellanic Stream in Modified Newtonian Dynamics

0.28 MB·English

by  Hossein Haghi

The Magellanic Stream in Modified Newtonian Dynamics
Hossein Haghi 1, Sohrab Rahvar 1,2, Akram Hasani-Zonooz 3
[email protected]
[email protected]
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2 ABSTRACT

n The dynamicsofthe MagellanicStream(MS) asaseriesofcloudsextending fromthe Magel-
a
lanic Clouds (MCs) to the south Galactic pole is affected by the distribution and the amount of
J
matter inthe Milky Way. We calculate the gravitationaleffectof the Galacticdisk onthe MSin
9
the framework of modified Newtonian dynamics(MOND) and compare with observations of the

3 Stream’sradialvelocity. WeconsiderthetidalforceoftheGalaxy,whichstripsmaterialfromthe
v MCstoformtheMS,and,usingano-halomodeloftheGalaxy,weignoretheeffectofthedragof
6 the Galactic haloon the MS. We also comparethe MONDian dynamics with that in logarithmic
4
andpower-lawdarkhalo modelsandshowthatthe MONDtheoryseems plausiblefordescribing
2
the dynamics of satellite galaxies such as the MCs. Finally, we perform a maximum likelihood
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0 analysis to obtain the best MOND parameters for the Galactic disk.
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Subject headings: dark matter –Galaxy: halo –gravitation – Magellanic stream – MOND.
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/
h 1. Introduction toward the Large and Small Magellanic Clouds
p
are most probably due to self-lensing by the
- It has long been known that Newtonian grav-
o Clouds (Sahu 2003; Tisserand & Milsztajn 2004;
r ity is unable to describe the dynamics of galaxies Rahvar 2004).
t
s and clusters of galaxies correctly and that there
Other observations, such as of the abundances
a is a discrepancy between the visible and the dy-
: of the light elements from nucleosynthesis in the
v namical masses of galaxies. Nowadays, most as-
earlyuniverseandrecentdatafromtheWilkinson
i tronomers believe that the universe is dominated
X Microwave Anisotropy Probe (WMAP) data, also
by dark matter andthat we can observethe grav-
r rule out baryons as the dark matter. The best-fit
a itationaleffects ofthis matter inlarge-scalestruc-
baryon abundance, based on WMAP data, to a
ture. Many observations have been dedicated to
ΛCDM model is Ω h2 = 0.0237+0.0013, implying
measuring the amount and nature of the dark b −0.0012
a baryon-to-photon ratio of η = 6.5+0.4 10−5
matter. Gravitationalmicrolensingis one ofthese −0.3 ×
(Spergel et al. 2003). For this abundance, stan-
experiments, designed to indirectly detect the
dardbigbangnucleosynthesis(Burles et al. 2001)
dark matter of the halo in the form of MACHOs1
implies a primordial deuterium abundance rela-
(Alcock et al. 2000; Paczyn´ski1986). After al-
tive to hydrogen of [D/H] = 2.37+0.19 10−5,
most two decades of microlensing experiments, it −0.21 ×
which is in the same range as observed in Lyα
seems that MACHOs contribute a small fraction
clouds(Kirkmanetal. 2003): analysisofQSOHS
of the halo mass and that the microlensing events
1243+3047yieldsaD/Hratioof2.42+0:35 10−5.
−0:25×
WIMPs2, as the other candidate for the so-
1Department of Physics, Sharif University of Technol-
called the cold dark matter (CDM), are moti-
ogy,P.O.Box11365–9161, Tehran,Iran
2InstituteforStudiesinTheoreticalPhysicsandMath- vated from non-standard particle physics. The
ematics,P.O.Box19395–5531, Tehran,Iran problem with WIMPs is their small cross section
3Department of Physics, Azarbaijan University of Tar-
biatMoallem,Azarshahr,Tabriz,Iran 2Weaklyinteractivemassiveparticles
1Massivecompact haloobject
1
with ordinary matter, which would makes them spherically symmetric perturbation, in the frame-
difficult to observe, to date, no significant signal work of MOND, the gas temperature and density
from experiments aimed at detecting such parti- profilesevolvetoauniversalformthatisindepen-
cles has been reported (Pretzl 2002; Green 2004; dentofboththeslopeandamplitudeoftheinitial
Goodman & Witten 1985; Jungman et al. 1996; density profile. While the obtained density profile
Mirabolfathi 2005). is compatible with results from XMM Newton
−
Another growingapproachto explain the miss- and Chandra observations of clusters, the tem-
ingmatterintheuniverseis apossiblealternative peratureprofilesdonotreconcilewiththe MOND
theory of gravity. In these models, modification prediction.
of the gravity law compensates for the lack of Another approach to examining MOND could
matter. One of the most famous alternative the- be through the dynamics of the satellite galax-
ories is modified newtonian dynamics (MOND), ies around the Milky Way (MW). Read & Moore
which was been introduced by Milgrom (1983). (2005)comparedthe precessionofthe orbitofthe
According to this phenomenological theory, the Sagittarius dwarf in a thin axisymmetric disk po-
flat rotation curves of spiral galaxies at large ra- tential under MOND with that resulting from a
dial distances can be explained with a modifica- thin disk plus nearly-spherical dark halo. They
tion of Newton’s second law for accelerations be- find thatimproveddataonthe leadingarmofthe
low a characteristic scale of a 1 10−8cms−2 Sagittarius dwarf,which samples the Galactic po-
0
≃ ×
(Milgrom 1983; Bekenstein and Milgrom 1984; tential at large radii, could rule out MOND if the
Sanders & McGaugh 2002). This theory, in ad- precessionoftheorbitalpolecanbedeterminedto
dition to the acceleration parameter,a , employs accuracy of a degree. Here we extend this work,
0
an interpolating function to connect and MON- usingthedynamicsoftheMagellanicStream(MS)
Dian regimes. It has been shown that this to test MOND. The MS is a narrow band of neu-
simple idea may explain the dynamical motion tral hydrogen clouds that extends along a curved
of galaxies without resorting to dark matter path,startingfromtheMagellanicCloudsandori-
(Pointecouteau & Silk 2005;Baumgardt et al. 2005; entedtowardthesouthGalacticpole(vanKuilen-
Scarpa et al. 2004). Theparametersofthismodel burg 1972; Wannier & Wrixon 1972; Mathewson
can be fixed by various observations, such as et al. 1974). We use the MONDian gravitational
the rotation curves of spiral galaxies. Recently, effectofathinKuzmindisktorepresentthelumi-
Famaey & Binney (2005) studied MOND in the nous part of the MW, and we obtain the dynam-
context of the rotation curves of spiral galaxies, ics of the MS, treating it as a quasi-steady flow.
using different choices for the MOND interpolat- Using a maximum likelihood analysis, we find the
ing function. Bekenstein (2004) presented a rela- bestmodelparametersfromananalysisoftheMS
tivistic version of MOND, considering a Lorentz- radialvelocitydatareportedbytheParkesObser-
covarianttheoryofgravity. Zhao&Famaey(2006) vatory (Bru¨ns et al. 2005).
also examined different interpolating functions by The paper is organized as follows: In 2, we
fitting to a benchmark rotation curve, and they givea brief reviewofthe darkhalo models f§or the
proposeda newsetofinterpolatingfunctions that Galaxy and of MOND theory as an alternative to
satisfy both weak and strong gravity. One of dark matter. In 3, we introduce the Magellanic
the main tests of MOND is its ability to produce Stream and the m§ain theories describing the dis-
powerspectra that are compatible with the struc- tributionofgasalongit. In 4,wecomparetheob-
tures that we observe. Sanders (2001) showed served dynamics of the MS§with predictions from
thatperturbations onsmallcomovingscalesenter CDM models and MOND theory, considering the
the MOND regime earlier than those on larger MS as a quasi-steady flow. We conclude in 5.
scales and, therefore, evolve to large overdensities §
sooner. Taking the initial power spectrum from 2. Dynamics of The Milky Way: MOND
the cosmic microwave background,Sanders found Versus CDM
that the evolved power spectrum resembled that
of the standard CDM universe. However, studies Aspointedoutintheprevioussection,thereare
byNusser&Pointecouteau(2006)showthatfora twoapproachesfordealingwiththeproblemofthe
2
rotation curves of spiral galaxies. The first is to For a given distribution of matter on the right-
consider a dark halo around the galaxy that is 10 hand side of equation (3), the dynamics of the
times moremassivethanits luminouscomponent, structure, depending on the interpolating func-
and the second is to invoke an alternative theory tion, can be found as
suchasMOND.Inthefirstpartofthissection,we
a
introduce MOND theory and consider a Kuzmin µ( )a=G G(x x′)ρ(x′)dx′3+ h(x) (4)
diskforthemassdistributionoftheGalaxy,andin a0 Z − ∇×
thesecondpartvariousdarkmattermodelsforthe
whereG(x)istheGreen’sfunction,Gisthegrav-
halo mass distribution in the MW are presented.
itationalconstant,andh(x)is anarbitraryvector
field. UsingtheGreen’sfunction,equation(4)can
2.1. MOND
be written in the form
As mentioned in 1, the shortage of luminous
§ a
matterinspiralgalaxiesandlarge-scalestructures µ(| |)a=g + h (5)
N
a ∇×
hasmotivatedsomeastrophysiciststoapplyalter- 0
native theories. One possible approach to solve where the first term on the right-hand side cor-
thedarkmatterproblemisachangeinNewtonian responds to the Newtonian gravitationalaccelera-
dynamics at small accelerations, so-called modi- tionandresultsfromthedistributionofthematter
fied Newtonian dynamics(Milgrom 1983). In this density. Since, withrespecttothe right-handside
model, Newton’s second law is altered to read of equation (5), ( h)=0, we can conclude
∇· ∇×
fromGauss’stheoremthatonaboundaryatinfin-
a
µ(| |)a= ΦN, (1) ity, h falls faster than 1/r2, in other words,it
a0 −∇ ∇×
canbe ignoredincomparisonwith the Newtonian
where ΦN is the Newtonian gravitational poten- acceleration (Bekenstein and Milgrom 1984). For
tial, a is the acceleration, a0 is an acceleration the interpolating function µ(x) given in equation
scale below whichthe dynamics deviates from the (2), we can obtain a in terms of gN as follows:
standardNewtonianform,andµ(x) isaninterpo-
lating function that runs smoothly from µ(x)=x 1 1 2a 1/2
at x ≪ 1 to µ(x) = 1 at x ≫ 1 (i.e., the latter a=gN(cid:20)2 + 2r1+(aN0)2(cid:21) . (6)
conditionrecoversNewtoniandynamicsatacceler-
ationsthatarelargecomparedwitha ). Anysuc- Here we model the Galactic disk as a simple, in-
0
cessfulunderlyingtheoryfor MONDmustpredict finitesimally thin Kuzmin disk (Kuzmin 1956),
an interpolating function. The functional form of for which the corresponding Newtonian potential
µ(x) and the value of free parameters such as a (Binney & Tremaine 1987) is given by
0
are usually chosen phenomenologically. The stan-
GM
dardinterpolatingfunction usedto fit with obser- φ (R,z)= − , (7)
N
vations was introduced by Bekenstein & Milgrom R2+(a+ z )2
| |
(1984): p
x whereaisthedisklengthscaleandM isthemass
µ(x)= . (2)
√1+x2 ofthedisk. Thequestioncanbeaskedwhetherwe
canexpressthe accelerationinthe MOND regime
Otherfunctionshavebeensuggestedforµ(x),such
in terms of the gradient of a scalar potential. To
asthoseof(Famaey & Binney 2005;Bekenstein 2004;
answer this question, we write the MONDian ac-
Zhao et al. 2006). The typical value for the pa-
celeration in terms of the Newtonian potential
rameter a , obtained from analysis of a sample of
0 from equation (5) as follows:
spiral galaxies with high-quality rotation curves
is a = 1.2 0.27 10−8cms−2 (Begeman et al.
0 ± × a= ϕNf( ϕN ). (8)
1991). Taking the divergence of both sides of ∇ |∇ |
equation (1) and substituting Poisson’s equation Theaccelerationiscompatiblewithascalarpoten-
in the right-hand side of equation (1) yields tialifthecurloftheright-handsideofequation(8)
a vanishes. The condition [ ϕNf( ϕN )]=0
(µ( )a)= 4πGρ. (3) ∇× ∇ |∇ |
impliesthat f( ϕ ) ϕ =0,sointhecase
∇· a − N N
0 ∇ |∇ | ×∇
3
of a Kuzmin disk, for which ϕ = ϕ2 /GM, scale, h is the scale height, and Σ is the column
|∇ N| N 0
we have f( ϕ ) ϕ , which means that density of the disk. The parameters of the disk
N N
∇ |∇ | ∝ ∇
a = 0. A technique for constructing more models are also indicated in Table 1. The Galac-
∇ ×
generaldensity-potentialpairsinMONDhasbeen tic bulge is consideredto be a point-like structure
introduced recently by Ciotti et al. (2006). with mass (1 3) 1010M⊙ (Zhao & Mao 1996).
− ×
2.2. CDM Halo Models 3. The Magellanic Stream
In this subsection, we review the various halo
The MS is a narrow band of neutral hydro-
models for the dark matter structure of Galaxy. gen clouds with a width of about 8◦ that extend
Generic classes of these models are axisymmet-
along a curved path, starting from the Magel-
ric ”power-law”and ”logarithmic” models for the
lanicClouds(MCs)andorientedtowardthesouth
dark halo of the MW (Evans 1993, 1994). The
Galactic pole (Van Kuilenburg 1972; Wannier &
gravitational potentials of the power-law and log-
Wrixon 1972; Mathewson et al. 1974). This
arithmic models are given by
structure comprises six discrete clouds, labeled
MS I to MS VI, lying along a great circle from
V2R β
Φ = a c (9) (l = 91◦,b = 40◦) to (l = 299◦,b = 70◦).
G β(R 2+R2+z2q−2)β/2 − −
c The radial velocity and column density of this
structure have been measured by many observa-
and
tional groups. Observations show that the ra-
Φ = 1V 2log(R 2+R2+z2q−2) (10) dial velocity of the MS with respect to the Galac-
G −2 a c tic center changes monotonically from 0kms−1
at MS I to 200kms−1 at MS VI, where MS
respectively. Poisson’s equation gives the density −
I is located nearest to the center of the MCs
distribution of the halo in cylindrical coordinates
and the MW, with a distance of 48 kpc, and
as ∼
MS VI is at the farthest distance. Another fea-
V 2R β ture of the MS is that the mean column density
a c
ρ(R,z)= (11)
4πGq2 × fallsfromMSI toMSIV (Mathewson et al.1977;
Mathewson et al. 1987 ; Bru¨ns et al. 2005).
R 2(1+2q2)+R2(1 βq2)+z2[2 (1+β)/q2]
c
− − , The MS is supposed to be an outcome of the
(R 2+R2+z2/q2)(β+4)/2
c interaction between the MCs and the MW, where
where the case β = 0 corresponds to a logarith- the gravitational force of the MW governs the
mic model R is the core radius, and q is the dynamics of the MS (Mathewson et al. 1987 ;
c
flattening parameter (i.e., the axial ratio of the Westerlund 1990;Fujimoto & Murai 1984;Wayte 1989).
concentricequipotential). Theparameterβ deter- There are several models for the origin and
mines whether the rotation curve asymptotically existence of the MS, such as tidal stripping
rises(β <0),asymptoticallyfalls(β >0),orisflat (Lin & Lynden-Bell 1982;Gardiner & Noguchi 1996;
(β = 0). At asymptotically large distances from Weinberg 2000),rampressure(Moore 1994;Sofue 1994)
the center of the Galaxy in the equatorial plane, and continued ram pressure stripping (Liu 1992).
therotationvelocityisgivenbyV R−β. The In the diffuse ram pressure model, there is a dif-
circ
∼
parameters of our halo models are given in Table fuse halo aroundthe Galaxythatproduces adrag
1. on the gas between the MCs and causes weakly
boundmaterialtoescapefromtheregiontoforma
For the Galactic disk, we use for the density of
tail. The discrete ram pressure models are based
the disk the double exponential function
on collision and mixing of the density enhance-
Σ R R z ments of the halo with the gas between the MCs
0 0
ρ(R,z)= exp − exp | | (12)
2h (cid:20)− R (cid:21) (cid:20)− h (cid:21) to form the MS, and in the scenario of continued
d
ram pressure stripping (Liu 1992), the MS is in a
(Binney & Tremaine 1987), where z and R are state of quasi-steady flow (i.e., ∂v = 0). In the
∂t
cylindrical coordinates, R0 is the distance of the tidal model, the condition for quasi-steady flow
Sun from the center of the MW, R is the length
d
4
Model : S A B C D E F G
Medium Medium Large Small Elliptical Maximal Thick Thick
Description disk disk halo halo halo disk disk disk
β – 0 -0.2 0.2 0 0 0 0
q – 1 1 1 0.71 1 1 1
v (kms−1) – 200 200 180 200 90 150 180
a
R (kpc) 5 5 5 5 5 20 25 20
c
R (kpc) 8.5 8.5 8.5 8.5 8.5 7 7.9 7.9
0
Σ0(M⊙pc−2) 50 50 50 50 50 80 40 40
R (kpc) 3.5 3.5 3.5 3.5 3.5 3.5 3 3
d
z (kpc) 0.3 0.3 0.3 0.3 0.3 0.3 1 1
d
Table1:ParametersoftheGalacticmodels. Thefirstlineindicatesthedescriptionofthemodels;thesecond
line, the slope of the rotation curve (β = 0 flat, β < 0 rising and β > 0 falling); the third line, the halo
flattening (q =1 representspherical); the fourth line, (v ), the normalizationvelocity; the fifth line, R , the
a c
halo core; the sixth line, the distance of the Sun from the center of the Galaxy; the seventh line, the local
column density of the disk (Σ =50 for canonical disk, Σ =80 for a maximal thin disk and Σ =40 for a
0 0 0
thick disk); the eighth line, the disk length scale; and the ninth line, the disk scale hight.
may be fulfilled by means of the tidal force of the the Galaxy, which has a halo of size R and
G
Galaxy,which strips the HI clouds fromthe MCs. mass M . The acceleration at the center of this
G
The Stream material then follows the same orbit structure is GM /R2, while the acceleration at
G G
astheMCs,suchthatthe localvelocityoftheMS a distance r from its center toward the Galaxy
remains constant. is GM /(R r)2, for R >> r, the differ-
G G G
−
Recently,Bru¨nset al. (2005)reportedonhigh- ence between these will be gG = 2GMGr/RG3.
resolution observations of the radial velocity and On the other hand, the acceleration imposed to-
column density of the MS. In the next section,we ward the structure is gs = GMs/r2. So, mate-
use these MS data to compare with the expected rial will move away in the direction of the Galaxy
dynamics from the MOND and CDM scenarios. if gG > gS, which implies a stability radius of
Comparingwiththeobservationswillenableusto rtidal =RG(2MMGs)1/3 (von Hoerner 1957). Forthe
place constraints on the parameters of the model. deep MOND regime, µ(x) x and the MONDian
≃
and Newtonian accelerations are related through
4. Dynamics of The Magellanic Clouds a (a0gN)1/2. The tidal acceleration, similarly
≃
to the Newtonian case, obtains as the difference
In this section, we obtain the dynamics of the between the accelerationin the MCs and ata dis-
MCs in MONDian theory, considering for the tance r from them as
structure of the Galaxy an infinitesimally thin g = GM a 1/2r/R2. On the other hand,
G G 0 G
Kuzmin disk without a dark halo component. In
we have the gravitational acceleration toward the
this model, since we have ignored the Galactic MCsatthedistancer,whichisg =GM a 1/2/r,
s s 0
halo, there is no drag force from the halo on the
and so for the tidal radius we obtain r =
tidal
MS, and the main factor in the stripping of the R (MS)1/4. If we take the same mass-to-light ra-
MS is the tidal forceexertedby the Galactic disk. g MG
tio for the Galaxy as for the Clouds, we can say
Using the gradient of the gravitational force im-
thatthetidalradiusinthe MONDianregimeisof
posed by the Galaxy on the MCs, we can make
the same order as that in the dark matter model.
an estimate of the tidal radius, the distance from
Let the ratio of the luminous matter of the MCs
theMCsbeyondwhichallthematerialcanescape to that in the Galactic disk be of order 10−2; for
from the structure.
R 50kpc, we obtain in the MONDian regime
G
≃
In the Newtonian case, we consider a struc- a tidal radius of about 15kpc,versus 13kpc in
≃ ≃
ture with mass Ms rotating in an orbit around the dark matter model.
5
Model S A B C D E F G MOND MOND (M. L.)
χ2 526 816 2285 437 576 1511 562 315 810 356
Table 2: Minimum χ2 from fitting with various Galactic models. The Cold Dark matter halo models are
labeled with of S, A, ... G with the corresponding values for the parameters of the models, indicated in
Table 1. The last column corresponds to the minimum χ2 from the Maximum Likelihood analysis in the
MOND model.
Model M(M⊙) a b c v0 Rc q Mb(M⊙) a0(ms 2) χ2
−
MOND 1.2 1011 4.5 - - - - - - 1.2 10−10 356
× ×
CDM 1.2 1011 4.5 - - 175 13 1 - - 343
×
f095CDM 1.2 1011 4.5 - - 175 13 0.95 - - 349
×
f09CDM 1.2 1011 4.5 - - 175 13 0.9 - - 361
×
f1.25CDM 1.2 1011 4.5 - - 175 13 1.25 - - 341
×
L05 1 1011 6.5 0.26 0.7 171 13 0.9 3.53 1010 - 362
× ×
Table3:ComparisonofMSdatawiththe MONDandCDMGalactichalomodels. Thefirstlineistheusual
MOND model, the second to the fifth lines correspond to the CDM models (Johnston et al. 2005 and Law
et al. 2005) The parametersa, b,c andR arein kpc and v is in terms ofkm/s. See Readand BenMoore
c 0
(2005) for more details of the models.
According to the quasi-steady flow model, the drically symmetric distribution of matter around
MSwilllieinthesameorbitalpathastheMCs,so the z-axis. To solve the equation of motion, we
the location and velocity of the MS can represent take the location and velocity of MS I as the
the dynamics of the MCs. Here we are looking to initial conditions for the dynamics of the MCs.
the radial velocity of the MS toward the Galactic Since MS I is located closest to the center of
centertocomparewithobservations. Asaresultof Galaxy, it almost has only a transverse veloc-
hydrodynamic friction, the MCs could have their ity component with respect to the Galactic cen-
motion damped and fall toward the Galaxy in a ter. Although, there are no indicators at MS I
spiral path. However, since in this model there is by which to measure its transverse velocity, we
no halo around the Galaxy, we can ignore friction can nevertheless use the transverse velocity of
forces. Substituting the Newtonian potential of the MCs measured from the kinematics of plan-
theKuzmindisk(eq. [7]),theMONDianpotential etary nebulae in the LMC, v = 281 41kms−1
θ
±
in cylindrical coordinates is obtained (van der Marel et al. 2002) as an estimate of the
transverse velocity of MS I. We utilize the con-
√GMa
φ = 0 ln(R2+(a+ z )2), (13) servation of angular momentum for the dynamics
M
2 | |
of the MCs (i.e., r v =r v ), which
MCs MCs MSI MSI
as(Read & Moore 2005;Brada & Milgrom 1995). provides anMS I transversevelocity on the order
For convenience, we carry out a coordinate trans- of v (r ) = 320 50kms−1.Here the distance of
θ 0
±
formation from the cylindrical to the spherical MS I has been taken to be r =48 1kpc.
0
±
frame and obtain the components of the accelera- Inwhatfollows,wecomparethedynamicsofthe
tion as follows: MCs with observation. Figure 1 shows the ob-
acosθ+r served radial velocity of the MS in terms of the
a = MGa ( ) (14)
r − 0 2arcosθ+r2+a2 angular separation of from the MCs in a frame
p
asinθ centered on the Galaxy. Since we are interested
aθ = − MGa0(2arcosθ+r2+a2),(15) in comparing the global motion of the MS with
p the model, the internal dynamics of the structure
a = 0, (16)
φ
has been ignored. The radial velocity of the MS
where a is zero as a consequence of the cylin-
φ
6
250 A
B
C
DEF 10.5 0.6 0.7 0.8 0.9 11
200 G
S
MOND 0.9 0.9
Km/s) 0.8 0.8
(150
Radialvelocity100 Likelihood000...567 000...567
e
v
50 ati0.4 0.4
el
R0.3 0.3
0 0.2 0.2
0 0.5 1
Angulardistance(radian)
0.1 0.1
0 0
0.5 0.6 0.7 0.8 0.9 1
Fig. 1.— Radial velocity v vs. angular dis- M a
r 0 0
tance θ along the MS in the center of Galacto-
centric frame. The observational data (Bru¨ns et
al. 2005) are compared with theoretical curves 10 0.25 0.5 0.75 11
obtained from MONDian and CDM models. The
0.9 0.9
minimumχ2 correspondsingtoeachcurveisgiven
inTable2.[SeetheelectroniceditionoftheJournal 0.8 0.8
for a color version of this figure.] d0.7 0.7
o
o
elih0.6 0.6
k
Li0.5 0.5
is calculated by averaging over the data at each ve
point of the Stream and the error bars on the ra- elati0.4 0.4
dial velocity result from the dispersion velocity of R0.3 0.3
the structure. This velocity dispersion is caused 0.2 0.2
by substructural motions of the MS such as the
0.1 0.1
stochastic behavior of gas, and for different real-
0 0
izations of the MS, the dispersion velocity will be 0 0.25 0.5 0.75 1
a
different at each point. The observed data are a
continuumdistributionofradialvelocitiesinterms
of angular separation; however, we collected data Fig. 2.— Marginalized likelihood functions of the
for each cluster of gas and obtained the average parameters of the MOND and Kuzmin disk. Top,
velocity and the corresponding dispersion veloc- product of the disk mass M0 (1011M⊙) and the
ity. We use χ2 fitting, MOND acceleration scale a0 (10−10ms−2); bot-
tom, length scale of the Kuzmin disk. The in-
N Vi Vi tersection of the curves with the horizontal solid
χ2 = ( th− obs)2, (17) and dashed lines give the bounds at 1σ and 2σ
σ
Xi i confidence levels, respectively. [See the electronic
edition of the Journal for a color version of this
for comparison of the observed data and dynam-
figure.]
ics of the MS from the MOND and CDM models
(see Fig. 1). We should take into account that
the CDM and MOND models in these cases have
different mass distributions.
7
15
14
13
12
)UN11
MS
10010
1
M(0 9
8
7
6
5
0.8 1.2 1.6 2
a (10-10ms-2)
0
Fig. 4.— ConstraintsonM a =8.4+0.5 fromthe
Fig. 3.— Joint confidence intervals for 0 0 −0.2
maximum likelihood analysis, with 1σ confidence
sMca0lae0(a1(0k1p1Mc),⊙w×it1h0−11σ0m(ss−o2li)danlidnet)h,e2dσisk(dleanshgethd interval (M0 in terms of 1010M⊙ and MONDian
accelerationscale a in terms of 10−10ms−2). Us-
line), and 3σ long-dashed line confidence levels. 0
ing the range a = 0.8 1.4 implies M = 6 11
The minimum value of χ2 corresponds to a = 0 − 0 −
forthemassoftheKuzmindisk.[Seetheelectronic
0.2+0.45kpc and M a = 0.84+0.05. [See the elec-
−0.1 0 0 −0.02 edition of the Journal for a color version of this
tronic edition of the Journal for a color version of
figure.]
this figure.]
As can be seen from the results of the maximum
ThedarkmattermodelsarelabeledS,A,B,C,
likelihood analysis, in MOND the length scale of
D,E,F,andGwiththeircorrespondingparame-
the Kuzmin disk favor small values, in the range
ters in Table 1. The minimum χ2 values from fit-
a<1kpc with a wide error bar. We can interpret
ting with the CDM and MOND models are listed
thisasaconsequenceoftheMSbeinglocatedata
in Table 2. We performed a maximum likelihood
large distance from the Galaxy compared with a,
analysis to find the best parameters for MOND.
which means that the dynamics of the MS is not
Comparing the minimum value of χ2 from the
sensitive to this length scale. M a is also con-
modified dynamics with those of the dark mat- 0 0
fined to a range of about 0.8 to 0.9. In order to
ter models shows that MOND results in a bet-
breakthedegeneracybetweenthemassofthedisk
ter fit than the CDM models, except for model
and the acceleration scale of MOND (a ), we use
G. The best parameters for a MONDian poten- 0
theluminousmassoftheMWtoconstrainthepa-
tialandthe characteristicaccelerationparameter,
rameter a . The total luminosity of the Galactic
from the maximum likelihood analysis, are found 0
to be M0a0 = 0.84+−00..0052 and a = 0.2+−00..415kpc, danisdkiifsw1e.2a×ssu1m01e0Lan⊙a(vBeirnangeeyst&ellTarremmaasins-eto1-9li8g7h)t,
where M is the mass of the disk in terms of
0 ratioofΥ=5,the totalstellarmassofthe Galac-
1te0r1m1Ms ⊙of, 1a00−1is0mths−e2MaOndNDa iasctcheelerleantigotnhssccaallee ionf tic disk should be about M∗ ≃6×1010M⊙; using
the constraint M a 0.84 then results in a
the Kuzmindisk. The marginalizedrelativelikeli- 0 0 ≃ 0 ≃
1.4. Comparing the value a = 1.2 0.27 from
hood functions for M0a0 and a are shown in Fig- analysis of the rotation curv0es of sp±iral galaxies
ure2,Figure3showsthejointconfidenceintervals
(Begeman et al. 1991), with that from our analy-
for M a and a with 1σ to 3σ confidence levels.
0 0
8
50
250 CDM
f09CDM
f095CDM
L05
MOND
40
200
1010M)SUN30 velocity(Km/s)150
M(020 Radial100
10 50
0
0 0.5 1
0.5 1 a (10-10ms-2)1.5 2 Angulardistance(radian)
0
Fig. 6.— Same as Fig. 1, but for recent CDM
Fig. 5.— Constraints on M a considering 2σ er-
0 0
models of the Galactic halo form studies of the
ror bars on the initial value for the transverse ve-
tidaldebrisfromtheSagittariusdwarfgalaxy(see
locityofMSI. UsingthepreferredvalueM∗ 6
1010M⊙ fortheGalacticdiskyieldsa0 =0.8≃1.×4; Table3). [See the electronic edition of the Journal
− for a color version of this figure.]
alternatively,adoptingaMONDaccelerationscale
in the range a =0.8 1.4 implies a disk mass in
0
−
the rangeM0 =(2.5 23) 1010M⊙.[See the elec-
− ×
tronic edition of the Journal for a color version of 0.84+1.01 whenconsideringthisuncertaintyonthe
−0.53
this figure.] initial value for MS I’s transverse velocity. Us-
ing the preferred value of M∗ 6 1010M⊙ for
≃ ×
theGalacticdiskconfinestheaccelerationscaleto
the range a = 0.8 1.4. Alternatively, consider-
sis of the MS, we see good agreement. If, in addi- 0 −
ing the literature acceleration scale for MOND of
tiontothecontributionofthedisktotheGalactic
a = 0.8 = 1.4 provides a Galactic disk mass of
mass, we add the lower and upper limits for the 0
mass of the Galactic bulge, the total mass of the M0 =2.5−23×1010M⊙,alargerrangethanthat
Galaxy lies in the range (4.3 12.8) 1010M⊙ of our previous analysis. For the standard value
− × of a = 1.2, the mass of the Galactic disk is con-
(Sackett 1997; Zhao et al. 1995), which confines 0
the value of a0 to 0.65 1.95. In addition, for the fined to M0 =3−15×1010M⊙. Higher accuracy
− observationsofthe transversevelocityofthe MCs
range of the MOND acceleration scale adopted in
willprovideabetter constraintonthe parameters
the literature, a =0.8 1.4,the mass of the disk
0
is found to be M0 = (6− 11) 1010M⊙ (see Fig. of the model.
− ×
4). Finally, we compare the dynamics of the MS
in MOND with recentCDM Galactic halo models
In the likelihood analysis here, we used as an
proposed to interpret the dynamics of the Sagit-
initial condition the central value of the velocity
tarius dwarf galaxy around the MW (Read &
of MS I without taking into account the corre-
Moore 2005). Figure 6 compares the observed ra-
sponding error bar. In order to check the sen-
dialvelocityoftheMSwiththeseGalacticmodels,
sitivity of our results to the uncertainty in the
and Table 3 indicates the best-fit value of χ2 for
velocity of the MCs, we repeated the likelihood
each model and shows that the MOND result is
analysis for an MS I transverse velocity in the
almost compatible with the CDM models.
range v = 320 100 with 2σ error bars. Fig-
MSI
±
ure 5 shows that M a is constrained to M a =
0 0 0 0
9
5. Conclusion manuscript and providing very useful comments,
and M. Nouri-Zonoz, M. A. Jalali, and M. S.
Inthis work,we usedrecentobservationaldata
Movahed for improving the text. We would also
on the radial velocity of the Magellanic Stream
like to thank an anonymous referee for very use-
to examine modified Newtonian dynamics. The
ful comments. S.R. thanks Alain Omont and the
MS is considered to be a trace of the Magel-
Institut d’astrophysique de Paris, where a part of
lanic Clouds produced by the interaction of the
this work was done, for their hospitality.
Clouds with the Galaxy. We considered the
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10