M31 Transverse Velocity and Local Group Mass from Satellite Kinematics

0.43 MB·English

by  Roeland P. van der Marel

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
M31 TRANSVERSE VELOCITY AND LOCAL GROUP MASS FROM SATELLITE KINEMATICS
Roeland P. van der Marel
SpaceTelescopeScienceInstitute, 3700SanMartinDrive,Baltimore,MD21218
Puragra Guhathakurta
UCO/LickObservatory,DepartmentofAstronomyandAstrophysics,UniversityofCaliforniaatSantaCruz,1156HighStreet,Santa
Cruz,CA95064
Draft versionFebruary 2, 2008
8
ABSTRACT
0
0 We present severaldifferent statistical methods to determine the transversevelocity vector of M31.
2 The underlying assumptions are that the M31satellites on averagefollow the motionof M31 through
space,andthatthe galaxiesinthe outerpartsofthe LocalGrouponaveragefollowthe motionofthe
n
LocalGroupbarycenterthroughspace. Weapplythemethodstotheline-of-sightvelocitiesof17M31
a
satellites, to the proper motions of the 2 satellites M33 and IC 10, and to the line-of-sight velocities
J
of 5 galaxies near the Local Group turn-around radius, respectively. This yields 4 independent but
3
mutually consistent determinations of the heliocentric M31 transverse velocities in the West and
2
North directions, with weighted averageshv i=−78±41kms−1 and hv i=−38±34kms−1. The
W N
uncertainties correspond to proper motions of ∼ 10µasyr−1, which is unlikely to be within reach of
]
h directobservationalverificationwithinthenextdecade. TheGalactocentrictangentialvelocityofM31
p is42kms−1,with1σconfidenceintervalV ≤56kms−1. TheimpliedM31–MilkyWayorbitisbound
tan
o- if the total Local Group mass M exceeds 1.72+−00..2265×1012M⊙. If the orbit is indeed bound, then the
r timingargumentcombinedwiththeknownageoftheUniverseimpliesthatM =5.58+−00..8752×1012M⊙.
t This is on the high end of the allowed mass range suggested by cosmologically motivated models for
s
a theindividualstructureanddynamicsofM31andtheMilkyWay,respectively. Itisthereforepossible
[ that the timing mass is an overestimate of the true mass, especially if one takes into account recent

results from the Millennium Simulation that show that there is also a theoretical uncertainty of 41%
2
(Gaussian dispersion) in timing mass estimates. The M31 transverse velocity implies that M33 is in
v
7 a tightly bound orbit around M31. This may have led to some tidal deformation of M33. It will be
4 worthwhile to search for observational evidence of this.
7 Subject headings: galaxies: kinematics and dynamics — Local Group — M31.
3
.
9 1. INTRODUCTION et al. 2005; hereafter L05)?
0
In the present paper we show that it is possible to ob-
7 The Local Group is dominated by two spiral galax-
tainastatisticaldeterminationofthetransversevelocity
0 ies, M31 and the Milky Way. These galaxies have com-
ofM31usingtheobservedvelocitiesofitssatellites. The
: parable properties, with M31 generally believed to be
v analysisassumesthatonaveragethe satellitesfollowthe
slightly more massive (e.g., Klypin, Zhao & Somerville
i motion of M31 through space, with some velocity dis-
X 2002). The next most luminous galaxies, M33 and the
persion. The ensemble ofline-of-sightvelocities,and the
Large Magellanic Cloud, are some 10 times fainter (e.g.,
r individualpropermotionsavailableforselectedgalaxies,
a van den Bergh 2000). The dynamics and future of the
thenyieldindependentestimatesoftheM31velocity. We
Local Group are therefore determined primarily by the
also revisit the method previously explored by Einasto
relative velocity of M31 with respect to the Milky Way.
&Lynden-Bell(1982),whichisbasedontheassumption
Unfortunately, this velocity is poorly known. The line-
that the galaxies in the outer parts of the Local Group
of-sightvelocityofM31canbemeasuredextremelyaccu-
followthemotionoftheLocalGroupbarycenterthrough
rately using the Doppler shift of a large variety of trac-
space. We apply the different methods to the currently
ers. However, even after a century of careful attempts
availabledatafortherelevantLocalGroupgalaxies,and
(starting with, e.g., Barnard1917) still no useful proper
we combine the results to obtain an accurate determi-
motion measurement exists to constrain the transverse
nation of the M31 transverse velocity. We then use this
velocity. This limits our ability to answer severalfunda-
determination to address the aforementioned questions.
mental questions. For example, do M31 and the Milky
The structure of the paper is as follows. Section 2
Way indeed form a bound system, as is usually assumed
discusses the constraints onthe M31 transversevelocity,
(e.g., van den Bergh 1971)? What is the exact mass of
based on: the line-of-sight velocities of an ensemble of
the Local Group implied by the so-called timing argu-
17 M31 satellites (Section 2.1); the recent high accuracy
ment (e.g., Kahn & Woltjer 1959; Kroeker & Carlberg
proper motion determinations of the M31 satellites M33
1991; Lynden-Bell 1999)? What is the expected future
and IC 10 by Brunthaler et al. (2005, 2007) from VLBI
evolution of the M31–Milky Way system (e.g., Cox &
observations of water masers (Section 2.2); and the line-
Loeb 2007)? And how has the structure of M33 been
of-sightvelocities ofan ensemble of galaxiesin the outer
influenced by possible interaction with M31 (e.g., Loeb
parts of the Local group (Section 2.3). The different
2 van der Marel & Guhathakurta
constraints are compared and combined in Section 2.4.
Section 3 discusses the implications of the inferred M31 TABLE 1
M31SatelliteGalaxySample
velocityfor the relativeorbitofM31 andthe Milky Way
(Section 3.1) and for the relative orbit of M31 and M33
(Section 3.2). Section4 discusses how the results for the Name Type dρeg dΦeg kmvloss−1
M31 transverse velocity (Section 4.1), the total mass of (1) (2) (3) (4) (5)
theLocalgroupasimpliedbythetimingargument(Sec-
M31 SbI-II 0.00 ... -301±1
tion 4.2), and the Local Group turn-aroundradius (Sec- M32 E2 0.40 -179.10 -205±3
tion 4.3) compare with theoretical predictions and other NGC205 dSph 0.60 -46.61 -244±3
observationalstudies. Section5 presentsthe conclusions AndIX dSph 2.69 43.32 -211±3
AndI dSph 3.27 169.84 -380±2
of our work.
AndIII dSph 4.97 -163.13 -355±10
In the analysis below we use several coordinate sys- AndX dSph 5.62 48.96 -164±3
tems. Observational systems have their three principal NGC185 dSph/dE3 7.09 -5.07 -202±7
axes aligned with the line-of-sight, West, and North di- NGC147 dSph/dE5 7.43 -12.29 -193±3
AndV dSph 8.03 35.31 -403±4
rections, respectively, for a given position on the sky.
AndII dSph 10.31 136.82 -188±3
WealsouseaGalactocentriccoordinatesystemcentered M33 ScII-III 14.78 131.78 -180±1
on the Milky Way and a barycentric coordinate system AndVII/Cas dSph 16.17 -47.93 -307±2
centered on the Local Group barycenter. The analysis IC10 dIrr 18.37 -9.11 -344±5
AndVI/Peg dSph 19.76 -143.64 -354±3
requires transformations between the positions and ve-
LGS3/Pisces dIrr/dSph 19.89 165.44 -286±4
locities in these systems. Many of the necessary nota- Pegasus dIrr/dSph 31.01 -143.37 -182±2
tions and derivations can be found in van der Marel et IC1613 dIrrV 39.44 171.30 -232±5
al. (2002; which presents a study of the kinematics of
the Large Magellanic Cloud), and are given here with- Note. —Thesample ofM31 satellites usedinthemodel-
out further reference. Heliocentric velocities are gener- ingofSection2.1. M31itselfislistedonthefirstlineforcom-
parison. Column(1)liststhegalaxynameandcolumn(2)its
ally denoted with a vector ~v, Galactocentric velocities
type. Columns (3) and (4) define the position on the sky: ρ
with a vector V~, and Local-Group barycentric velocities
istheangulardistancefrom M31 andΦistheposition angle
with a vector U~. Velocities and proper motions in the with respect to M31 measured from North over East. These
observationalsystemsaregenerallyheliocentric(i.e.,not angles were calculated from the sky positions (RA,DEC) as
correctedforthe reflexmotionofthe Sun), unlessstated in vanderMarel et al. (2002). Thesatellites in thetableare
otherwise. sorted by their value of ρ. Column (5) lists the heliocentric
line-of-sight velocity and its error. Velocities and sky posi-
2. THE M31 TRANSVERSE VELOCITY tions (RA,DEC) for most satellites were obtained from the
compilation of Evans et al. (2000), except for the satellites
2.1. Constraints from Line-of-Sight Velocities of M31 And IX and X which had not yet been discovered in 2000.
Satellites Forthoseweusedthevelocitymeasurementsand(RA,DEC)
given by Chapman et al. (2005) and Kalirai et al. (2007),
The velocity vector of an M31 satellite galaxy can be
respectively.
written as the sum of the M31 velocity vector, and a
peculiar velocity
Marel&Cioni2001;vanderMareletal.2002). Because
~v =~v +~v . (1)
sat M31 pec satellitesarenotlocatedonthesamepositiononthesky
We assume that any one-dimensional component of~vpec as M31, the vector ~vM31 has a different decomposition
is a random Gaussian deviate with dispersion σ. More in line-of-sight and transverse components than it does
formally, this is true if the velocity distribution of the for M31. More specifically, the line-of-sight velocity of a
satellites is both isotropic and isothermal. The first satellite is
assumption finds some support from studies of galaxy
v =v cosρ+v sinρcos(Φ−Θ )+v . (3)
satellite systems (e.g., Kochanek 1996) and clusters of sat,los sys t t pec,los
galaxies (e.g., van der Marel et al. 2000). The second The factor cosρ in the first term indicates that only a
assumption is reasonable in view of the fact that the fractionofv isseenalongtheline-of-sight. Thesecond
sys
gravitational potentials of dark halos are approximately termisanapparentsolidbodyrotationcomponentinthe
logarithmic. Either way, in the present analysis there is v velocity field on the sky, with amplitude v sinρ
sat,los t
only a very weak dependence on the accuracy of these and kinematic major axis along position angle Θ . The
t
assumptions(bycontrasttostudiesofthemassdistribu- last term merely adds a scatter σ on top of the velocity
tion of M31; e.g., Evans et al. 2000). field defined by the first two terms.
ThevelocityvectorofM31canbedescribedbyquanti- Itfollowsfromequation(3)thatthetransversevelocity
tiesv ,v andΘ ,wherev istheline-of-sightvelocity, of M31 affects the line-of-sight velocities of its satellites.
sys t t sys
v is the transversevelocity, andΘ is the position angle Therefore, a study of the line-of-sight velocities of the
t t
ofthetransversemotiononthesky. Thevelocitiesinthe satellites can constrain the M31 transverse velocity. To
directions of West and North are this end we compiled the sample in Table 1, which con-
v ≡v cos(Θ +90◦), v ≡v sin(Θ +90◦). (2) sistsofM31satelliteswithknownvelocities. Thesample
W t t N t t isbasedonthatusedbyEvansetal.(2000),butwithad-
Thepositionofasatelliteontheskycanbedescribedby ditionof the more recently discoveredAndromeda dwarf
the angles (ρ,Φ), where ρ is the angular distance from satellites And IX and X. The satellites And XI and XIII
M31 and Φ is the position angle with respect to M31 (Martin et al. 2006, who also reports the finding of an
measured from North over East (as defined in van der unusuallydistantglobularcluster)andAndXVandXVI
M31 Transverse Velocity and Local Group Mass 3
(Ibata et al. 2007) are not included because no line-of- 0
sight velocity measurements have yet been reported for
them. We fitted equation(3) to these data by determin-
ing the values of (v ,v ,v ) that minimize the scat-
sys W N
ter in v . The resulting scatter is the dispersion σ
pec,los
-200
of the satellite population. After the best-fitting model
was identified, we calculated error bars on the fitted pa-
rametersusing Monte-Carlosimulations. Many different
pseudodatasetswerecreatedwiththe samesatellitesat
the same positions, but with velocities drawn from the
-400
best-fitmodel,withv drawnasarandomGaussian
pec,los
deviate with dispersion σ. The pseudo data sets were
then analyzedsimilarly as the realdata set. The disper-
sions in the inferred model parameters are a measure of
their formal 1-σ error bars.
-600
Themodelingprocedureyieldsvsys =−270±19kms−1, 0 100 200 300
v = −136± 148kms−1, v = −5±75kms−1, and
W N
σ = 76±13kms−1 (see also Table 3 below). The data
and representative predictions of the best-fitting model Fig. 1.— Comparison of heliocentric line-of-sightvelocity data
ofM31satellites andthepredictionsofequation (3),forthebest-
are shown in Figure 1. Color-coding indicates the dis-
fit values of the heliocentric M31 transverse velocity (vW,vN) =
tance from M31. The inferred transverse velocity cor- (−136,−5)kms−1. Thecurvesshowthepredictionsasfunctionof
responds to sinusoidal variations that are less than the positionangle Φ, forangular distances fromM31of ρ=3◦ (red),
observedscatterinthedata. Nosinusoidalvariationwith ρ = 9◦ (cyan), ρ = 15◦ (green), ρ = 29◦ (black). Data points
are from Table 1. They are color coded based on bins in angular
angle is discernible in the data by eye; indeed, the case distanceρ,from0–6◦ (red),6–12◦ (cyan),12–18◦ (green),and18–
of zero transverse velocity (amplitude zero for the sinu- 40◦ (black). Predictions with zero transverse velocity (amplitude
soidal variations) is also statistically consistent with the zero for the sinusoidal variations) are also statistically consistent
data. However,thisisnotanullresult. Largetransverse with the data. However, a large transverse velocity for M31 of
velocitiesofhundredsofkms−1wouldhaveinducedlarge hundredsofkms−1(correspondingtosinusoidalvariationsofmuch
largeramplitudethanshowhere)areruledout.
sinusoidalvariations that are not seen in the data. Such
transverse velocities are therefore ruled out. We note 2.2. Constraints from Proper Motions of M31
that we could also have kept v fixed in the fit at the Satellites
sys
observed value of v = −301±1kms−1 (Courteau &
sys Water masers can be observed at high spatial reso-
van den Bergh 1999). We verified that such a fit yields
lution with VLBI techniques. This makes them a valu-
verify similar results, namely v = −123±159kms−1
W abletoolforpropermotionstudiesofLocalGroupgalax-
and v =−33±79kms−1, which is well within the un-
N ies. Brunthaler et al. (2005, 2007) recently determined
certainties quoted above.
the proper motions for two galaxies in the M31 group,
Our analysis does not assume that the satellites are
namely M33 and IC 10. Unfortunately, no water masers
necessarily bound to M31. For M33, orbit calculations
have yet been found in M31 and, at present detection
dosuggestthatitprobablyisbound(seefigure3ofLoeb
limits, may be none should be expected (Brunthaler et
etal.2005),butthishasnotbeenestablishedwithconfi-
al. 2006). So the direct application of this technique to
dence for mostof the other satellites. We decided notto
M31maynotbe possibleintheforeseeablefuture. How-
include the recently discovered dSph galaxies And XII
ever, the measurements for M33 and IC 10 can be used
and And XIV in our sample. They lie at similar posi-
to constrain the transverse velocity of M31 indirectly.
tion angles, Φ = 171.94◦ and 170.49◦, respectively, and
Equation (1) implies that the unknown velocity vector
at distances ρ = 6.96◦ and 11.71◦, that are not atypi-
ofM31 canbe estimated fromthe knownvelocityvector
cal for the rest of our sample. However, their velocities
of a satellite as ~v = ~v −~v . Since the peculiar
of −556±5 and −478±5kms−1, respectively, are 3.3σ M31 sat pec
velocityis unknown,it actsas a Gaussianuncertaintyof
and2.3σ awayfromthe observedvelocityofM31. Ithas
size σ in each velocity component.
been suggested that they form a system that is falling
In analogy with equation (3), one can write for the
into M31 for the first time (Chapman et al. 2007; Ma-
transverse velocity components of the satellite
jewski et al. 2007), which would not make them useful
additions to our analysis. As a test we did repeat our vsat,2=−vsyssinρ+vtcosρcos(Φ−Θt)+vpec,2,
analysiswiththesegalaxiesincludedinthesample. This v =−v sin(Φ−Θ )+v . (4)
sat,3 t t pec,3
yieldedv =−296±27kms−1,v =−85±200kms−1,
sys W
v =41±108kms−1, and σ =106±18kms−1. This is Herethe unit vectors2and3onthe plane ofthe skyare
N related to the directions of West and North, all at the
consistent with the result for our main sample to within
position of the satellite, according to a rotation
the uncertainties. This illustrates that the results are
fairly robust against the inclusion or removal of individ- v −sinΓ −cosΓ v
ual galaxies. By contrast,modeling of the parentgalaxy (cid:18) vssaatt,,WN (cid:19)=(cid:18) cosΓ −sinΓ (cid:19) (cid:18)vssaatt,,23 (cid:19), (5)
massbasedonsatellitescanbequitesensitivetoassump-
tions about the bound state of individual galaxies (e.g., where the rotation angle Γ is determined by
Kochanek 1996).
cosΓ=[sinδcosδ cos(α−α )−cosδsinδ ]/sinρ,
0 0 0
sinΓ=[cosδ sin(α−α )]/sinρ. (6)
0 0
4 van der Marel & Guhathakurta
Here (α,δ) are the RA and DEC of the satellite, and tic Rotation. For the set of outer Local Group galaxies
(α ,δ ) are the RA and DEC of M31 (i.e., the position i=1,...,N we calculate the unit vector~r in the direc-
0 0 i
with ρ=0). tionofeachgalaxy. IfthevelocityvectoroftheSunwith
Given values of (vsat,los,vsat,W,vsat,N), the equa- respect to the Local Group barycenter is U~⊙, then one
tions(2)–(6)uniquelyconstrainthethreeunknowncom- has
ponents (vsys,vW,vN) of the M31 velocity vector. We (vlos,i+[U~⊙·~ri])~ri =0, (7)
solve these equations for each of the two satellites M33 Xi
and IC 10. We take v from Table 1. To obtain
sat,los
where v is the heliocentric line-of-sight velocity for
v and v we write each velocity component as los,i
sat,W sat,N
eachgalaxy. This canbe written asa 3×3matrix equa-
v = (0.0047404Dµ ) − δv , where µ is the ob-
served proper motioobns in µarostyr−1, D isobsthe satellite tion for the components of the vector U~⊙. The best-
distance in kpc, and δv is a correction for the in- fit values and their formal errors are easily obtained us-
rot
ternal rotation of the galaxy under study. We take ing standard techniques (Einasto & Lynden-Bell 1982).
D =794±23kpcforM33(McConnachieetal.2004)and Once this solution is obtained one can calculate the ve-
D =660±65kpcforIC10(Evansetal.2000). Theother locityofthe Milky Waywithrespectto the LocalGroup
quantities follow from Brunthaler et al. (2005, 2007): Barycenter, U~MW = U~⊙ − V~⊙. Here V~⊙ is the veloc-
(µ ,µ ) = (−4.7±3.2,−14.1±6.4)µasyr−1 for ity vector of the Sun in the Galactocentric rest frame.
obs,W obs,N
M33; (µ ,µ )=(−6.0±5.0,23.0±5.0)µasyr−1 For the circular velocity of the Local Standard of Rest
obs,W obs,N
for IC10; (δvrot,W,δvrot,N)=(70±23,−81±23)kms−1 (LSR) we use the standard IAU value V0 = 220kms−1
forM33;and(δv ,δv )=(−25±19,9±19)kms−1 (Kerr&Lynden-Bell1986),towhichweassignanuncer-
rot,W rot,N tainty 10kms−1 (none of our results depend sensitively
for IC 10. We do not include the Brunthaler et al. cor-
on this quantity). For the residual velocity of the Sun
rections for the reflex motion of the Sun, since we deal
with respect to the LSR we adopt the values of Dehnen
with that issue separately in Section 3.1. We add Gaus-
&Binney(1998). Ifweassumethatallofthemassofthe
sian random deviates in our calculations to reflect the
LocalGroupresidesintheMilkyWayandM31,thenthe
uncertainties. We take each component of ~v to be a
pec
Gaussianrandomdeviatewithdispersionσ =76kms−1, barycenter is simply the mass-weighted average of their
position vectors. This implies in the Galactocentric rest
as determined in Section 2.1. For each combination of
frame that the velocity of M31 is V~ = −U~ /f ,
(v ,v ,v ) we solve the equations to obtain M31 MW M31
sat,los sat,W sat,N
where f ≡ M /(M +M ). The heliocentric
(v ,v ,v ) and we repeat this in Monte-Carlo fash- M31 M31 M31 MW
sys W N
ion. We adopt the average and dispersion of the results velocity of M31 is therefore ~vM31 = V~M31 −V~⊙. After
as our final estimate for the M31 velocity vector and its substitution of the previous equations this yields
error. Using M33, we obtain the following estimates for
M31: v = −183±76kms−1, v = −48±80kms−1, ~vM31 =−U~⊙/fM31+V~⊙[(1/fM31)−1]. (8)
sys W
vN = 71±84kms−1. Using IC 10, we obtain the fol- This heliocentric vector can be decomposedinto compo-
lowing estimates for M31: vsys = −346 ± 76 kms−1, nents along the line-of-sight and in the West and North
v = −16±80kms−1, v =−47±81kms−1 (see also directionsfollowingthemethodologyofvanderMarelet
W N
Table 3 below). al. (2002).
Application of equation (8) requires that we assume
2.3. Constraints from Line-of-Sight Velocities of Outer a value for f . Einasto & Lynden-Bell (1982) used
M31
Local Group Galaxies the Tully-Fisher relation to constrain this quantity, and
built this constraint directly into their matrix solution
The Local Group contains not only the virialized sub-
groupsof galaxiessurrounding the Milky Way and M31, for U~⊙. However, the mass ratio of M31 and the Milky
but also a number of unattached galaxies that populate Way isn’t actually all that well known observationally,
the outer regions of the Local Group. On average,these and different arguments for estimating it have yielded
galaxies are expected to follow the motion of the Local different results. Our aim here is to constrain~vM31 from
Groupbarycenterthroughspace. Theirheliocentricline- observationaldata,whileimprintingaminimumamount
of-sightvelocityvectors,averagedinathree-dimensional of theoretical prejudice into the result. So we simply
sense,thereforestatisticallyequalthereflexmotionofthe assume that fM31 is homogeneously distributed between
Sun with respect to the Local Group barycenter. Since 0.39and0.67(i.e.,MM31/MMW =0.8–2.0). Thisencom-
the velocity of the barycenter is itself determined by the passes most of the values that have been quoted in the
relative velocity of M31 with respect to the Milky Way, literature (see discussion in Section 4.2). We then solve
this yields a determination of the latter. Variations of equation (8) in Monte-Carlo fashion, while simultane-
thismethodhavebeenappliedinthepastby,e.g.,Yahil, ously adding in the random errors in U~⊙ and V~⊙. This
Tammann,&Sandage(1977),Lynden-Bell&Lin(1977) yields both the best fit result for~v and its statistical
M31
andEinasto&Lynden-Bell(1982). Itis nowworthwhile uncertainties.
torevisitthismethod,sinceavailableinformationonthe We base our analysis on the sample of Local Group
membership and distances of Local Group galaxies has dwarf galaxies (there are no giant galaxies in the outer
evolved significantly in the past decades. partsoftheLocalGroup)compiledbyGrebel,Gallagher,
We adopt a Cartesian coordinate system (X,Y,Z), & Harbeck (2003). From their table 1 we removed all
with the originat the Galactic Center, the Z-axispoint- galaxies listed as being (potentially) part of the Milky
ingtowardstheGalacticNorthPole,theX-axispointing Way or M31 subgroups. We obtained the heliocentric
inthedirectionfromthesuntotheGalacticCenter,and line-of-sight velocities of the galaxies from the NASA
the Y-axis pointing in the direction of the sun’s Galac- Extragalactic Database (NED). We removed the Cetus
M31 Transverse Velocity and Local Group Mass 5
TABLE 2
Outer LocalGroupGalaxySample
Name Type RA DEC vlos D Dbary
deg deg kms−1 kpc kpc
(1) (2) (3) (4) (5) (6) (7)
WLM dIrrIV-V 0.49234 -15.46093 -122±2 945±40 802
Aquarius/DD0210 dIrr/dSph 311.71585 -12.84792 -141±2 950±50 940
LeoA dIrrV 149.86025 30.74639 24 ±... 800±40 953
Tucana dSph 340.45667 -64.41944 130 ±... 870±60 1068
SagDIG dIrrV 292.49573 -17.67815 -79±1 1060±100 1152
Note. — The sample of outer Local Group galaxies used in the modeling of Section 2.3. Column (1) lists the galaxy name
andcolumn(2)itstype. Columns(3)and(4)givethepositiononthesky. Column(5)liststheheliocentricline-of-sightvelocity
and, where available, its error. Velocities and sky positions (RA,DEC) were obtained from the NASA Extragalactic Database
(NED). Column (6) lists the heliocentric distance and its error, from Grebel et al. (2003). Column (7) lists the Local Group
barycentric distance Dbary, calculated assuming the M31 distance listed in Section 3.1 and a mass fraction fM31 = 0.53. The
galaxies in thetable are sorted bytheir valueof D .
bary
dwarf from the sample, since it has no line-of-sight ve-
locity available. We restricted our primary sample to TABLE 3
M31Heliocentric Velocity Estimates
galaxies with a Local Group barycentric distance less
than ∼ 1.2 Mpc. The resulting sample consists of 5
Method vsys vW vN
galaxies, which are listed in Table 2. With this sample kms−1 kms−1 kms−1
the analysis yields for M31: vsys = −405±114kms−1, (1) (2) (3) (4)
v =−126±63kms−1, v =−89±50kms−1 (see also
W N M31Satels. -270±19 -136±148 -5±75
Table3below). Thefityieldsanestimateσ =22kms−1 M33PM -183±76 -48±80 71±84
fortheone-dimensionaldispersionofthegalaxiesaround IC10PM -346±76 -16±80 -47±81
the space motion of the Local-Group barycenter. The OuterLGGals. -405±114 -126±63 -89±50
WeightedAv. -273±18 -78±41 -38±34
small value of this dispersion is due to the fact that
Note. —EstimatesoftheheliocentricvelocityofM31es-
the sample galaxies reside at an average Local Group
timated using different methods, as indicated in column (1).
barycentricdistanceof0.98Mpc,whichisconsistentwith
ThemethodbasedontheM31satelliteensembleline-of-sight
the Local Group turn-around radius (e.g., Karachentsev
velocities is described in Section 2.1, and that based on the
etal.2002),wherethe velocitywithrespecttothe Local observedpropermotions(PMs)ofM33andIC10isdescribed
Group barycenter is zero by definition. inSection2.2,andthatbasedontheline-of-sightvelocitiesof
When comparing the analyses in Section 2.1 and the thesatellites inouterregionsoftheLocalGroupisdescribed
present section, there is an important difference in the inSection2.3. Column(2)liststheestimatedM31systemtic
expected galaxy velocities. The velocities of satellites line-of-sight velocities. Columns (3) and (4) lists the esti-
around M31 are virialized, so the expectation value of a mated M31 transverse velocities in the West and North di-
satellitevelocitywithrespecttoM31iszero,independent rections, respectively. The bottom line of the table lists the
weighted average of theresults from thedifferent methods.
of where the satellite is located on the sky. By contrast,
the motions of the outer Local-Group galaxies around
theLocalGroupbarycenterarenotvirialized. Therefore, nearby structures. These reside at barycentric distances
theexpectationvalueofagalaxyvelocitywithrespectto of 1.6–1.9 Mpc. Analysis of the combined sample of
the Local Group barycenter is zero only if the galaxy is 11 galaxies yields for M31: v = −608±154kms−1,
sys
nearthe turn-aroundradius. If this is not the case,then v =−82±138kms−1, v =−46±82kms−1. The fit
W N
equation (7) is valid only if the galaxies are distributed yieldsanestimateσ =50kms−1 fortheone-dimensional
homogeneously around the Local Group. In reality, the
dispersionofthegalaxiesaroundthespacemotionofthe
distribution is both non-homogeneous and the number
Local-Groupbarycenter. Thelargedeviationofv from
sys
of galaxies is small. Therefore, addition to the sample
the observedvalue as well as the increasedσ and formal
of galaxies beyond the turn-around radius is expected
errors support our assertion that adding these distant
to add both bias and shot-noise to the estimate of the
galaxies decreases the quality of the results. Nonethe-
M31 velocity. Moreover, galaxies significantly outside
less, the results for v and v are consistent with those
W N
the turn-aroundradius do not necessarily need to follow
inferred from the smaller sample, within the errors. So
the Local Group barycenter motion.
we conclude that the results for the M31 transverse mo-
The sample in Table 2 consists of a rather small num-
tion are quite robust, and not very sensitive to the com-
ber of galaxies. So despite the aforementioned disad-
position of the sample. This is further supported by the
vantages, we did study the effect of adding more Local
factthatourresultsareconsistentwithintheerrorswith
Group galaxies at larger barycentric distances. In par-
the preferred solutions obtained by Einasto & Lynden-
ticular, we tried to add the only other 6 Local Group
Bell (1982), despite their use of a sample that is only
galaxies within 2Mpc (namely: NGC 3109, Antlia, Sex-
partially overlapping with ours.
tans A and B, IC 5152 and KKR 25; Grebel et al. 2003)
that are not believed to be associated with any other 2.4. Comparison and Combination of Constraints
6 van der Marel & Guhathakurta
The v and v for M31 inferred from the different
W N
methods and listed in Table 3 are shown in Figure 2
as colored data points with error bars. The weighted
averages of all four of the independent estimates are
hv i = −78±41kms−1 and hv i = −38±34kms−1.
W N
This is shown in the figure as a black data point with
error bars.
The χ2 that measures the residuals between the indi-
vidual measurements in Table 3 and the weighted av-
erages is χ2 = 8.1 for N = 9 degrees of freedom (12
measurements minus 3 parameters). Therefore, the re-
sults for the v and v from the different methods are
W N
consistent within the errors. Among other things, this
implies that there is no evidence that the dispersion of
the peculiar velocities of M31 satellites in the transverse
direction, which enters into the analysis of Section 2.2,
is largerthan the value σ =76kms−1 derivedfromline-
of-sight velocities in Section 2.1. This is consistent with
the assumptionthat was made about the isotropyof the
Fig. 2.— Estimates of the M31 heliocentric transverse velocity
peculiar velocities.
in the West and North directions. Data points with error bars
Theinferredweightedaverageofthethreeindependent arefromTable3,basedontheM31satelliteensembleline-of-sight
systemic velocityestimates ishv i=−273±18kms−1. velocities (blue), the proper motion of M33 (green), the proper
sys
motion of IC 10 (red), the line-of-sight velocities of outer Local
This differs at the 1.6σ level from the observed M31 ve-
groupsatellites(magenta),ortheweightedaverageoftheseresults
locityvsys =−301±1kms−1(Courteau&vandenBergh (black). Thestarredsymbolindicates thetransversevelocitythat
1999), but the agreement in an absolute sense is better corresponds to a radial orbit for M31 with respect to the Milky
than the formal uncertainties in (hv i,hv i). So this Way. The cyan rectangle approximates the region that is ruled
W N
outinallofthetheoreticalmodelsexploredbyLoebetal.(2005),
comparison provides no reason to mistrust our assump-
because the resulting relative orbit of M31 and M33 would have
tions that the M31 satellites (Sections 2.1 and 2.2) and produced more disruption of the M33 disk than is observed. The
theouterLocalGroupgalaxies(Section2.3)moveonav- small dots are the 18% of 30,000 samplings from the error ellipse
erage through space with the same velocity as M31 and belongingtotheblackdatapointthatareconsistentwiththethe-
oreticalconstraint.
the Local Group barycenter, respectively (see Lynden-
Bell 1999 for an earlier discussion of this). and IC10 enter not only in the analysis of Section 2.2,
The analyses in the previous sections make assump- but also in that of Section 2.1. However, this is not
tionsaboutthevelocitydistributionsofthesatellites,but a very important issue. We verified that if M33 and
not about their spatial distributions. These spatial dis- IC10areremovedfromthe line-of-sightvelocityanalysis
tributionsareknowntobeinhomogeneous. Inthecaseof in Section 2.1, then the changes in the results are well
the M31 subgroup, there are more satellites on the near within the uncertainties.
sideofM31(i.e.,betweenM31andtheMilkyWay)than
onthe farside (e.g.,McConnachie&Irwin2006). More- 3. ORBITS
over, recent studies have suggested that (some of) the
3.1. M31–Milky Way Orbit
satellitesareconcentratednearaplanesurroundingM31
(e.g.,Koch&Grebel2006). Thesefactsbythemselvesdo To calculate the velocity of M31 in the Galactocen-
not affect our analysis at all, as long as the velocity dis- tric rest frame we adopt the same Cartesian coordinate
tributions remainisotropic. However,the analysiswould system (X,Y,Z) as in Section 2.3. We adopt a dis-
be affectedifthe satellite ensemblepossessedameanro- tance D =770±40kpc for M31 (Holland 1998; Joshi et
tation. This is possible in some of the scenarios that al. 2003; Walker 2003; Brown et al. 2004; McConnachie
have been suggested for a possible disk-like distribution et al. 2005; Ribas et al. 2005). The position of M31 is
of satellites (but not necessarily the favored scenarios; then ~r = (−379.2,612.7,−283.1)kpc. To calculate the
see, e.g., Metz, Kroupa, & Jerjen 2007). In the method reflex motion of the Sun at the position of M31 we use
ofSection2.1wefitanapparentsolid-bodyrotationfield the same solar velocity as in Section 2.3 and we use the
(eq.[3])totheobservedsatellitevelocities. Anyintrinsic standard IAU value R0 = 8.5 kpc for the distance of
rotation would therefore bias the inferred transverse ve- the Sun from the Galactic Center (Kerr & Lynden-Bell
locity. However,if this had been the case then we might 1986),towhichweassignanuncertaintyof0.5kpc(none
haveexpectedthe M31transversevelocityresultsofthis of our results depend sensitively on this quantity). The
methodtobeinconsistentwiththeresultsfromtheother velocity of the Sun then projects to (vsys,vW,vN)⊙ =
methodsthatwehaveused,andinparticularthemethod (172,128,71)kms−1 at the position of M31. Since one
ofSection 2.3 basedonouter LocalGroupgalaxies. The observes the reflex of this, these values must be added
fact that the results from the different methods are ac- to the observed M31 velocities to obtain its velocity in
tually statistically consistent therefore suggests that our theGalactocentricrestframe. Thevelocityvectorcorre-
results are not affected by potential rotation of the M31 sponding to the observed velocity component vsys given
satellite system. inTable 1andthe inferred(hvWi,hvNi) giveninTable3
In the comparisons of the different estimates listed in is then V~ = (97±35,−67±26,80±32)kms−1. The
obs
Table 3 it should be noted that they are not completely errors (which are correlated between the different com-
independent, because the line-of-sight velocities of M33 ponents) were obtained by propagation of the errors in
M31 Transverse Velocity and Local Group Mass 7
theindividualpositionandvelocityquantities(including ies are bound to each other (E <0) if M ≥M =
bind,min
those for the Sun) using a Monte-Carlo scheme. |~r||V~|2/2G. Theorbitoftheseparationvector~risthena
If the transverse velocity of M31 in the Galactocen- Keplerellipsewitheccentricitye2 =1+(2EL2/G2M2µ3)
tric rest frame, Vtan, equals zero, then M31 moves andsemi-majoraxislengtha=L2/[GMµ2(1−e2)]. The
straight towards the Milky Way on a purely radial pericenter separation is r = a/(1−e). The period is
peri
(head-on collision) orbit. This orbit has (v ,v ) =
W N rad T =2π(a3/GM)1/2. The orbitcanbe parameterizedus-
(−127,−71)kms−1 (this is approximately the reflex of
ingtheeccentricanomalyη as(e.g.,Kibble1985;Binney
the velocity of the Sun quoted above, because the lines
& Tremaine 1987)
from the Sun to M31 and from the Galactic Center to
M31 are almost parallel). The radial orbit is indicated r=a(1−ecosη),
as a starred symbol in Figure 2. The velocity V~obs cal- t=(a3/GM)1/2(η−esinη). (10)
culated in the previous paragraph has tangential and
radial components Vtan,obs = 59kms−1 and Vrad,obs = In this parameterization η = 0 corresponds to the peri-
−130kms−1. The total velocity is |V~ |=142kms−1. center passage at t=0, while η =2π correspondsto the
obs
The value V is an unbiased estimator of the next pericenter passage at t = T. The radial and tan-
rad,obs
true radial velocity V . The associated uncertainty gentialvelocitiescanbe similarlyparameterizedas(e.g.,
rad
can be calculated in straightforward fashion using the Kochanek 1996)
previously described Monte-Carlo scheme. This yields
V = −130±8kms−1. The uncertainty is due mostly Vrad=(GM/a)1/2(esinη)/(1−ecosη),
rad
to the uncertainty in our knowledge of the circular ve- V =(GM/a)1/2(1−e2)1/2/(1−ecosη). (11)
tan
locity of the LSR in the Galactic plane. The values of
V and|V~ |aremoredifficulttointerpret,because ItisgenerallyassumedthattheLocalGroupisabound
tan,obs obs
they are not unbiased estimators of the true velocities system that, due to its overdensity, decoupled from the
V and |V~|. This is because the area coverage of V Hubble expansion at fairly high redshift. Since then the
tan tan orbital evolution of M31 and the Milky Way has been
valuesinthe(v ,v )planescalesas2πV dV ,which
W N tan tan governed by Newtonian dynamics. Given this scenario
produces a bias in the sense that any measurement er-
and the orbital description given by equation (10), the
ror in (v ,v ) tends to yield overestimates of V and
W N tan BigBangmusthavecorrespondedtoa previouspericen-
|V~|. Toquantify andcorrectthese biaseswe usedBayes’
ter passage, which we can take to be t=0. The current
theorem, which yields the identity
timetthencorrespondstotheageoftheUniverse,which
P(V |V )∝P(V )P(V |V ) (9) hasbeentightlyconstrainedusingdatafromtheWilkin-
tan tan,obs tan tan,obs tan sonMicrowaveAnisotropyProbetobet=13.73+0.16Gyr
−0.15
We are interested in the quantity on the left-hand side,
(Spergelet al. 2007). With measurements of the current
which is the probability distribution of V , given our
tan Galactocentric distance r and velocities V and V ,
rad tan
measurement. The quantity P(V ) on the right-hand
tan theequations(10)and(11)canbe solvedforthe quanti-
side is the Bayesian prior probability of V , which we
tan tiesM,a,eandη. Thequantityηmustbeintheinterval
assume to be flat (i.e., homogeneous). The quantity
[π,2π] (so that M31 and the Milky Way are falling to-
P(V |V ) is the probability of measuring a value
tan,obs tan wards each other for the first time), since unplausibly
V iftheactualvalueisV . Thislatterdistribution
tan,obs tan high masses M are otherwise required.
iseasilycalculatedusingMonte-Carlodrawings,because
This methodology for modeling the Local Group is
themeasurementuncertaintiesareknown. Oncethedis-
commonly called the “timing-argument”. It has been
tributions P(V |V ) have been pre-calculated for
tan,obs tan widelyappliedanddiscussedintheliterature(e.g.,Kahn
all V , it is straightforward to obtain a Monte-Carlo
tan & Woltjer 1959; Lynden-Bell 1981, 1999; Einasto &
samplingofthe probabilitydistributionP(V |V ).
tan tan,obs Lynden-Bell 1982; Raychaudhury & Lynden-Bell 1989;
To this end one draws a random deviate V , and then
tan Kroeker & Carlberg 1991; Kochanek 1996; Loeb et
accepts this value with probabilityP(V |V ). The
tan,obs tan al. 2005), mostly to obtain joint estimates for the age
top left panel of Figure 3 shows the probability dis-
of the Universe and the mass of the Local Group, often
tribution P(V |V ) thus obtained. The median
tan tan,obs assuming a purely radial orbit. We can now do a more
Vtan = 42 kms−1. The fact that this is smaller than accurateanalysis,bothbecausewehaveanobservational
Vtan,obs = 59kms−1 quantifies the aforementioned bias. estimateofVtan,andbecausetheageoftheUniversecan
The 1σ confidence interval is Vtan ≤ 56kms−1. So the beassumedtobewell-knownfromindependentdata. We
radialorbitisconsistentwiththe dataatthisconfidence haveusedthis methodologyonour results. Monte-Carlo
level. These results are discussed in the context of pre- error propagation was performed as described above, to
vious model predictions in Section 4.1. When combined obtain full probability distributions of M, a, e, η, T,
withthevalueforVrad,the1σconfidenceintervalaround and rperi. The results are shown as histograms in Fig-
the median for the total velocityis |V~|=138+14kms−1. ure 3. The inferred 1σ confidence intervals around the
To get insight into the relative orbit of M−3111and the median are: M = 5.58+−00..8752×1012M⊙; a = 561+−2296kpc;
Milky Way we assume that they can be approximated η = 4.301+0.047 radians; r = 23kpc, with 1σ confi-
−0.045 peri
as point masses of mass MM31 and MMW, respectively. dence interval r ≤ 40.9kpc; T = 16.70+0.27Gyr; and
In the center-of-mass frame, their orbit then has en- peri −0.26
e = 0.959, with 1σ confidence interval (1−e) ≤ 0.072.
ergy E = 1µ|V~|2 − GµM/|~r| and angular momentum
2 The implications of this result for M are discussed in
L=µ|~r×V~|,wherethetotalmassisM =M +M Section4.2. Theuncertaintiesinthelistedquantitiesare
M31 MW
andthe reducedmassisµ=M M /M. Thegalax- due primarily to the uncertainties in V and the M31
M31 MW tan
8 van der Marel & Guhathakurta
Fig. 3.—Probabilityhistogramsof: M31Galactocentrictangential velocityVtan;minimumtotalmassMbind,minofM31andtheMilky
Wayifthegalaxiesarebound;andassumingaboundorbitandthetimingargument,theM31-MilkyWayorbitalpericenterdistancerperi,
orbital eccentricity e, orbital semi-major axis length a, orbital period T, current eccentric anomaly η, total mass Mtiming, and implied
turnaroundradiusr0fortest-particlesonradialorbitsaroundtheM31-MilkyWaysystem. Theprobabilitydistributionswereobtainusing
Monte-Carlo simulations as described in the text. The vertical scales are arbitrary. The blue curves take into account all observational
uncertainties in the distances and velocities of both M31 and the Sun in the Galactocentric rest frame, as well as the observational
uncertainties in the age of the Universe. The red curves also enforce the theoretical exclusion zone of Loeb et al. 2005 (cyan region in
Figure2),withinwhichmoretidaldeformationofM33wouldhavebeenexpected thanisobserved.
distance D. The uncertainties in these quantities con- and207(MMW/2.3×1012M⊙)kpc,respectively(Loebet
tribute moreorlessequallyto the uncertaintiesinM,a, al. 2005). The dark halos will intersect once the orbital
T, and η. The uncertainties in the distance have little separationbecomes smallerthanthe sums ofthese sizes.
effect on e and r . All quantities vary monotonically This will have two consequences. First, the orbits will
peri
withV andD. LargervaluesofV yieldlargervalues deviate from Kepler ellipses in the sense that the orbit
tan tan
of M, a, η, T, and r , and smaller values of e. Larger willhavelesscurvatureandlargerr thanindicatedby
peri peri
values of D yield larger values of M, a, e, T, and r , thepreviouslyderivedKeplerorbit. Second,therewillbe
peri
andsmallervaluesofη. Figure3alsoshowstheprobabil- dynamical friction, which will tend to increase the cur-
itydistributionofM ,theminimummassrequired vatureandwilltend todecreaser . Morecomplicated
bind,min peri
for a bound orbit. The inferred 1σ confidence intervals calculationsarenecessaryto properlycalculatethe orbit
around its median is Mbind,min =1.72+−00..2265×1012M⊙. once the dark halos of the galaxies start to overlap and
The anticipated collision between M31 and the Milky tostudythepropertiesoftheresultingmerger. Suchcal-
Way will happen at the next orbital pericenter, which culationswererecentlypresentedbyCox&Loeb(2007).
in the Kepler model is at T −t ≈ 3.0± 0.3Gyr from However,they adopted an orbit with V =132kms−1,
tan
now. The orbital pericenter distance is much smaller r = 450kpc, and e = 0.494. Comparison to the top
peri
than the sizes of the galaxies’ dark halos. If the galax- panel of Figure 3 shows that this orbit is not consistent
ies are assumed to have a logarithmic potential that re- with our measurement of the M31 transverse velocity.
produces the observed rotation curve amplitudes, then
their radialextents rt are 235(MM31/3.4×1012M⊙)kpc 3.2. M31–M33 Orbit
M31 Transverse Velocity and Local Group Mass 9
Loeb et al. (2005; hereafter L05) recently derived a approximation (e.g., Binney & Tremaine 1987, eq. [7-
theoretical constraint on the transverse motion of M31 55]),wherer is the impact parameter. Therefore,asone
from the fact that M33 is relatively undisturbed. This moves awayalong either diagonalin Figure 2, the struc-
appears to rule out orbits where M33 had a previous tureofM33willbelessperturbedbytheencounter. The
close interaction with M31. The exact shape of the re- excluded regionin Figure 2 can therefore be understood
gion in (v ,v ) space thus ruled out (being defined by to lowest order as the region where the orbital integral
W N
L05as: morethan20%oftheM33starswouldhavebeen overdE exceedssomethreshold,withrandV calculated
stripped)hasacomplexshapeanddependssomewhaton on the basis of the Kepler orbit.
the modeling assumptions, but it can be approximated Interestingly, the observationally implied transverse
bythesolidcyanrectangleinFigure2. Thisisanapprox- velocity of M31 from Table 3 falls right in the regionfor
imation to figure 2c of L05 (which quantified the trans- whichconsiderabletidal deformationofM33wouldhave
verse motion using Galactocentric rest-frame velocities been expected. This M31 velocity yields a Kepler orbit
V = −v −128kms−1 and V = v +71kms−1). for the M31–M33 separation vector with a = 127kpc,
αcosδ W δ N
In some of their models (see their figures 2a,b) even a r = 30kpc, T = 2.4Gyr, and e = 0.76. However,
peri
somewhat larger region is ruled out. theuncertaintiesintheobservationallyimpliedvelocities
The overall shape of the excluded region in (v ,v ) cannot be ignored. Upon performing Monte-Carlo sam-
W N
space can be understood with fairly simple calculations plingwefindthatasmuchas18%ofsamplingsfromthe
and arguments. In doing so, we ignore the observa- error ellipse fall outside the cyan rectangle in Figure 2,
tional uncertainties in the quantities of interest. This and therefore do not violate the M33 tidal stripping ar-
is sufficient for the scope of the present discussion, but gument. These 18% are shown as small black dots for a
should be included for more quantitative understanding total sample of 30,000 drawings. The black dots can be
of the M31-M33 orbit. Based on the data from Ta- viewed as a visual representation of the probability dis-
ble 1 and the M33 and M31 distances from Sections 2.2 tributionofM31’stransversevelocityobtainedbytaking
and3.1,respectively,thedistancebetweenM33andM31 into account not only the observational constraints de-
is203kpc. ThecenteroftheexcludedregioninFigure2is rived here, but also the theoretical M33 stripping argu-
at(v ,v )≈(−68,−11)kms−1. Thisdefinesthethree- mentofL05. Sincemostofthedotsfallclosetothecyan
W N
dimensional velocity vector of M31, while for M33 that rectangle, it is likely that there has been some tidal de-
vectorisknownfromBrunthaleretal.(2005). Theradial formationofM33byM31(althoughnotsufficienttopass
andtangentialvelocitycomponentsoftheseparationvec- L05’sthresholdforbeingconsideredexcluded). Thissug-
tor are then V = −71kms−1 and V = 126kms−1. geststhatitwillbeworthwhiletoperformdeepsearches
rad tan
The Kepler orbit of the separation vector can be cal- fortidaltailsandstructuresinthe outerregionsofM33,
culated similarly as in Section 3.1. In doing so, we similar to those that have already been performed for
assume that M = 3.0 × 1012, which is based on M31 (e.g., Ferguson et al. 2002).
M31
M =5.58×1012M⊙ (Section3.1)andfM31 ≈0.54(com- Our results for the relative M31–M33 orbit involve a
pareSection2.3). WeassumethattheM33-to-M31mass small amount of circular reasoning, since we have as-
ratioisequaltothevalue∼(118/250)4suggestedbythe sumed a priori that the residual space motion of M33
Tully-Fisher relation and the galaxy’s circular velocities withrespectM31hasthesamedispersion(σ =76kms−1
(Corbelli& Schneider 1997;Klypinet al. 2002). The or- per coordinate,as derivedfrom line-of-sightvelocities in
bitthenhasa=120kpc,r =27kpc,T =2.2Gyr,and Section 2.1) as do the other M31 satellites. To avoid
peri
e = 0.77. The value of a is close to the minimum that thiscircularreasoningonecouldignoretheresultlabeled
this quantity can attain (i.e., maximum binding energy) “M33 PM” in Table 3 (green cross in Figure 2). The
asafunctionof(v ,v ),whichisa =113kpc. Given weighted average of the remaining M31 transverse mo-
W N min
the value of a, M33 moves inside of the M31 dark halo tionestimatesthenbecomeshv i=−89±47kms−1and
W
formostofitsorbit. OurassumptionthatalloftheM31 hv i= −60±37kms−1. This differs by only 24kms−1
N
massresidesatitscenterthereforeoverestimatesthecur- from what we have used so far (last line of Table 3),
vatureoftheorbit. Thelistedpericenterdistanceshould and this difference is well within the uncertainties. This
thusbeinterpretedasalowerboundontheactualvalue. modified result still falls well inside the region excluded
Moredetailedcalculations,asinL05,arerequiredtoget by L05. Therefore, our conclusions about the M31–M33
a proper estimate of this quantity. orbit are not influenced by the fact that we have used
As one moves from (v ,v ) ≈ (−68,−11) kms−1 the M33 space velocity as one of the estimators of the
W N
along the diagonal that runs from the bottom left to M31 space velocity.
the top right in Figure 2, the Kepler orbit value of r L05 only considered models with M = 2.6–3.4×
peri M31
doesn’tchangemuch. However,aincreases,whichmeans 1012 M⊙. As is discussed in Section 4.2 below, it is
that the orbits become more eccentric and less bound possible that the M31 mass is actually lower than this.
(and ultimately unbound). The galaxies therefore spend For example, Klypin et al. (2002) advocate M =
M31
less time in close vicinity of each other, and the rela- 1.6× 1012 M⊙. It follows both from the simple argu-
tive velocity at pericenter increases. By contrast, as one ments of Section 3.2 and from the detailed calculations
moves from (v ,v )≈ (−68,−11)kms−1 along the di- of L05 (their figure 2) that the amount of past tidal de-
W N
agonalthatrunsfromthe topleftto the bottomrightin formation of M33 is smaller for smaller values of M .
M31
Figure2,theKeplerorbitvalueofedoesn’tchangemuch. This would reduce the area of the excluded rectangular
Butagain,aincreases,whichmeansthatthe orbitshave region in Figure 2, and would reduce the concern that
largerpericenterseparationsandbecomelessbound(and our observational estimate of the M31 transverse veloc-
ultimately unbound). The energy dissipated during an ity falls in the region of parameter space that was disfa-
encounterscalesasdE ∝1/(r4V2)intheimpulsivetidal vored by L05. Also, the models of L05 (and hence the
10 van der Marel & Guhathakurta
cyanregioninFigure2)donotaccountexplicitlyforthe served galaxy velocities starting from their observedpo-
observational uncertainties in the assumed distances of sitions(Peeblesetal.1989;Peebles1994),tofitobserved
M31 and M33. If the actual distances differ at the 1 or galaxy distances starting from their observed velocities
2σ levelfrom the canonicalvalues, then this could affect (Shaya, Peebles, & Tully 1995; Schmoldt & Saha 1998),
the locationoftheexcludedregionintransverse-velocity or to fit observed galaxy velocities and distances simul-
space. taneously (Peebles et al. 2001, hereafter P01; Pasetto &
It is straightforward to include the theoretical con- Chiosi2007). Thegeneralpredictionfromthetheoretical
straintof L05into the M31-MilkyWay timing argument work that includes tidal torques is that M31 tangential
calculations of Section 3.1. To address this, we applied velocities of V .200kms−1 are expected in plausible
tan
the same Monte-Carlo scheme as in that section, but models.1
now with rejection of all Monte-Carlo drawn velocities Figure6ofP01showstheM31transversevelocityvec-
with (v ,v ) combinations in the region excluded by tors predicted in 30 minimum-action solutions for the
W N
L05. This yields the probability histograms shown in nearby Universe. The reason that there are multiple
redinFigure3. ThedistributionsofV ,eandr be- possible solutions is due to the absence of observational
tan peri
come rather non-Gaussian, as can be easily understood knowledge of most galaxy proper motions. P01 char-
from the distribution of points in Figure 2. However, acterized the M31 velocity using supergalactic angular
theotherdistributionsremainclosetoGaussian,andare coordinates in the Galactocentric rest frame. These are
notverydifferent fromthose obtainedwithout using the related to the heliocentric transverse velocity v and the
t
LO5 M33 stripping argument. The same is true for the position angle Θ of the transverse motion on the sky,
t
inferred 1σ confidence intervals around the median val- defined as in equation (2), according to
ues. For example, for the Local Group mass we obtain
M = 5.50+−10..1746×1012M⊙, which is similar to the result vSGL=−vtcos(Θt+34.60◦)−131kms−1,
of Section 3.1. v = v sin(Θ +34.60◦)−64kms−1. (12)
SGB t t
Our weighted average velocity (hv i,hv i) given in Ta-
W N
4. DISCUSSION ble 3 corresponds to (v ,v ) = (−55,−21)kms−1.
SGL SGB
4.1. M31 Tangential Velocity The 68.3% confidence ellipse around this measurement
encloses 8 of the 30 viable solutions presented by P01.
Many previous studies of the M31–Milky Way system
Our measurement is therefore fully consistent with their
haveassumedthattheirorbitcanbeapproximatedtobe
theoretical work. The action method is based on the as-
radial (V ≈ 0). This simplifies analyses based on the
tan sumptions that mass follows light and that the galaxy
timing argument and, in the absence of a reliable V
tan peculiar velocities are due to their mutual gravitational
measurement, was a reasonable guess based on simple
interactions. Our M31 transverse motion determination
cosmological arguments. In the absence of mutual grav-
therefore provides no reason to doubt these assump-
itational interactions, peculiar velocities with respect to
tions (although cosmological N-body simulations sug-
the Hubble flow decrease with time as (1+z). In the
gest that these assumptions are at best only approxi-
M31–MilkyWaysystemthereismutualgravitationalat-
mately satisfied; Martinez-Vaquero, Yepes, & Hoffman
tractionalongthegalaxyseparationvector. Thischanges
2007). The mass M + M assumed in the P01
theradialvelocityV frompositive(receding)athighz M31 MW
to negative (approarcahding) at the present time, as quan- models is 5.16×1012 M⊙, which is consistent with the
range calculated in Figure 3 based on the timing argu-
tified by the timing argument. However, the angular
ment. The mass was not varied independently in P01,
momentum is conserved in a two-body system without
but is within the factor ∼ 2 range of masses for which
external perturbations. Therefore, a significant present-
the action method yields plausible solutions (Peebles et
day tangential motion in such as system implies an un-
al. 1994). Pasetto & Chiosi (2007) obtained a best-fit
realistically high peculiar velocity at high redshift.
solution from their action modeling that corresponds to
The situation is more complicated when the possibil-
a heliocentric velocity (v ,v ) = (−142,−41)kms−1.
ityofangularmomentumexchangeistakenintoaccount. W N
This value is near the edge of our 68.3% confidence el-
Tidaltorques canlead to exchangebetween the spin an-
lipse, and is therefore also consistent with our measure-
gular momentum of the galaxies and their orbital angu-
ment. The mass M +M assumed in their models
larmomentum. This processmayhavecontributedboth M31 MW
to the observedspins ofM31and the Milky-Way,andto is 5.36×1012M⊙, which is also within the range calcu-
lated in Figure 3 based on the timing argument.
thetangentialvelocitycomponentintheirorbitalmotion
Insummary,our measurementofthe transverseveloc-
(Gott&Thuan1978;Dunn&Laflamme1993). Moreim-
ity of M31 is consistent with the most recent theoretical
portantly,tidaltorquesexertedbythegalaxiesoutsideof
models. Moreover, the fact that 73% of P01’s action-
the Local Group also induce a tangential velocity com-
methodsolutionsdonotfallwithinour68.3%confidence
ponent in the M31–Milky Way system (Raychaudhury
ellipse suggests that our measurement has sufficient ac-
& Lynden-Bell 1989). A useful approach to study this
curacy to provide meaningful constraints on the allowed
effect, and more generally, the orbits of all the galaxies
in the nearby Universe,is based on the principle of least
1 The mass M = MM31 +MMW calculated with the timing
action (Peebles 1989). This assumes that nearby galax- argument is a monotonically increasing function of Vtan. So if
ies arrived at their present configuration through grav- one pre-assumes an upper limit to M, then one also obtains an
itational interactions from a nearly homogeneous high- upper limit to Vtan. L05 used this approach to obtain Vtan .
redshiftstatewithnegligiblepeculiarvelocities. Allowed 120kms−1 based on the assumption that M ≤ 5.6×1012 M⊙.
However, no physical motivation was provided for this assumed
solutions are those that minimize the relevant Hamilto-
mass limit. Larger values of Vtan are not inconsistent with the
nian action integral. This method can be used to fit ob- timingargument,buttheydorequirehighermasses.