Determining the Nature of Dark Matter with Astrometry

0.17 MB·English

by  Louis E. Strigari

Draftversion February4,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
DETERMINING THE NATURE OF DARK MATTER WITH ASTROMETRY
Louis E. Strigari1,2 James S. Bullock1, Manoj Kaplinghat1,
Draft versionFebruary 4, 2008
ABSTRACT
Weshowthatmeasurementsofstellarpropermotionsindwarfspheroidalgalaxiesprovideapowerful
probe of the nature of dark matter. Allowing for general dark matter density profiles and stellar
velocity anisotropy profiles, we show that the log-slope of the dark matter profile at about twice the
stellar core (King) radius can be measured to within 0.2 when the proper motions of 200 stars are
±
added to standard line-of-sight velocity dispersion data. This measurement of the log-slope provides
7
a test of Cold and Warm Dark Matter theories at a sensitivity not possible with line-of-sightvelocity
0
0 dispersion measurements alone. The upcoming SIM PlanetQuest will have the sensitivity to obtain
2 the required number of proper motions in Milky Way dwarf spheroidal galaxies.
Subject headings: Cosmology: dark matter, theory–galaxies: kinematics and dynamics–Astrometry
n
a
J
1. INTRODUCTION stellarLOSvelocitieswithpropermotionmeasurements.
9 Weshowthat 200propermotionsat 5kms−1 accu-
1 In a Universe dominated by collisionless, Cold Dark ∼ ∼
racy can break the relevant degeneracies and determine
Matter (CDM), the central densities of galaxy halos are
1 high, and rise to form cusps with ρ r−1, as r 0 the dark matter distribution in the vicinity of the dSph
∼ → stellar core radius, r . Specifically both the dark halo
v (Dubinski & Carlberg 1991; Diemand et al. 2005). In king
density and the local log-slope of the dark halo density
1 contrast,alargeclassofalternativestoCDM,heregener-
8 ically called Warm Dark Matter (WDM), are expected profile at ∼ 2rking may be determined with & 5 times
5 tohavelowercentraldensitiesandconstantdensitycores higher precision than from LOS velocity dispersion data
1 at small radii (Dalcanton & Hogan 2001; Kaplinghat alone. Using two well-motivated examples of a cusp and
0 core halo,we show that the locallog-slope measurement
2005; Cembranos et al. 2005; Strigari et al. 2006a). In
7 can distinguish between them at greater than the 3-σ
the majority of WDM models, phase space arguments
0 level. We discuss our results in the context of NASA’s
predict that density cores will be more prominent in
/ SIM PlanetQuest (Space Interferometry Mission) which
h lower mass halos. Current observations of the small-
will have the sensitivity to measure proper motions of
p est rotationally-supported galaxies are somewhat am-
- biguous. Some galaxies prefer cored halos, some pre- stars at this accuracy in multiple dSphs.
o Astrometry as a means to constrain the central den-
fer cusps, and others are well fit by either (Simon et al.
r sity profiles of dSphs was previously considered by
t 2005; Kuzio de Naray et al. 2006). Moreover,the effects
s Wilkinson et al. (2002). Using a two-parameter family
of non-circular motions and other systematic issues are
a
of models for the dSphs, these authors show that proper
: yet to be sorted out for these systems.
v motions will be enough to completely reconstruct the
The dwarf spheroidal (dSph) satellite galaxies of the
i profileiftheunderlyingshapefollowstheiradoptedform.
X Milky Way provide a potentially superior laboratory for
However, CDM halos, and likely their WDM counter-
studying the nature of dark matter. Observationally,
r parts, are described by a minimum of four unknown pa-
a their proximity ( 100 kpc) allows kinematic studies
∼ rameters: a scale density, a scale radius, an asymptotic
of individual stars. Theoretically, their small masses
innerslope,andanasymptoticouterslope. Bymarginal-
make them ideal candidates for prominent WDM cores.
izing over all of these parameters as well as the velocity
Moreover, the CDM prediction of cuspy central density
anisotropyof the stars,we demonstrate that the asymp-
profiles is robust for these satellite systems because the
totic slopes are never well-determined, even with proper
cusps are stable to any tidal interactions that may have
motions. We show that what is best constrained is the
occurred (Kazantzidis et al. 2004). Unfortunately, ob-
taining dSph density profile slopes has been difficult. log-slope,γ(r∗)≡−dlnρ/dlnr|r=r∗,ataradiusr∗ com-
The best currentconstraints ontheir mass profiles come parable to the King core radius, specifically r∗ 2rking.
∼
Wenotethatunliketheasymptoticr 0slope,thelog-
fromanalysesof 200line-of-sight(LOS)velocities(e.g.
→
L okas et al. 2005∼; Munoz et al. 2005). The stellar veloc- slope at r∗ 0.4 kpc is known from CDM simulations.
≃
This radius corresponds to 1% of the relevant halo
ity anisotropy is a major source of degeneracy in LOS
∼
virial radius, and is well-resolvedin current simulations.
velocitydispersionmodeling,and,asweshowbelow,the
log-slopeoftheunderlyingdensityprofileremainspracti- 2. MASSMODELING
cally unconstrainedevenif the number of observedstars
is increased by a factor of 10. We define the three-dimensional (spatial, not pro-
InthisLetter,weexamine∼theprospectsofconstraining jected) components of a star’s velocity to be vr, vθ, and
the dark matter density profiles of dSphs by combining vφ. The velocity component along the line of sight is
then v = v cosθ +v sinθ, where ~z ~r = cosθ and
los r θ
·
1CenterforCosmology,Dept. ofPhysics&Astronomy,Univer- ~z is the line-of-sight direction. The components parallel
sityofCalifornia,Irvine,CA92697 andtangentialto the radiusvector R~ in the plane of the
2McCueFellow
sky are v =v sinθ+v cosθ and v =v , respectively.
R r θ t φ
2 Strigari et al.
Foreachcomponent,thevelocitydispersionisdefinedas set of the five shape parameters. The first is the log-
σ2 v2 . We will assume σ2 =σ2. slopeofthedarkmatterdensityprofile,definedasγ(r)=
i ≡h ii θ φ
The velocity dispersion for each observed component dlnρ(r)/dlnr. Forthe densityprofileinEq.(4)thisis
−
canbe constructedby solvingthe Jeansequationfor the given by γ(r) = a (a c)(r/r )b/[1+(r/r )b]. Other
0 0
− −
three-dimensional stellar radial velocity dispersion pro- quantities of physical interest are the integrated mass
file σ (r) and integrating along the line of sight. We within a given radius, M(r), and the physical density at
r
note that even in the case of tidally disturbed dwarfs, a givenradius, ρ(r), which are clearly obtained for a de-
Klimentowski et al. (2006) have shown that dSph veloc- generate set of shape parameters. Below, we show that
ity dispersions are well modeled by the Jeans equation, while the shape parameters are not well constrained by
as long as unbound, interloper stars are removed with dSph velocity data, the physical quantities of interest at
standard procedures. We derive the three resulting ob- the scale of the stellar core radius, r 2r , may be
⋆ king
≃
servable velocity dispersions: constrained to high precision.
2 ∞ R2 ν σ2rdr 3. FORECASTINGERRORSONPARAMETERS
σ2 (R)= 1 β ⋆ r , (1)
los I⋆(R)ZR (cid:18) − r2 (cid:19)√r2 R2 Our goal is to estimate the accuracy with which the
2 ∞ R2 −ν σ2rdr velocity components of stars in dSphs can be used to
σ2(R)= 1 β+β ⋆ r , (2) probe the underlying dark matter distribution. We will
R I⋆(R)ZR (cid:18) − r2 (cid:19)√r2 R2 consider a model with six independent parameters: a,
−
2 ∞ ν σ2rdr b, c, ρ , r , and β = constant. We will consider gen-
σ2(R)= (1 β) ⋆ r . (3) 0 s
t I⋆(R)ZR − √r2 R2 eralized β(r) forms below. In order to keep the profile
− shape relatively smooth (as is expected for dark matter
Here β(r) =1 σ2/σ2 is the stellar velocity anisotropy, halo profiles) we restrict the range of b and c by adding
− θ r Gaussian priors of 2.
I⋆(R) is the surface density of stars, and ν∗(r) is the ±
The errors attainable on these parameters will de-
three-dimensional density of stars. It is clear from in-
pend on the covariance matrix, which we will approx-
spectionthateachcomponentdependsonβ inadifferent
imate by the 6 6 Fisher information matrix F =
fashion, and therefore can be used together to constrain ı
×
1it9s6v2a)l,uwe.hFicohriIs⋆c(hRa)raanctderνi⋆z(erd)bwyeausceoraeKraindgiupsr,orfile(,Kainndg ho∂f2tlhneLF/i∂shpıe∂rpmia(tKrixe,ndFa−ll1,&pSrtouvaidrtes1a9n69e)s.timThaeteinovfetrhsee
king
tidal radius, rt. We adopt values that describe the sur- covariance between the parameters, and Fı−ı1 approx-
face density of Draco: rt = 0.93 kpc and rking = 0.18 imates the error in the estimate on thepparameter pı.
kpc. Note that the remaining dSphs have similar King The Cramer-Rao inequality guarantees that F−1 is
concentrations, r /r 5, with Sextans having the ıı
t king ∼ the minimum possible variance on the ıth parapmeter for
largest ratio rt/rking ∼ 10. Our results do not change an unbiased estimator. Using F−1 in place of the true
significantlyaswevaryr /r (equivalenttolookingat
t king covariance matrix involves approximating the likelihood
different dSphs).
function of the parametersas Gaussiannear its peak, so
In Eqs. (1), (2) and (3), the radial stellar velocity dis- F−1willbeagoodapproximationtotheerrorsonparam-
persion, σ , depends on the total mass distribution, and
r eters that are well-constrained. The Fisher matrix also
thus the parameters describing the dark matter density
providesinformationaboutdegeneraciesbetweenparam-
profile. We will consider the following general parame-
eters but obviously should not be trusted for estimates
terization of the dark matter density profile,
of the error along these degeneracy directions.
ρ
0 We pick large radial bins to compute the velocity dis-
ρ(r)= . (4)
(r/r )a[1+(r/r )b](c−a)/b persions and check that this uncorrelates the different
0 0
bins. Then the elements of F are given by
Here, the value of a sets the asymptotic inner slope, and
differentcombinationsofbandcsetthetransitiontothe 1 ∂σ2 ∂σ2
F = Mℓ Mℓ . (5)
asymptoticouterslope. Forthe specific choice(a,b,c)= ı ǫ2 ∂p ∂p
(1,1,3), we have an NFW profile (Navarro et al. 1996). XM,ℓ Mℓ ı 
We denotethisasourcuspcasebelow. Forourcorecase The sum is over ℓ radial bins and M refers to the three
we use (a,b,c) = (0,1.5,3), corresponding to a Burk- velocity “methods” – one line-of-sight and two compo-
ert profile (Burkert 1995). We take these two models to nents in the plane of the sky. The errors on the velocity
be representative of the predictions of CDM and WDM dispersion are represented by ǫMℓ. We choose bins of
models. The Burkert profile for the core case is moti- equalwidth in distance, so that there are approximately
vated by the expectation that WDM halos will mimic anequalnumberofstarsineachradialbin. Aslongaswe
CDM halos at large radius. This was seen in the WDM distribute equal number of stars in each bin, the results
simulations of Col´ın et al. (2000). The Burkert choice is we present below are insensitive to the binning scheme,
also conservative compared to the often-used isothermal except in the limit of very few bins, or in the limit of
core with (a,b,c) = (0,2,2), which is more divergent in small numbers of stars per bin.
shape from an NFW and would be easier to distinguish Tomodelthe errorsonthe velocitydispersions,wede-
observationally. Regardless, our methods are robust to fine ǫ2 = [σ2 σ2 ]2 . We assume that the errors
Mℓ h Mℓ−h Mℓi i
changes in the underlying form of the density profile. on the velocity of each star are Gaussian and that the
Eq. (4) allows considerable flexibility in overall form, theory error (from the distribution function) and exper-
andthe fiveshape parameters(a,b,c,r ,ρ ) arein many imental error are summed in quadrature,
0 0
cases degenerate. However, there are a number of phys- 2(n 2) 2
ically relevant quantities that may be derived for any ǫ2Mℓ = n−2 σMt ℓ2+σMmℓ2 , (6)
h i
Determining the Nature of Dark Matter 3
TABLE 1
FisherMatrix 1-σ errorson parameters.
Parameter FiducialValue ErrorQuoted 200LOS 1000LOS 200LOS 1000LOS 1000LOS
+0SIM +0SIM +200SIM +200SIM +500SIM
γ(r⋆=2rking) 1.3(0.32) ∆γ⋆ 2.0(2.2) 0.91(1.0) 0.28(0.24) 0.21(0.23) 0.16(0.15)
M(r⋆=2rking)[107M⊙] 1.3(5.3) ∆M⋆/M⋆ 0.27(0.38) 0.12(0.17) 0.11(0.11) 0.07(0.08) 0.06(0.05)
ρ(r⋆=2rking)[107M⊙ kpc−3] 4.0(2.5) ∆ρ⋆/ρ⋆ 1.6(0.73) 0.73(0.33) 0.16(0.16) 0.13(0.12) 0.09(0.09)
Vmax [kms−1] 22(28) ∆Vmax/Vmax 1.6(1.7) 0.71(0.76) 0.68(0.93) 0.45(0.62) 0.38(0.52)
β 0(0) ∆β 2.1(1.4) 0.97(0.62) 0.16(0.16) 0.15(0.15) 0.10(0.10)
a 1(0) ∆a 4.4(4.6) 2.0(2.1) 0.78(0.87) 0.55(0.66) 0.44(0.51)
ρ0 [107M⊙ kpc−3] 1.0(3.2) ∆ρ0/ρ0 20(9.2) 9.1(4.2) 5.9(2.0) 4.0(1.5) 3.3(1.1)
r0 [kpc] 2.0(1.5) ∆r0/r0 8.3(5.1) 3.8(2.3) 3.3(1.8) 2.2(1.2) 1.8(1.0)
Note. —Errorsrefertothe1−σ rangeobtained fromafullmarginalizationoverthe6inputparameters (a,b,c,r0,ρ0,β). Quantities
without (with) parenthesis refer to the cusp (core) model described in the text. In the last 5 columns, LOS refers to the number of
line-of-sightstarsusedtoderivethequotederrors,andSIMreferstothenumberpropermotions.
4. CONSTRAININGTHELOG-SLOPE
The log-slope of the density profile varies with radius,
and so does the minimal error attainable on it. We are
interested in searching for the radius, r , where the log-
⋆
slopeisbestconstrained. Fig.1showsthe1 σ erroron
−
γ(r)computedasafunctionofradiususing1000line-of-
sightvelocitiesand200propermotionsforthecusp(left)
andcore(right) models. Inboth cases,the errorreaches
a minimum of ∆γ 0.2 at 2r and we explicitly
king
≃ ∼
adopt r =2r for the rest of the discussion. We find
⋆ king
thatthevalueofr andtheminimumerrordependsome-
⋆
Fig. 1.— The 1-σ error on the log-slope of the density profile what on the fiducial model. As shown in Fig. 1, profiles
asafunctionofthe radiusfor1000 LOSvelocities combined with with cusps are slightly better constrained than profiles
200propermotions. Theleftandrightpanelsshowourresultsfor
with cores, and r occurs at slightly smaller radii for
the cusp and corecases, respectively. The value of r⋆ is setto be ⋆
wheretheerrorisminimized. cusped models. We also find that as rt/rking increases,
the minimum error decreases and r increases. We have
⋆
wherenisthe numberofstarsineachbin(takenhereto chosen a conservatively small ratio here, r /r 5.
t king
be constant from bin to bin). Here σt is the dispersion Table 1 summarizes our results. We list 1-σ err≃ors at-
ℓ
in each bin determined from Eqs. (1)-(3), and σm rep- tainable on severalquantities determined by marginaliz-
Mℓ
resents the measurement error in the velocities of stars ingoverthe6fitparametersdiscussedabove. Wepresent
inthatbin. ForLOSvelocities,thespectroscopicresolu- errors for five different combinations for the number of
tion implies errors σm 1kms−1, which are negligible LOS stars and proper motion stars. In all cases, the
compared to the undloesrly∼ing dispersion, σt 10kms−1. errors on the “physical” parameters, M(r⋆), ρ(r⋆), and
SIM can achieve 10µas yr−1 proper mot∼ions for faint γ(r⋆),decreasesignificantlywhenpropermotionsarein-
stars3. At100kp∼c,thistranslatestoanaccuracyof 5 cluded. Quantitieslistedwithout(with)parenthesescor-
kms−1. Wewilltakethisasatypicalmeasurementer∼ror respondtothecusp(core)halocase. Notethattheerrors
for velocities in the plane of the sky. on the halo shape parameters b and c are essentially the
The derivatives in Eq. (5), and thus the errors at- same as the priors we set on them of 2 and hence we
±
tainable on any of the parameters, depend on the lo- do not list them in the table.
cation in the true set of parameter space. To exam- A quantity that is well constrained without the addi-
ine how the errors vary as a function of the true set of tion of proper motion information is the mass M(r⋆),
parameters, we choose two fiducial models. Both mod- which can be determined to 27% (38%) for cusped
els produce a “typical” dSph velocity dispersion profile, (cored) models with current data. The mass can be de-
which is roughly flat at 10kms−1 out to r . This terminedtoaremarkable7 8%accuracywhen200SIM
requirement does not com∼pletely fix the profilte. For stars are added. The maxi−mum circular velocity is rel-
our cusp model (NFW) we add the further restriction atively unconstrained by LOS data alone, but becomes
that r and ρ fall within the expected CDM range determinedtowithin45%(62%)forcusped(cored)mod-
0 0
(Strigari et al. 2006b). For our core model (Burkert) els with 200 proper motions. More interesting for the
we set r and ρ to give a central phase space density natureofdarkmatteris the log-slopeofthe density pro-
0 0
Q 10−5M⊙pc−3(kms−1)−3, which can be produced fileatr⋆. Evenwith1000LOSvelocities,thelog-slopein
in ≃some WDM models (Strigari et al. 2006a). The val- thecuspcaseisvirtuallyunconstrained,butwiththead-
ues of r and ρ for both models are listed in Table 1 dition of 200 SIM stars the log-slope is well-determined,
0 0
as are the implied log-slopesγ(r⋆)=1.3 (cusp) and0.32 e.g. γ(r⋆)=1.3±0.2 for the case of anunderlying cusp.
(core) at the characteristic radius r , which is set below The power of velocity dispersions to constrain the
⋆
at the value 2rking =0.36 kpc. density and mass at r⋆ ∼ rking is relatively sim-
ple to understand by examining the observable veloc-
3 http://planetquest.jpl.nasa.gov/SIM/sim AstroIndex.cfm ity dispersion components (Eqs. 1, 2 and 3). These
4 Strigari et al.
1 two-dimensionalplane of γ⋆-β. This figure clearly shows
1000 LOS the utility of combining both LOS and proper motions.
0 Due to the degeneracy with β, a sample of 1000 line-
∼
β of-sight stars is unable to distinguish between our two
-1 fiducialmodels, thoughthe models canbe clearlydistin-
Cusp Core guishedwiththeadditionofpropermotions. Intheβ-γ
⋆
-2
plane,theprincipalconstrainingpowercomesfromcom-
10 00 LOS biningtheLOSandRvelocitydispersionsbecausetheβ
+200 SIM
0 dependence in these two quantities comes with opposite
β signs (Eqs. 1, 2).
-1 Finally, we address the fact that β could in princi-
ple vary significantly with radius. We have repeated
-2 our analysis assuming that β(r) = β +β r2/(r2+r2),
0 1 β
0 1 2 0 1 2 3 and marginalize over β , β and r . We find very small
0 1 β
Log-Slope Log-Slope changes to the γ(r ) error. For reasons similar to those
⋆
given above, the three different velocity dispersion pro-
Fig. 2.— The 68% and 95% confidence regions for the dark
files constrain β(r = r ) to high accuracy, breaking the
halo density profile slope measured at r⋆ = 2rking and velocity ⋆
β-γ degeneracy.
anisotropyβ. Theleft(right)panelscorrespondtothecusp(core)
halo model and the small x’s indicate the fiducial values. Upper
panels show the errors with 1000 LOS velocities and the bottom
panelsshowtheerrorsforanadditional200propermotions. 5. SUMMARY
We have shown that the measurement of proper mo-
observable quantities depend on the three-dimensional
stellar velocity dispersion, which scales as σ2(r) = tions for 200 stars in a typical dSph can be combined
ν−1r−2β rtGν (r)M(r)r2β−2dr r2−γ⋆ for arpower- with curr∼ent line-of-sight velocity measurements to con-
⋆ r ⋆ ∝ strainthe darkhalo log-slopeto 0.2andnormalization
law stellaRr distribution ν⋆(r) and constant β. The ma- to 15%atabouttwicetheKing±radius. Thisisafactor
jority of stars reside at projected radii rking . R . rt, of ± 5 better than currently possible with LOS velocity
wherethestellardistributionisfallingrapidlyν r−3.5. ∼
⋆ dispersiondataalone. Theresultsfromsuchobservations
∼
In this case, for β = 0, the LOS component scales as
will provide a very sensitive test of the CDM paradigm
σl2os(R)∝ Rrtr−0.5−γ⋆(r2−R2)−1/2dr andis dominated andanincisive toolforinvestigatingthe microscopicna-
by the masRs and density profile at the smallest relevant ture of dark matter. We estimate that with 100 days
radii, r ∼rking. For R.rking, ν⋆ ∝r−1 and σl2os is sim- of observing time over the 5 year lifetime of S∼IM, it will
ilarly dominated by r rking contributions. Therefore be possible to obtain the proper motions of 200 stars
we expect the stronges∼t constraints on the dark matter in multiple dSphs (Strigari et al. 2007). ∼
profile at r r . For β = 0 similar arguments hold
⋆ king
∼ 6
but there is a manifest degeneracy between β and the
log-slopeγ . AsdiscussedaboveandillustratedinTable 6. ACKNOWLEDGMENTS
⋆
1, this degeneracycanbe brokenby including both LOS We thank S. Kazantzidis, S. Kulkarni, S. Majewski,
information (Eq. 1) with tangential information (Eqs. and R. Munoz for useful discussions. LES is supported
2, 3). inpartbyaGaryMcCuepostdoctoralfellowshipthrough
With the constraints on γ , are we able to distinguish theCenterforCosmologyattheUniversityofCalifornia,
⋆
cored from cusped models? To answer this question, in Irvine. JSB,LES,andMKaresupportedinpartbyNSF
Figure 2 we present 68% and 95% error contours in the grant AST-0607746.
REFERENCES
Burkert,A.1995,ApJ,447,L25+ L okas,E.L.,Mamon,G.A.,&Prada,F.2005,MNRAS,363,918
Cembranos, J. A., Feng, J. L., Rajaraman, A., & Takayama, F. Munoz,R.R.etal.2005,Astrophys.J.,631,L137
2005, PRL,95,181301 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462,
Col´ın,P.,Avila-Reese,V.,&Valenzuela,O.2000,ApJ,542,622 563
Dalcanton, J.J.&Hogan,C.J.2001,ApJ,561,35 Simon, J. D., Bolatto, A. D., Leroy, A., Blitz, L., & Gates, E. L.
Diemand,J.,Zemp,M.,Moore,B.,Stadel,J.,&Carollo,M.2005, 2005,ApJ,621,757
MNRAS,364,665 Strigari, L. E., Kaplinghat, M., & Bullock, J. S. 2006a, astro-
Dubinski,J.&Carlberg,R.G.1991,ApJ,378,496 ph/0606281
Kaplinghat,M.2005,Phys.Rev.D,72,063510 Strigari, L. E., Kaplinghat, M., Bullock, J. S., Kazantzidis,
Kazantzidis,S.,Mayer,L.,Mastropietro,C.,Diemand,J.,Stadel, S., Munoz, R. R., Majewski, S., & Kaufmann, T. 2007, in
J.,&Moore,B.2004,ApJ,608,663 preparation
Kendall, M. G. & Stuart, A. 1969, The Advanced Theory of Strigari,L. E., Koushiappas, S. M.,Bullock, J. S., & Kaplinghat,
Statistics (London: Griffin) M.2006b,astro-ph/0611925
King,I.1962, AJ,67,471 Wilkinson, M. I., Kleyna, J., Evans, N. W., & Gilmore, G. 2002,
Klimentowski,J.,Lokas, E.L.,Kazantzidis, S.,Prada,F.,Mayer, MNRAS,330,778
L.,&Mamon,G.A.2006, astro-ph/0611296
KuziodeNaray,R.,McGaugh,S.S.,deBlok,W.J.G.,&Bosma,
A.2006, ApJS,165,461