What can Gaia (with TMT) say about Sculptor's Core?

0.45 MB·English

by  Jarah Evslin

What can Gaia (with TMT) say about Sculptor’s Core?
Jarah Evslin
Institute of Modern Physics, CAS, NanChangLu 509, Lanzhou 730000, China
Walker et al.’s Magellan/MMFS Survey survey identified 1355 red giant candidates in the dwarf
5
spheroidal galaxy Sculptor. We find that the Gaia satellite will be able to measure the proper
1
motions of 139 of these with a precision of between 13 and 20 km/s. Using a Jeans analysis
0
and 5-parameter density model we show that this allows a determination of the mass within the
2
deprojectedhalf-lightradiustowithin16%andameasurementofthedarkmatterdensityexponent
y
γ to within 0.68 within that radius. If, even at first light, the TMT observes Sculptor then the
a
M combined observations will improve the precision on these proper motions to about 5 km/s, about
5 years earlier than would be possible without Gaia, further improving the precision of γ to 0.27.
7 Using a bimodal stellar population model for Sculptor the precision of γ improves by about 30%.
1 This suggests that Gaia (with TMT) is capable of excluding a cusped profile of the kind predicted
by CDM simulations with 2σ (4σ) of confidence.

]
A
G According to the standard cosmological model, Σmp (cid:72)km(cid:144)sec(cid:76)
. most of the matter in the universe is cold dark mat- 45
h
40
p ter (CDM), which at long distances only interacts
35
- gravitationally. Direct and indirect dark matter de-
o 30
r tection experiments are a major industry and so far 25
t 20
s have rapidly excluded large regions of the CDM pa-
a rameter space, as has the Large Hadron Collider. 17.5 18.0 18.5 19.0G
[
HoweversuchtestscanonlyconfirmCDM,anysuch
3 FIG.1: TheprecisionwithwhichGaiacanmeasurethe
attempt to falsify this paradigm can be evaded by
v proper motion of a Sculptor member with Gaia magni-
3 fine-tuning. On the other hand, CDM makes fal- tude G and color V −I =1.35
C
0 sifiable predictions for the density profiles of suffi-
5
ciently dark matter dominated systems. They must
7
0 fallas1/r3 atlargeradiiandasatleast1/r atsmall
. radii, a scaling known as a cusp. While there is lit-
1
0 tle consensus over just how much dark matter dom- again. This yields proper motions of hundreds or
5 ination is sufficient, studies such as Ref. [1, 2] sug- thousandsstarsataprecisionofabout4km/sec,al-
1
: gestthatgalaxieswithstellarmassesbeneathabout lowing a clean resolution to the cusp/core problem
iv 106.5−7 M(cid:12) should be cusped while Ref. [3] argues around 2030. We will argue that the Gaia satellite
X that energy conservation implies that baryonic ef- can yield a 2σ hint of the cored/cusped nature of
r fects cannot remove cusps in the Milky Way’s dwarf Sculptor within 5 years and then, when its position
a
spheroidal (dSph) satellites except under quite spe- measurements are combined with TMT’s first ob-
cial circumstances. Thus the discovery that a dSph servations of Sculptor in under 10 years, a definitive
is cored, not cusped, would pose a strong challenge exclusionofacuspedprofilewillalreadybepossible.
to CDM.
In Ref. [5] the authors report the observation of
In this letter, following a strategy similar to that 1541 objects in the part of the sky occupied by the
of Ref. [4], we determine just when it will be known dSph Sculptor, as part of their MMFS survey using
whether the dSph Sculptor, with a stellar mass of theMagellan/ClayTelescope[24]. Ofthese,wecon-
2.3×106 M ,iscored. Commonwisdomstatesthat sider the 1355 objects for which they assign a 90%
(cid:12)
oneneedfirstwait10yearsfora30meterclasstele- or greater membership probability in Sculptor. For
scope, such as the Thirty Meter Telescope (TMT), each object they provide the V magnitude and the
toobservesuchagalaxy. Thenonemeasuresthepo- I magnitude, which we identify with the Johnson-
sitions of the stars, waits 5 more years and observes Cousins magnitude I . We use these to calculate
C
2
the Gaia magnitude, G, following Ref. [6] NumberofStars
25
G = V −0.0257−0.0924(V −I)
20
−0.1623(V −I)2+0.0090(V −I)3. (1) 15
10
As described in the Gaia Science Performance doc- 5
Σm (cid:72)km(cid:144)sec(cid:76)
ument [7] and updated in [8] to reflect post-launch 15 20 25 30 p
performance,Gaiacanmeasurepropermotionswith NumberofStars
40
an end of mission precision, in µas/yr, of
30
(cid:113)
σm0= −1.6+680.8(10)0.4(G−15)+32.7(10)0.8(G−15) 20
p
×0.526(0.986+(1−0.986)(V −I)). (2) 10
Σm (cid:72)km(cid:144)sec(cid:76)
Therearealsoposition-dependentcorrectionstothis 3 4 5 6 p
precision. Using the location of Sculptor and the
FIG. 2: Number of stars in Sculptor whose proper mo-
precision corrections expected [7], we find an im-
tion can be measured with a given precision by Gaia
provement in this precision of roughly 7 µas/yr (14
alone (top) and Gaia with TMT (bottom). The inclu-
µas/yr) at G = 17 (G = 18) corresponding to an sionofTMTimprovestheastrometricprecisonbyabout
improvement of nearly 20% over the average preci- a factor of 4.
sion[8]of39µas/yr(72µas/yr). Inourcalculations
below we conservatively approximate this improve-
menttobe15%andsothefinalprecisionwithwhich TMT will be precisely 20 µas, whereas Gaia deter-
proper motions can be measured is mines positions with a precision of 1.41σm µas [7].
p
The combination of Gaia and TMT separated by 6
σm =0.85σm0. (3)
p p yearsthereforedeterminesthepropermotionwitha
precision of
This function of G is drawn in Fig. 1 for a typical
(cid:113)
color V −I.
400+2(σm)2 µas
p
As is explained in Ref. [9], at Sculptor’s ecliptic σTMT = (4)
p 6 years
latitude of 36.5◦ south this improvement is a com-
bination of a geometric parallax factor and a higher leading to the precisions summarized in the bottom
thanaveragenumberoftransits. However,thenum- panel of Fig. 2.
ber of focal plane transits is also unusually high for OneobservesthattheinclusionofTMTimproves
Sculptor’s ecliptic latitude, as a result of its longi- the astrometric precision by about a factor of 4, an
tude and Gaia’s transit pattern. Assuming a dis- approximation which we will adopt in our analysis.
tancetoSculptorof79kpc[10],theresultingproper We have verified that, as the astrometric precisions
motion precisions are summarized in the top panel arenowsmallerthanthedispersion,theuncertainty
of Fig. 2. with which the velocity dispersion can be measured
We will also determine the precision with which is reasonably insensitive to these precisions.
thepropermotioncanbeobtainedbyGaiaincluding In this letter we will answer the following ques-
asurveyofSculptorbyTMTin2022outtoaradius tions: Given the line of sight velocities of the 1355
of 700 pc using 10 second exposures of IRIS. This members in the MMFS catalog and proper motions
will allow an astrometric precision of better than 20 measuredwiththeprecisionsshowninFig.2: (1)To
µas down to magnitude K = 20, which includes what extent can Sculptor’s dark matter density pro-
AB
thestarswhosepositionsarewell-measuredbyGaia. file be determined? (2) To what extent could the
" "
IRIS has recently be redesigned to have a 34 ×34 CDM paradigm in principle be falsified?
field of view, so the required observing time will be While there are many approaches to this problem
25hours. Inouranalysisbelow,wemaketheconser- intheliterature,suchasRefs.[11–13],ourapproach
vative assumption that the astrometric precision at to these questions will be similar to that of Ref. [4].
3
However, while we still use an assumed stellar pro- precisemeasurementsthisdegeneracycanbebroken
file to determine the expected dispersion, we per- using,inparticular,theazimuthaldependenceofall
form our fit using stars in the MMFS catalog with, of the velocity dispersions. However, given Gaia’s
asdescribedabove,propermotionsdeterminedwith expected precision we expect that this degeneracy
aprecisiongivenbyEq.(3). Thedarkmatterprofile willnonethelessincreasetheuncertaintiesinthehalo
ρ(r)isassumedtolieina5-parameterfamily,corre- parameterssomewhatbeyondthosereportedbelow.
sponding to the generalization [14] of the Hernquist We obtain the radial-dependence of the radial ve-
profile [15] locity dispersion σ2 = (cid:104)v2(r)(cid:105) by integrating the
Jeans equation
(cid:16)r (cid:17)a(cid:32) (cid:18) r (cid:19)b(cid:33)(a−c)/b
ρ(r)=ρ0 r0 1+ r0 (5) σ2(r)= G (cid:90) rtρs(R)M(R)R2β−2dR (6)
ρs(r)r2β
r
where r is the distance from the center of Sculptor.
We assume both a constant orbital anisotropy whereM(R)istheintegratedmasswithintheradius
β = 1 − (cid:104)v2(cid:105)/(cid:104)v2(cid:105) and a spherical halo. Ref. [4] R, approximated to be just the dark matter mass.
θ r
finds that the effect of the first assumption is min- ρs(R) is the 3d stellar density at a radius R, not
imal. In Ref. [16] the authors used the axisymmet- just the density of the stars in the catalog, which is
ric Jeans equation to determine the elipticity of 6 taken to follow a King profile [17] with core radius
nearby dSphs, including Sculptor. They found that r = 0.28 kpc and tidal radius r = 1.63 kpc [18].
k t
not only are these systems not spherically symmet- Theobservedlineofsight,radialandtangentialdis-
ric, but in fact their ellipticities are far higher than persionscanbefoundbyintegrating(6)overtheline
those found in CDM simulations. However, this of sight
study assumed that the radial and major-axis ve-
2 (cid:90) ∞(cid:18) r2 (cid:19)Rρs(R)σ2(R)dR
locity anisotropies are equal, which in the spherical σ2 (r)= 1−β √
los ρs (r) R2 R2−r2
caseisequivalenttoassumingavanishinganisotropy 2D r
β. Indeed,theeffectofellipticityonthelineofsight σ2(r)= 2 (cid:90) ∞(cid:18)1−β+β r2 (cid:19)Rρs√(R)σ2(R)dR
velocity dispersion, as described by the authors, is r ρs2D(r) r R2 R2−r2
quite similar to that obtained by varying β. Thus it 2 (cid:90) ∞ Rρs(R)σ2(R)dR
σ2(r)= (1−β) √ (7)
seems likely that the statistical significance of these t ρs (r) R2−r2
2D r
high ellipticities would be greatly diminished were
where ρ2 (r) is the stellar density integrated along
the restriction on the anisotropies relaxed. This ef- 2D
thelineofsightatr, thedistancefromthecenterof
fect will be compounded by relaxing their other as-
Sculptor in the transverse plane.
sumptions, such as the parallel orientations of the
To determine the precision with which the mass
dark matter and stellar distributions.
profile can be determined, we use the Fisher matrix
Nonetheless, the results of Ref. [16] demonstrate
that nonspherical CDM halos do provide a system-
atic modification of the dispersion profiles. This (cid:32) (cid:33)
shiftsthehaloparametersobtainedusingthespher- F = 1(cid:88) 1 ∂σn2(ri)∂σn2(ri)
ical Jeans equations. Our goal is not to obtain the ab 2 i,n (cid:0)σn2(ri)+σim,n2(cid:1)2 ∂θa ∂θb
parametersthemselves,butmerelytodeterminethe (8)
precision with which they can be determined. By
assuming that the halos are spherical, we will ob- where the index i runs over the stars in the MMFS
tain only a lower bound on the uncertainty on the catalog,theindexnrunsoverthe3directionslos, r
halo parameters. The true uncertainty will be in- and t, r is the projected distance from the center
i
creasedbythedegeneracybetweenthesphericalhalo of Sculptor to the ith star, θ are the 5 parameters
a
parameters and the anisotropy parameters. If only of the dark matter mass model (5) and σm is the
i,n
line of sight velocities are available, this degeneracy measurement uncertainty on the nth component of
is a serious problem, invalidating such an analysis. the velocity of the ith star. The line of sight mea-
Inafuturepaper,wewillshowthatwithsufficiently surement uncertainties are taken to be 1 km/sec.
4
The derivatives with respect to the parameters θa δM(r1/2)/M(r1/2) δγ(r1/2)
are evaluated at the fiducial NFW model [18] LOS only 69% 4.1
LOS only, c=3 18% 1.6
a=b=1, c=3, r =0.5, ρ =8, β =0 (9)
0 0
Gaia 16% 0.68
where r0 and ρ0 are measured in units of kpc and Gaia, c=3 10% 0.66
107 M(cid:12)/kpc3 respectively. Gaia+TMT 10% 0.27
The parameter θa can be measured with a pre- Gaia+TMT, c=3 7% 0.27
(cid:112)
cision (F−1) . By the chain rule, a quantity q
aa
TABLE I: The precision of measurements of the mass
which depends upon the θ may be measured with
a
within r =375 pc and the dark matter density slope
a precision 1/2
within r that can be expected with only LOS data
1/2
(cid:114) ∂q ∂q andalsoatGaiawithandwithoutTMT.Precisionsare
δq = (F−1) . (10)
ab∂θ ∂θ givenusingasinglecomponentKingprofileforthestellar
a b
distributionwithandwithoutimposingthatc=3. The
We will consider two quantities q. The first is linearized Fisher matrix approach cannot be trusted for
M(r1/2), the mass within the deprojected half-light entries of order or greater than unity.
radiusr =375pc. UsingtheJeansequation,this
1/2
can be determined directly from the stellar density
profile and radial velocity dispersion at r = r , been described above, with and without the condi-
1/2
theirderivatives,andβ andsoisquiterobusttodif- tion c=3. In the last two cases we add a survey of
ferentchoicesofdarkmatterdensityprofile[19,20]. Sculptor by the TMT in 2022.
The second is These results are all summarized in Table I. We
ρ(r) havealsotriedincludingestimatesofr andρ from
γ(r)=−3+4πr3 (11) 0 0
M(r) CDMsimulations,assummarizedinRef.[18]. With
these constraints we find that Gaia can determine
which is essentially a weighted average of the expo-
γ(r ) with a precision of 0.57.
nentsofther-dependenceofthedensityatradiibe- 1/2
tween0andr. Notethat,inthecaseofapowerlaw In the standard ΛCDM paradigm one expects
ρ(r),ityieldsthelogarithmicslope. Ingeneralitde- [1,2]thatadark-matterdominatedgalaxywiththe
pendsonthederivativeofM(r),whichisdetermined luminosity of Sculptor will have γ equal to about -1
less precisely than M(r) itself, but unlike standard whereas other dark matter models, such as a Bose-
definitionsitdoesnotdirectlydependuponthesec- Einstein condensate or giant monopole model, sug-
ond derivative of M(r) and so is nonetheless fairly gestγ closertozero. Ifγ indeedisclosetozero,then
wellconstrained. Belowwewillreportthefractional by assuming the CDM condition c=3 and measur-
uncertainty in M(r ) and the total uncertainty in ing γ ∼0 one may hope to exclude CDM with 2σ of
1/2
γ(r ). confidence when the Gaia mission is completed in 5
1/2
These quantities will be determined in 6 cases. years and with 4σ of confidence when TMT begins
The first case just uses the line of sight velocities. observing in a decade. One should note however
Inthesecondcaseweimposetheconditionc=3by that the CDM simulations on which these conclu-
adding a large number to F . This condition is sat- sions rest consider field galaxies, of which Sculptor
cc
isfied by both cold and warm dark matter models, is not an example.
and in fact by any particulate dark matter model In Ref. [21] it was observed that the Sculptor
with no nongravitational long range interactions. dwarf contains at least two ancient populations of
Thus, while it is certainly less general, if one is in- stars, a metal rich population near the center and a
terested in testing CDM then it suffices to impose more spatially extended metal poor population. In
c = 3 and test to see if γ ∼ −1 as is suggested by Ref. [11] the authors found that the stellar density
dark matter simulations with the low baryonic con- and metallicity distribution is well fit by the sum
tent of the Sculptor dwarf. of two Plummer profiles of projected half-light radii
In the next two cases we include proper motion 167pcand302pc,having53%and47%ofthemem-
measurements from Gaia’s 5 year mission, as has bersrespectively. Wehaveredoneouranalysisusing
5
δM(r )/M(r ) δγ(r )
1/2 1/2 1/2
LOS only 30% 1.3
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Gaia 13% 0.47 tensen, S. H. Oh, A. M. Brooks, T. Quinn and
Gaia, c=3 13% 0.44 S. Shen et al., “Cuspy No More: How Outflows
Affect the Central Dark Matter and Baryon Dis-
Gaia+TMT 9% 0.22
tribution in Lambda CDM Galaxies,” Mon. Not.
Gaia+TMT, c=3 6% 0.21
Roy.Astron.Soc.422(2012)1231[arXiv:1202.0554
[astro-ph.CO]].
TABLE II: As in Fig. I but using a 2-component Plum-
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mer model for the stellar density profile.
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phys. J. 759 (2012) L42 [arXiv:1207.2772 [astro-
measured, as is summarized in Table II.
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