次世代位置天文衛星による次世代位置天文衛星による銀河系ポテンシャル測定銀河系ポテンシャル測定

次世代位置天文衛星による次世代位置天文衛星による銀河系ポテンシャル測定銀河系ポテンシャル測定

T.T. 　　　　 SumiSumi 　　　 　　　 (Nagoya STE)(Nagoya STE) K.V. K.V. Johnston (Columbia)Johnston (Columbia) S.S.　　　　　　 Tremaine (IAS) Tremaine (IAS) D.N.D.N.　　 SpergelSpergel (Princeton)(Princeton) S. Majewski (Virginia) S. Majewski (Virginia) Sumi et al. Sumi et al.

20092009

T.T. 　　　　 SumiSumi 　　　 　　　 (Nagoya STE)(Nagoya STE) K.V. K.V. Johnston (Columbia)Johnston (Columbia) S.S.　　　　　　 Tremaine (IAS) Tremaine (IAS) D.N.D.N.　　 SpergelSpergel (Princeton)(Princeton) S. Majewski (Virginia) S. Majewski (Virginia) Sumi et al. Sumi et al.

20092009

KPKP ：： Taking Measure of the Milky Taking Measure of the Milky Way: Way: Proposed Scope (1999)Proposed Scope (1999)

o Mass Potential of Galaxy (tidal tails & Mass Potential of Galaxy (tidal tails & satellites in halo)satellites in halo) o Mass and Mass Distribution (shape, radial profile) Mass and Mass Distribution (shape, radial profile) of MWof MW

o ““Lumpiness” of the HaloLumpiness” of the Haloo Dynamics of the DiskDynamics of the Disk

o Surface Mass Density (Surface Mass Density (Oort limitOort limit))o Milky Way Milky Way Rotation CurveRotation Curve

o Dynamics of the Central GalaxyDynamics of the Central Galaxyo Dynamics of BulgeDynamics of Bulgeo Orientation and Motions of the BarOrientation and Motions of the Bar

o Fundamental and Legacy MeasurementsFundamental and Legacy Measurementso Proper motions of every known MW Proper motions of every known MW satellite galaxy & globularsatellite galaxy & globularo Proper motions for large number of selected Proper motions for large number of selected open clustersopen clusterso Solar rotation speed, Solar rotation speed, & dynamical & dynamical distance to GCdistance to GC

methodmethod1, generate stars following model density 1, generate stars following model density Σ, mean v and dispersion σ, at R=0-25kpcΣ, mean v and dispersion σ, at R=0-25kpc

2, assign observational err in p2, assign observational err in pphotphot

3, select sample3, select sample

4, observe p4, observe ptritri, μ, v, μ, vloslos with err with err

5, modeling by MCMC5, modeling by MCMC

Spiral arms modelSpiral arms model

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Φ(R,φ, t) = Φa (R)cos m(φ − Ω p t) + c logR

R0

+ φ0

⎡

⎣ ⎢

⎤

⎦ ⎥

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vRs(R,φ, t) = vRa (R)cos m(φ − Ω p t) + c logR

R0

+ φ0

⎡

⎣ ⎢

⎤

⎦ ⎥

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vRa (R) =m(Ω − Ω p )

ΔkΦaF,

Δ = κ 2 − m(Ω − Ω p )[ ]2

Φa: amplitude of spiral armR0: distance to GCm: number of spiral armsk=C/R: radial wave numberp: pattern speed: epicyclic frequencyF: reduction factor16 parameters in total

Potential:

Radial verlocity:

~10% of local disk surface density

Sampling & observationSampling & observationM-giant: MV=-2 magPhotometric parallax: pphot=0.15p

Sample uniform in RIn R=4-20kpc

ptri=10μasμ=0.2+0.6ptri

vlos=1km/s

APOGEE, APOGEE, H-H-band RVs with band RVs with <0.5 km/s for 1-<0.5 km/s for 1-2 x 102 x 105 5 starsstars

Markov Chain Monte CarloMarkov Chain Monte Carlo

Conditional probability:PConditional probability:P

Σ:number density of stars ε: error functionV:volume ~p-4(p-3) S: selection functionUx: phase space distribution

Likelihood:

€

L(x) = Πi=1

N

P(ptrii

,μoi ,v los,o

i | pphoti , l i ,x)

€

=S(pphot , l)

N ldp

dμ

pdv los∑−∞

∞

∫−∞

∞

∫ (p, l)V (p)Ux (μ

p,v los)0

∞

∫

ε(ptri | p)ε(pphot | p)ε(μo |μ)ε(v los,o | v los)€

P =P(pphot , ptri,μo,v los,o | l ,x)

P(Pphot | l,x)

Recovery by MCMC. Recovery by MCMC. (N=850, fit R(N=850, fit R00))

δphot=15%δtri=10μas

Likelihood surface. Likelihood surface. (N=810)(N=810)

68%,95%CL.68%,95%CL.

Accuracy vs. Number of starsAccuracy vs. Number of stars

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δM /(109Msun × 500 /N ) =1.7, 2.7

Accuracy vs. parallax accuracyAccuracy vs. parallax accuracyN=850, fix R0

Disk starsDisk stars

Accuracy v.s. Accuracy v.s. N=850

SummarySummary Our method is immune to bias in sample selectionOur method is immune to bias in sample selection

Next generation astrometoric survey can constrain Next generation astrometoric survey can constrain Mass distribution in ~1% at 4-20 kpc (currently Mass distribution in ~1% at 4-20 kpc (currently ~10% at<8kpc) with N=a few 100~1000~10% at<8kpc) with N=a few 100~1000

δδM M does NOT depend on δdoes NOT depend on δptri ptri until ~100μasuntil ~100μas GAIA does good work.GAIA does good work.

Measure RMeasure R00 in ~2% in ~2%

maxmax should be >60 should be >60

(Knowing the error distribution is important)(Knowing the error distribution is important)

MOA-II1.8m telescopeMOA-II1.8m telescope（（ New Zealand/Mt. John Observatory New Zealand/Mt. John Observatory at NZ, 44at NZ, 44SS ））

Mirror : 1.8mMirror : 1.8mCCD : 8kCCD : 8k xx 10k pix. 10k pix. FOV : 2.2 square deg.FOV : 2.2 square deg.

Mirror : 1.8mMirror : 1.8mCCD : 8kCCD : 8k xx 10k pix. 10k pix. FOV : 2.2 square deg.FOV : 2.2 square deg.

the Galactic Bar structure the Galactic Bar structure the Galactic Bar structure the Galactic Bar structure (face on, from North)

8kpc

G.C.

Obs.

1, 1, Microlensing Optical depth, Microlensing Optical depth,

(Alcock et al. 2000; Afonso et al.2003; Sumi et al. 2003;Popowski (Alcock et al. 2000; Afonso et al.2003; Sumi et al. 2003;Popowski et al. 2004; et al. 2004; HamadacheHamadache et al. 2006;Sumi et al. 2006) et al. 2006;Sumi et al. 2006)

M=1.61010M,

axis ratio (1:0.3:0.2),

~20

1, 1, Microlensing Optical depth, Microlensing Optical depth,

(Alcock et al. 2000; Afonso et al.2003; Sumi et al. 2003;Popowski (Alcock et al. 2000; Afonso et al.2003; Sumi et al. 2003;Popowski et al. 2004; et al. 2004; HamadacheHamadache et al. 2006;Sumi et al. 2006) et al. 2006;Sumi et al. 2006)

M=1.61010M,

axis ratio (1:0.3:0.2),

~20

2.Red Clump Giants2.Red Clump Giants2.Red Clump Giants2.Red Clump Giants Metal-rich horizontal branch starsMetal-rich horizontal branch stars Small intrinsic width in luminosity Small intrinsic width in luminosity function (~0.2mag)function (~0.2mag)

Stanek et al. 1997

=20-30=20-30, axis ratio 1:0.4:0.3, axis ratio 1:0.4:0.3

Streaming motions of the Streaming motions of the barbar

Streaming motions of the Streaming motions of the barbar

Sun

faint brightfaint bright

Vrot=~50km/s

Color Magnitude Diagram