Chap.1 Kinematics and dynamics of a stellar system

• Orbits of stars– Orbits in spherical and non-spherical potentials– Orbits in Stäckel potentials, Action integrals

• Kinematics of stars– (U,V,W), LSR, solar motion, Galactic constants– Galactic gravitational potential

• Distribution functions of stars– Schwarzschild, modeling distribution functions

• Jeans equations– Jeans theorem, spherical system, asymmetric drift– Virial theorem 1

1.1 Orbits of stars

• Spherical potential Φ(r)L=r×dr/dt=const. ⇒ confined to the orbital plane– E.g. Kepler potential (by a point mass: M)Φ(r) = －GM/r, L= r2dφ/dt: integral of motiona(1-e2)/r = 1+e cos(φ-φ0)

equation for an ellipse (orbit)a=L2/GM(1-e2): semi-major axis,e: eccentricity, φ-φ0: true anomaly

peri-center: rperi=a(1-e), apo-center: rapo=a(1+e)e=(rapo-rperi/(rapo+rperi)

a

b

ae

φ-φ0

2

Galaxy structure: superposition of many stellar orbits

3

Δφ=2π over one period ofradial oscillation⇒closed orbit Δφ=π

closed orbit

Kepler motion(orbit in a point-mass potential)

Orbit inside a uniform sphere

M(<r) = 4πρr3/3F(r) = - GM(<r)/r2

∝ - rFx ∝ -x, Fy ∝-ySimple oscillator

π<Δφ<2πRosette orbit(non-closed)

Orbit in a gravitational potentialprovided bygeneral spherical mass distribution

4

・2D axisymmetric Φ(R) ・2D non-axisymmetric Φ(x,y)(Bar potential)

loop orbits

loop orbits

box orbitsOrientation of a bar

Box orbit Short-axis tube orbit

Outer long-axistube orbit

Inner long-axistube orbit

Orbits in a (nonrotating) triaxial potential

Statler 1987, ApJ, 321, 1135

• Stäckel potential– Hamilton-Jacobi eq. is separable ⇒ eq. of motions is solvable

independently in each spatial coordinate– Integral of motion Ｉi(x,v) i=1,3

Only regular orbits exist (de Zeeuw 1985, MNRAS, 216, 273)explicit expressions for Ｉ1(x,v) (=E), Ｉ2(x,v), Ｉ3(x,v)

• Action integralsJi(E,I2,I3) i=1,3

– Adiabatic invariance– Ji ≥ 0 for bound orbits– Phase volume

Canonical transformation (q,p) → (θ,J)

( )∫ ∫ ∫===D D D

JdJddvxddV 333333 2πθ

∫= iii dqpJπ21

0, =∂∂

−==∂∂

=••

iii

ii

HJJH

θωθ qi

pi

6

Action integrals for a Kepler motion

7

∫

∫

∫∫

==

−=−=

−

−=−=+−==

φφφ

φ

θ

θ

φθ

φπ

θθπ

ππ

pdpJ

JLdp

LJ

eLL

EGMdr

rGM

rLEdrpJ

r

r

r

rrr

21

sin1

11

122

21

max

min

max

min

max

min

2

22

22

2

L=|Jφ|+Jθ : adiabatic invariance⇒ Orbital eccentricity: e is an adiabatic invariance as well(a conserved quantity when the change of a gravitational potential is sufficiently slowcompared to its dynamical time scale)

8

B S

O I

loop

box

dvpJ

dupJ

va

ur

∫

∫

=

=

π

π

21

21

・2D non-axisymmetric Φ(x,y)⇒Φ(u,v) ・3D triaxial Φ(λ,µ,ν)

• Observed kinematics– Line of sight velocity: Vrad,– Distance: D (pc) = 1 / π (arcsec)

or D (kpc) = 1 / π (mas, milli-arcsec) – Proper motion: μ= [(μαcosδ)2+(μδ)2]1/2

[unit: arcsec (″) /yror mas (milli-arcsec: 10-3″) /yr]

With Distance:D given,tangential velocity:Vtan=4.74 D(pc) μ(arcsec/yr)

= 4.74 D(kpc) μ(mas/yr)– (α,δ),D,Vrad,(μα,μδ)

→3d position＋3d velocity

1.2 Kinematics of stars

9

north pole

Description of stellar kinematics observed from the Sun

10

Hipparcos Gaia Magnitude limit 12 mag 20 mag Completeness 7.3 – 9.0 mag 20 mag Bright limit 0 mag 6 mag Number of objects 120,000 26 million to V = 15 250 million to V = 18 1000 million to V = 20 Effective distance

1 kpc 50 kpc

Quasars 1 (3C 273) 500,000

Galaxies None 1,000,000 Accuracy 1 milliarcsec 7 µarcsec at V = 10 10 – 25 µarcsec at V = 15 300 µarcsec at V = 20 Photometry

2-colour (B and V) Low-res. spectra to V = 20

Radial velocity None 15 km s-1 to V = 17 Observing

Pre-selected Complete and unbiased

1989~1993 2013~2021

Astrometry Satellites

11Gaia: 10µas = 10% error @distance 10kpc, 10µas/yr = 1km/s @20kpcHipparcos: 1mas = 10% error @distance 100pc, 1mas/yr = 5km/s @ 1kpc

• Description of 3d velocity– Velocity component of a star: (Π,Θ,Z)

in rectangular coordinates at the Sun– The Local Standard of Rest (LSR) (a circular orbit around the GC)

(Π,Θ,Z)=(0,Θ0,0)– Full space motion of a star : = LSR

+ motion relative to LSR (U,V,W)(U,V,W) = (Π,Θ-Θ0,Z) Θ0:given (~220km/s)

+ solar motion (Usun,Vsun,Wsun) (deviation from a circular orbit)(Usun,Vsun,Wsun) = (Πsun, Θsun-Θ0,Zsun)

– Actually observed motion (heliocentric)(u,v,w) = (U-Usun, V-Vsun, W-Wsun)

Galactic constants: Θ0, R0 & (Usun,Vsun,Wsun) 12

Galactic plane

Galactic center

Sun

Π

Θ

Ζ

• Determination of (Usun, Vsun, Wsun)<U>=0, <W>=0 by definition, <V>≠0 (<Vφ>≠Θ0)⇒ using many stars,(<u>,<v>,<w>) = ( - Usun, <V> - Vsun, - Wsun ) is estimatedUsun= -<u>, Wsun= -<w>, Vsun= -<v> in the limit of <V>=0V’sun≡ -<v> = Vsun-<V> = Vsun + cS2 (S2: velocity dispersion)

S2 → 0: Vsun is determined* Delhaye 1965 using A stars, K giants, M dwarfs

(Usun, Vsun, Wsun) = (-9, 12, 7) km/s, (l,b)=(53,25)* Feast & Whitelock 1997 using 227 Cepheids (HIP)

(-9.3, 11.2, 7.6) km/s* Dehnen & Binney 1998 using 11865 single MS stars (HIP)

(-10.0, 5.3, 7.2) km/s* More recent result (Schönrich +10, Coşkunoğglu+11)

(-11.10, 12.24, 7.25) km/s

VU

WVelocity relative to the LSR

13

(velocity dispersion)

Local stellar kinematics from Hipparcos data

Dehnen & Binney 1998, MN, 298, 387

-

14

Distance from the SunD<4kpc

-0.8<[Fe/H]≤-0.6

[Fe/H]≤-1.7× -1.7<[Fe/H]≤-0.8

(U,V) velocities for nearby stars

15

σU σV σW <V>[Fe/H]≤-1.7 150 km/s 110 km/s 100 km/s -200 km/s-0.8<[Fe/H]≤-0.6 60 km/s 60 km/s 40 km/s -30 km/s

16

Determination of Galactic Constants• Rotational velocity of the LSR: Θ0

– Oort constants (A,B) → R0→ Θ0=R0(A-B)– Motion relative to Pop II system, <Θ>=0 is assumed– Proper motion of Sgr A* → R0→ Θ0

if Sgr A* is fixed at the center and the LSR has Θ0=220km/s, then Θ0=4.74Dμｌ→ proper motion along Galactic long.: μl ~ 5.8 mas/yr

• Solar position: R0– The center of halo tracer populations (GCs, RR Lyr, Mira variables

in the bulge)– Parallax of Sgr A*: p(mas) = (D/kpc)-1 = 0.1 mas– Stellar motions near Sgr A* (“binary method”) Salim & Gould 1999

• Kerr & Lynden-Bell (1986, MN, 221, 1023)Θ0=220 km/s, R0=8.5 kpc (IAU standards)

• Recent trend:Θ0 > 220 km/s, R0 ~ 8 kpc

17

Determination of the rotation curve

Θ

+Θ

−≡

Θ

−Θ

≡

+=

+==

0

0

0

0

0

0

tan

21

21

74.42cos

)2cos(2sin

R

R

l

rad

dRd

RB

dRd

RA

BlABlADV

lADV

µ )(

/

0

000

BAdRd

BAR

R

+−=

Θ

−=Θ=Ω

⇒<<⇒Ω−Ω−Ω=

Ω−Ω=−==−

=+=Θ−Θ=Θ−Θ=

1/cos)(sin)(

cossin)90cos(/cos/)90sin(/sin

cossinsincos

0

00tan

00

0

00

0tan

0

DRDlRV

lRVDlRRR

RRRllVlV

rad

rad

αααα

αα

α

lRDlDRRDR

DRDVrad

coscos2

0//

0

020

22

=⇒−+=

=∂∂⇒∂∂

19

220個のセファイド星の銀経 l 方向の銀経依存性(Feast & Whitelock 1997)

μ ∝ Vtan/D = Acos 2l +B ⇒ A, B 決定

20

Rotation curve of the Milky Way

See also, Gunn, Knapp, Tremaine 1979, AJ, 84, 1181; Fich & Tremaine 1991, ARAA, 29, 409

Sofue et al. 2009, PASJ, 61, 227

Scaled with R0=8kpc, Θ0=200km/s

Feast & Whitelock 1997, MN, 291, 683

Galactic kinematics of Cepheidsfrom HIPPARCOS proper motions

Eisenhauer et al. 2003, ApJ, 597, L121

A Geometric Determination of theDistance to the Galactic Center

orbital eclipse vs. angular sep. → R0

B

A

μl

R0 =7.94±0.42kpc

21

22

Gillesen et al. 2009:16 years of monitoringthe orbits of 28 starsR0 = 8.33±0.35 kpc

Reid & Brunthaler 2004:µl(SgrA*)=6.379±0.026mas/yr

⇒(Θ0+Vsun)/R0=30.24 km/s/kpcThen if R0=8.3 kpc & Vsun=12.24 km/s⇒Θ0=239 km/s

Recent results

1.3 Distribution function of stars

• Schwarzschild (1907) model

( )22

23

23

22

22

21

21

3212/3

321321321 222exp

)2(),,(

iii vv

vvvdvdvdvdvdvdvvvvf

−≡

++−=

σ

σσσσσσπ

σ1

σ3

σ2Velocity ellipsoid

σi does not necessary match the direction of (U,V,W)

U

VVertex deviation

23

Modeling distribution functions

( )[ ]

( )( ) ( )222

333

32

33

,

1,1,)(

),(),,(,,),(

iiijijijjiiij

jijiii

vvvvvvvvvv

vfdvvn

vvvfdvn

vvfdxn

vxIvxIvxEvxddvxf

−=−=−−=

=== ∫ ∫ ∫σσ

(x,v) phase space

Integrals of motions

f(E,I2,I3) Jeans Theorem f(J1,J2,J3)

24

Some simple cases

• f(E) isotropic velocity distribution• f(E,Lz) Lz=Rvφ in axisymmetric Φ(R,z)

– σR2=σz

2 (but ≠σφ2) anisotropic

– but σR2≠σz

2 near the Sun → presence of Ｉ3(σU,σV,σW) ≈ (150,110,100) km/s for halo stars

• f(E,L) L: total angular momentum– vr=vcosη, vθ=vsinηcosψ, vφ=vsinηsinψ– vt

2=vθ2+vφ2=v2sin2η, L=|rvt|=|rvsinη|

– σθ2=σφ

2≠σr2 anisotropic

– β(r)=1- σθ2/σr

2, β≤1β>0: radially anisotropicβ<0: tangentially anisotropic 25

rGCβ>0 β<0

BHB stars as tracers of halo velocity fieldSommer-Larsen et al. 1997, ApJ, 481, 775

Line of sight velocitydistributionat four directions

26

Velocity anisotropy vs. radius

Sommer-Larsen et al. 1997, ApJ, 481, 775

22φθ σσσ +=t

β> 0radiallyanisotropic

β< 0tangentially anisotropic

Sun

27

recent resultsK12: Kafle et al. 2012

(using 4664 BHBs from SDSS)Red cross: Deason et al. 2013

(halo stars with HST proper motions)

1.4 Jeans equations ( )

( ) ( )

( )

( )( ) jijijjiiij

i i

ij

ji i

ji

j

jijijj

i jji

i

j

ii

i

vvvvvvvv

xxxv

vtv

vfdvvvvvfdvv

xvv

xtv

vxt

−=−−≡

∂

∂−

∂Φ∂

−=∂

∂+

∂

∂

==

=∂Φ∂

+∂∂

+∂

∂

=∂∂

+∂∂

∑∑

∫ ∫

∑

∑

2

2

33 1,1

0

0

σ

νσννν

νν

ννν

νν

0=∂

∂⋅Φ∇−∇⋅=

∂

∂+∇=

vIIv

dtvd

vI

dtxdI

dtdI Ｉ is a solution to steady-state

collisionless Boltzmann eq.

f(Ｉ(x,v)): a solution to steady-state collisionless Boltzmann eq.

28

0=∂

∂⋅Φ∇−∇⋅+

∂∂

vffv

tf

Boltzmann equation

Jeans equations

Jeans theorem

• Strong Jeans theoremPotential Φ allowing only regular orbits(no resonance among 3 orbital frequencies)→3 isolating integrals→ＤＦ depends only these 3 integralsf(Ｉ1,Ｉ2,Ｉ3) Ｉ1=E f(J1,J2,J3) Ji(Ｉi)

Ｉ1

Ｉ3

Ｉ2

J1

J3

J2

29

• Spherical system

• Asymmetric drift

( )

∫∞

−=⇒=

−=Φ

−=+

rr

rr

drrr

rGMrvconst

rrGM

drd

rv

drvd

ββ ννβ

βνν

22

22

2

22

)(.

)(21

( ) ( )( ) ( )

222

2

2

2

2222

222

lnln1

0

Rc

zR

R

R

RRc

RzRR

vvV

zvv

vR

Rv

vvvV

RRvv

vvz

vR

∝−⇒

∂∂

−∂

∂−−=−

=∂Φ∂

+−

+∂∂

+∂∂

φ

φφ

φ

νσ

νννν 2Rv2Rv

φvVc >

φvVc ≈

is large (old stars)

is small (young stars)

Application to the MW andexternal galaxies (later…)

Simplified cases for Jeans equations

30