Chap.1 Kinematics and Dynamics ofGalactic Stars• Orbits of stars– Orbits in spherical and tri-axial potentials– Orbits in Stäckel potentials, Action integrals• Kinematics of stars– (U,V,W), LSR, solar motion, Galactic constants• Distribution functions of stars– Schwarzschild, modeling distribution functions• Jeans equations– Jeans theorem, spherical system, asymmetric drift• Virial theorem1

1. Orbits of stars
Galaxy structure: superposition of stellar orbits• Orbits in a spherical potential Φ(r)L=r×dr/dt =const.
⇒ confined to the 2D orbital plane ⇒ coordinates (r, φ)
E.g. Kepler potential (by a point mass: M)bequation for
an ellipse (orbit)
semi-major axis

eccentricity
pericenter

apocenter

φ-φ0aaeKepler motion(orbit in a point-mass potential)

Δφ=2π over one periodof radial oscillation⇒ closed orbit

Orbit inside a uniform sphereM(<r) = 4πρr3/3F(r) = - GM(<r)/r2∝-rFx ∝ -x, Fy ∝-ySimple oscillator

Δφ=πclosed orbit

Orbit in a gravitational potential
provided bya general spherical mass distribution
π<Δφ<2π
Rosette orbit(non-closed)

3

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

loop orbits

loop orbits

box orbits
Orientation of a bar

Orbits in a (nonrotating) triaxial potential
Box orbit

Outer long-axis
tube orbit

Short-axis tube orbit

Inner long-axis
tube orbit

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=1,3Only regular orbits exist (de Zeeuw 1985, MNRAS, 216, 273)
explicit expressions for Ｉ1(x,v) (=E), Ｉ2(x,v), Ｉ3(x,v)• Action integrals

1Ji pi dqi
– Adiabatic invariance2– J ≥ 0 for bound orbits
V   d xd v   d Jd   2   d J– Phase volumeJi(E,I2,I3) i=1,3i

3

D33D

333

Dpi

Canonical transformation (q,p) → (θ,J)
HHi  i , J i  0J i
 iqi
6Action integrals for a Kepler motionr

Jr r

1 max

2 max

p dr  rrminrminJ 1 max

 minJ L2 p

2sin 
21p d  p
2

 1
L2 GMGME 2 dr 
 L  L
 1
22r
r
2E
 1 ed  L  J 

L=|Jφ|+Jθ : adiabatic invariance⇒ Orbital eccentricity: e is an adiabatic
invariance as well(a conserved quantity when the change ofa gravitational potential is sufficiently slow
compared to its dynamical time scale)7

・2D non-axisymmetric (x,y)⇒(u,v)

・3D triaxial (,,)loopBSOIbox

1Jr pu du2
1Ja pv dv
2

8

2. Kinematics of stars
Description of stellar kinematics observed from the Sun• Observed kinematical quantities– 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 () /yr
or mas (milli-arcsec: 10-3) /yr]

north poleTangential velocity:
Vtan(km/s)=4.74 D(pc) μ(arcsec/yr)= 4.74 D(kpc) μ(mas/yr)– (α,δ),D,Vrad,(μα,μδ)→3d position＋3d velocity→6d phase space

9Astrometry Satellites: Hipparcos & Gaia
1989~19932013~2021

Hipparcos

Gaia

Magnitude limit
CompletenessBright limit
Number of objects

12 mag
7.3 – 9.0 mag
0 mag
120,000

Effective distancelimit
Quasars
Galaxies
Accuracy

1 kpc
1 (3C 273)
None
1 milliarcsec

Photometry
photometry
Radialvelocity
Observing
programme

2-colour (B and V)
None
Pre-selected20 mag
20 mag
6 mag
26 million to V = 15250 million to V = 18
1000 million to V = 2050 kpc
500,000
1,000,000
7 µarcsec at V = 1010 – 25 µarcsec at V = 15300 µarcsec at V = 20
Low-res. spectra to V = 20
15 km s-1 to V = 17
Complete and unbiasedGaia:
10as = 10% error @distance 10kpc, 10as/yr = 1km/s @20kpc
Hipparcos: 1mas = 10% error @distance 100pc, 1mas/yr = 5km/s @ 1kpc

10The Local Standard of Rest (LSR) & UVW velocities••••The LSR: a circular orbit at (R=R0 , Vrot = Θ0)(U,V,W): star’s velocities relative to the LSR(Usun,Vsun,Wsun): solar motion relative to the LSR
Observed star’s velocities from the Sun (heliocentric)(U-Usun, V-Vsun, W-Wsun) ≃ (VR, VΦ - Θ0, Vz) in cylindrical coords.
Galactic constants: Θ0, R0, (Usun,Vsun,Wsun)

Determination of Galactic Constants
LSR(Θ0, R0)• Rotational velocity of the LSR: Θ0– Oort constants (A,B) → Ω0=A-B → R0 given → Θ0=R0 (A-B)– Motion relative to Pop II system, but <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 tracers (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* Delhaye 1965 using A stars, K giants, M dwarfs
Solar motion(Usun, Vsun, Wsun) = (-9, 12, 7) km/s, (l,b)=(53,25)(Usun, Vsun, Wsun)* More recent result (Schönrich +10, Coşkunoğglu+11)(Usun, Vsun, Wsun) = (-11.10, 12.24, 7.25) km/s

Determination of the rotation curve

αVrad  AD sin 2l

Vrad   cos    0 sin l

Vtan  D( A cos 2l  B)

Vtan   sin    0 cos lsin l / R  sin(90   ) / R0  cos  / R0R cos(90   )  R sin   R0 cos l  DVrad  (   0 ) R0 sin lVtan  (   0 ) R0 cos l  D
 R(RD  10<<1 ⇒0 /0-R)/RA cos 2l  B
l 
4.74
1    d  A   0  2  R0  dR  R0 1   0  d  B  
2RdR R0 
A, B: Oort Constants 0 0   0 / R0  A  B d   ( A  B )
dR

 R0
Vrad is max at R = R0sin l⇒Θ（R)𝜕𝑉= 0 ⇒ 𝜕𝑅/𝜕𝐷 = 0𝜕𝐷
𝑅 = 𝐷 + 𝑅 − 2𝐷𝑅 cos 𝑙⇒ 𝐷 = 𝑅 cos 𝑙 ⇒ R=R0sinl

Hipparcos proper motions toward Galactic longitude
for 220 Cepheids(Feast & Whitelock 1997)μℓ
AB

μℓ ∝ Vtan/D = Acos 2ℓ +B ⇒ A, B

14

Rotation curve of the Milky Way
Sofue et al. 2009, PASJ, 61, 227

Vrot
km/s

Scaled with R0=8kpc, 0=200km/s

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

15Effect of disk thickness on rotation curveat the disk plane
Vrot provided by an exponential disk: (R,z) = 0exp(-R/h)z(z)16Geometric determination of thedistance to the Galactic Center
orbital eclipse vs. angular sep.
→ R0
R0 =7.94±0.42kpc

Eisenhauer et al. 2003, ApJ, 597, L121

17

Using more orbits
Gillessen et al. 2009:16 years of monitoring
the orbits of 28 stars
R0 = 8.33±0.35 kpc
Reid & Brunthaler 2004:
ℓ(SgrA*)=6.379±0.026mas/yr⇒(0+Vsun)/R0=30.24 km/s/kpc
Then if R0=8.3 kpc & Vsun=12.24 km/s
⇒0=239 km/s
18

(U,V) velocities for nearby stars ( VR , Vϕ －Θ0 )Distance from the Sun
D<4kpcΘ0( VR )( Vϕ －Θ0 )-0.8<[Fe/H]≤-0.6× -1.7<[Fe/H]≤-0.8[Fe/H]≤-1.7

19

( VR )( Vϕ －Θ0 )( VZ )σUσVσW<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
20

3. Distribution function of stars• Schwarzschild (1907) modelvelocity dispersion

Velocity ellipsoidσi axis does not necessary matchthe direction of (U,V,W)

σ3
σ1

Vertex deviation

σ2VU

21

Modeling distribution functions
phase space
EI2

I3

Integrals of motions

Integrals of motions, I, are the solutions tothe steady-state collisionless Boltzmann equation⇒ Jeans Theorem: f(E,I2,I3), f(J1,J2,J3)

22

Some simple cases• f(E) isotropic velocity distribution• f(E,Lz) Lz=Rv in axisymmetric Φ(R,z)– σR2=σz2 (but ≠σ2) anisotropic– but σR2≠σz2 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, v=vsinηsinψ
– vt2=vθ2+v2=v2sin2η, L=|rvt|=|rvsinη|– σθ2=σ2≠σr2 anisotropic– β(r)=1- σθ2/σr2, β≤1GC
β>0: radially anisotropic
β<0: tangentially anisotropic

β>0

<0r

These velocity anisotropies reflect past merging/accretion histories

Velocity anisotropy parameter β(r)β>0<0rGC

Radial anisotropy:β(r)=1- (σθ2+σϕ2)/2σr2

βDip at r~20kpc
K12: Kafle et al. 2012(using 4664 BHBs from SDSS)
Red cross: Deason et al. 2013
(halo stars with HST proper motions)
Sunr (kpc)24Recent results on β(r)Bird et al. 2019using 5600 K giants fromLAMOST and Gaia DR2-1.8<[Fe/H]<-1.3: boundary
at r ~ 20kpc[Fe/H]<-1.8: no dip

Loebman et al. 2019using simulation resultsby Bullock & Johnston 2005
(hierarchical clustering process)β ~ 0.7
Radially anisotropic
over entire radii
Presence of temporal dips

4. Jeans equations
Continuity eq.
in phase space

Boltzmann equation

moment over vi

Jeans equations(hydrodynamical description)
26

Jeans theorem
Ｉ is a solution to steady-statecollisionless Boltzmann eq.
f(Ｉ(r,v)): a solution to steady-state collisionless Boltzmann eq.

・Strong Jeans Theorem
Potential 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, I2=Lz2/2
f(J1,J2,J3)
Ｉ3

J3

Ｉ2

Ｉ1Ji(Ii)

J2
J127

Nearby stars in (E,Lz,I3) phase space
SDSS-DR7 Calibration Stars + Gaia DR2
Carollo & Chiba 2021
(R,z) distributionof sample starsz (kpc)

ESunR (kpc)

LzWhat is the third integral, I3 ?
(2I3)1/2

rapoZmax
Rapo

arcsin (Z_max/r_apo) (deg)~ orbital inclination angle θ29

Nearby stars in (E,Lz,I3) phase space
SDSS-DR7 Calibration Stars + Gaia DR2(grey: all stars)

Carollo & Chiba 2021
(2I3)1/2

EE

halo30Simple cases for Jeans equations (I)• Spherical system

Example: Line-of-sight velocity dispersion profile in MW dwarf satellites

σlosLeo I

σlos(R)
σr(r)CarM(r)Dark matterSimple cases for Jeans equations (II)• Axisymmetric system

Example 1: R direction
At z~0Circular velocity

Asymmetric drift
is large (old stars)<is small (young stars)⋍

Example 2: z direction
At z~0>>1

vertical equilibrium5. Virial theoremx

Steady stateM
Rg: gravitational radius

Rh: half-mass radius ⋍ 0.4Rg