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