The Local Milky Way Color-Magnitude Diagram
15,630 stars
M0
red giants⇒
Distances from Hipparcos
trigonometric parallaxes
F0 G0 K0
⇐ red clump
A0
Hertzsprung gap
B0
d < 100 pc
age differences
Half of stars with
MV > 10 are not
detected.
metallicity differences
white dwarfs
J.M. Lattimer
AST 346, Galaxies, Part 2
Local Stellar Luminosity Function
Φ(M) =
number stars MV ±1/2
volume over which seen
Ψ(M) = ΦMS (M) Θ
τgal
τMS (M)
LV Φ(MV )
K giants
total
Θ(x) = 1 if x ≤ 0
Θ(x) = x if x > 0
τgal ' 10 Gyr
A, F
MS
MΦ(MV )
B, O
Dim stars hard to find; Bright
stars are rare.
Stars not uniformly distributed.
Stars in binaries mistaken for
brighter single stars.
J.M. Lattimer
⇐Wielen dip
• Hipparcos
◦ Reid et al. (2002)
AST 346, Galaxies, Part 2
Stellar Initial Mass Function
ξ(M)∆(M)= number stars born with mass between M and M + ∆M
−2.35
ξ0
M
Salpeter initial mass function ξ(M) = M
M
1.35 1.35 R Mu
M
ξ0
`
1− M
Total number: M` ξ(M)dM = 1.35 M`
Mu
0.35 0.35
R Mu
M
ξ0 M M`
1 − Mu
Total mass: M` Mξ(M)dM = 0.35
M`
2.15 2.15
R Mu
ξ0 L
M`
Mu
1 − Mu
Total luminosity: M` Lξ(M)dM = 2.15 M
Pleiades cluster
Salpeter mass function
ξ(M)
Mξ(M)
J.M. Lattimer
AST 346, Galaxies, Part 2
Distances From Motions
If Vr and Vt measured, and how they are related, we can find d.
Vt = µd,
µ(marcsec/yr) = Vt (km/s)/(4.74d(kpc))
Example: Distance to Galactic center, if orbit edge-on and perpindicular
2a = (s/1 arcsec)(d/1 pc) AU,
P 2 a3 = 4π 2 G
MBH
2
2
Vp (1−e) = Va (1+e), Vp −Va = 2G MBH a−1 (1 − e)−1 − (1 + e)−1
6
2a = sd
Galactic center
?
J.M. Lattimer
AST 346, Galaxies, Part 2
Distances From Motions
Example: Distance to supernova ring, if ring is circular
t− = R(1 − sin i)/c,
t+ = R(1 + sin i)/c
∆t = tA − t0 = t−
∆t = 0
@
I
@SN 1987a
-
J.M. Lattimer
∆t = tB − t0 = t+
AST 346, Galaxies, Part 2
Spectroscopic and Photometric Parallax
Assume stars of same spectral type and Z
Cheaper alternative is to estimate
have equal L.
spectral type from color, other
Correct for dust reddening.
indicators determine dwarf/giant.
Works to 10% for main sequence stars,
Works best for clusters, where
60% for K giants.
reddening is easier to determine.
Needs lots of observation time.
Photometric distances towards South Galactic pole
V − I colors of 12,500 stars with mV < 19
nZ ∝ e −|z|/hz
hhhh
J.M. Lattimer
5 < MV < 6
hhhh
hh
halo→
AST 346, Galaxies, Part 2
Mass-to-Light Ratio of Galactic Disc
hR,S is scale length of object of type S, hz,S is scale height.
nS (R, z) = nS (0, 0)e −R/hR,S e −|z|/hz,s
ΣS (R) = 2nS (0, 0)hz,S e −R/hR,S ,
IS (R) = LS ΣS (R)
2
LD,S = 2πIS (0)hR,S
is disc luminosity
Milky Way: LD ' 1.5 × 1010 L in V band, hR ' 4 kpc
ID (4 kpc) ' 20 L pc−2 ,
σD ' 40 − 60 M pc−2
M/LV = σD /ID ' 2 − 3 in disc; but about 1 for MS stars near the Sun
M/LV ∼ 0.74 for all stars except white dwarfs and gas
M/LV ∼ 5 from Galactic rotation
J.M. Lattimer
AST 346, Galaxies, Part 2
Analyzing Nearby Stars
I
Star Formation Rate
I
I
I
I
M/L ∼ 2, LD ∼ 1.5 × 1010 L =⇒ M ∼ 3 × 1010 M .
τgal ∼ 10 Gyr, so if half of gas is returned to stars, star formation
rate is 3 − 6 M /yr.
Gas mass is 5 − 10 × 109 M , enough for 1–3 Gyr.
V/V max test – a measure of the uniformity of a spatial distribution
I
I
An average of the ratio of the volume V from which an object could
be seen divided by the volume Vmax defined by the maximum
distance at which an object in the sample is detectable.
Assume sample is of equally luminous objects and limited by a
brightness limit which corresponds to the distance zmax and the
volume Vmax . A brighter object with m < mmax originates from a
smaller distance z < zmax
D. E
R
z3
3
zmax
V
Vmax
I
V
V max
I
When n(z) is uniform, hV/Vmax i = 1/2
=
=⇒
J.M. Lattimer
=
zmax 5
z dz
0R
zmax 2
z dz
0
3
zmax
AST 346, Galaxies, Part 2
Stellar Velocities and Scale Heights
Stars show the cumulative effects of acquiring random gravitational pulls
from the lumpy disk: molecular clouds and dense star clusters.
This translates into larger scale heights for older objects.
Note that the Sun must be moving upwards at 7 km s−1 .
• Z > 0.25Z
◦ Z < 0.25Z
Nearby F and G stars
J.M. Lattimer
AST 346, Galaxies, Part 2
Stellar Velocities and Scale Heights
J.M. Lattimer
AST 346, Galaxies, Part 2
Clusters and Associations
Disk stars are born from clouds large enough to gravitationally collapse.
I
Gould’s Belt (a Moving Group)
A ring centered 200 pc away
containing the Sun and stars
younger than 30 Myr. They lie in
a layer tilted by 20◦ about a line
along the Sun’s orbit at ` = 90◦ .
I
Open clusters
100 L < L < 3 × 104 L
M/L < 1M /L
0.5 pc < rc < 5 pc
σs < 0.5 km s−1
Globular clusters
104 L < L < 106 L
1M /L < M/L < 2.5M /L
0.5 pc < rc < 4 pc
25 pc < rt < 85 pc
4 km s−1 < σs < 20 km s−1
I
J.M. Lattimer
Hipparcos data
MV < 3, 100 pc < d < 500 pc
Gould’s Belt
IRAS data
M7
AST 346, Galaxies, Part 2
M22
blue giants, ≈ 5 M
(m − M)0 = 5.6
•
← binaries
↓
Pleiades
100 Myr isochrone
100 Myr, w/o dust reddening
16 Myr isochrone
J.M. Lattimer
AST 346, Galaxies, Part 2
pre-main sequence stars
Distances and Ages of Clusters
⇓
Globular Clusters
[Fe/H]= −0.71
[Fe/H]= −2.15
47 Tuc
Horizontal
Branch
Horizontal
Branch
13 Gyr
SM
C
re
d
gi
a
nt
s
12 Gyr
M92
12 ± 2 Gyr
J.M. Lattimer
AST 346, Galaxies, Part 2
13 ± 2 Gyr
Globular Cluster Distribution
Metal-poor [Fe/H]< −0.8
Metal-rich
` = 90◦
b = 0◦
` = 20◦
♦
disrupted dwarf galaxy remnants
Metal-poor clusters have highly eccentric, random orbits
and do not share the Galactic disk rotation.
Metal-rich clusters share Galactic rotation like stars in the thick disk.
J.M. Lattimer
AST 346, Galaxies, Part 2
Halo Properties
I
Oldest stars in halo a result of Galactic cannibalism.
I
I
I
I
I
I
Gravity slows galaxies during encounters and leads to mergers.
The Sagittarius dwarf galaxy is the closest satellite galaxy, and it is
partially digested.
ω Cen is a bizarre globular cluster with non-uniform metallicity;
possibly the remnant core of a digested satellite galaxy.
Magellanic Clouds will share the fate of Sagittarius dwarf galaxy in
3–5 Gyr.
Some believe the blue horizontal branch globular clusters joined the
Milky Way after their galaxies were cannibalized.
Metal-poor halo consists of globulars and metal-poor stars down to
[Fe/H]= 10−5
I
I
I
I
I
Most globulars have dissolved, forming many halo stars.
Palomer 5 has lost 90% of its stars and has tidal tails extending 10◦ .
Metal-poor halo stars form moving groups, including some carbon
stars and M giants stripped from Satittarious dwarf galaxy.
Total mass of halo metal-poor stars is about 107 M .
Halo is somewhat flattened but rounder than bulge.
J.M. Lattimer
AST 346, Galaxies, Part 2
Palomar 5
J.M. Lattimer
AST 346, Galaxies, Part 2
Sagittarius Dwarf Galaxy
J.M. Lattimer
AST 346, Galaxies, Part 2
The Halo as an Ensemble of Disrupted Satellites
J.M. Lattimer
AST 346, Galaxies, Part 2
The Bulge and Nucleus
I
Bulge best studied with infrared light
due to dust (31 mag extinction).
I
Bulge accounts for 20% of Galaxy’s
light with a size of order 1 kpc.
Bulge is not dense inner portion of halo:
stars are old but have average metallicty
> 0.5Z , Zmax ∼ 3Z .
I
I
Average rotation speed 100 km s−1
(half disk), stars have large random
motions.
I
Close to the center are dense, active
star forming regions.
I
Center has a 2 pc radius torus with
106 M of gas surrounding a
3 × 107 M star cluster within 1000
(0.2 pc) including > 30 massive stars.
I
Innermost stars < 0.05 pc from central
black hole, 4 × 106 M (Sag A∗ ).
J.M. Lattimer
AST 346, Galaxies, Part 2
Sag A∗
R0 = 8.28 ± 0.1 ± 0.29 kpc
MBH = 4.30 ± 0.20 ± 0.30 × 106 M
J.M. Lattimer
AST 346, Galaxies, Part 2
Galactic Rotation
Differential rotation in Galaxy noticed by
1900 and explained by Oort in 1927.
GC
?
φ
J.M. Lattimer
AST 346, Galaxies, Part 2
Geometric Relations
V0
R0 sin `
R0 cos `
φ+`+α
Vr (R, `)
=
=
=
=
=
V (R0 )
A
R sin(90◦ + α) = R cos α
R sin α + d
B
90◦
A+B
V (R) cos
α − V0 sin ` A−B
V (R) V0
= R0 sin `
−
R
R0
Vt (R, `) = V (R) sinα − V0 cos ` V (R) V0
V (R)
= R0 cos `
−
−d
.
R
R0
R
≡
≡
=
=
0
R V (R)
−
2
R
R0
1
0
− [RV (R)]R0
2R
−V (R)0R0
V0 /R0
When d << R,
0
V (R)
≈ R0 sin `
(R − R0 ) = d A sin(2`)
R
R0 0
V (R)
V (R)
(R − R0 ) − d
≈ R0 cos `
R
R
R0
= d[A cos(2`) + B]
Vr
Vt
J.M. Lattimer
AST 346, Galaxies, Part 2
φ
Galactic Rotation – A Simple Model
V (R) = V0 , equally luminous clouds in rings of radii 2, 3 . . . 14 km
spaced every degree in φ. Dot sizes indicate relative brightnesses.
J.M. Lattimer
AST 346, Galaxies, Part 2
Galactic Rotation Curve
For inner Galaxy, R < R0 , employ tangent point method. Vr (d, `) has a
maximum along the line-of-sight at the tangent point where α = 0:
R = R0 sin `,
V (R) = Vr + V0 sin `.
Method fails for R < 0.2R0 since gas follows oval orbits in Galactic bar.
For outer Galaxy, must
use spectroscopic or photometric parallax
measurements. J.M. Lattimer
AST 346, Galaxies, Part 2
Galactic Rotation Curve
Method of Merrifield (1992) for outer rotation curve. Local hydrostatic
equilibrium dictates that the thickness hz of the HI layer is a function of
R but not `. A ring with constant thickness
qhas a variable angular size:
θb = 2 tan−1 (hz /2d) ,
d = R0 cos ` + R 2 − R02 sin2 `. For circular
rotation, and assuming co-rotation, the radial velocity satisfies
Vr
R0
RVr = sin ` cos b(R0 V (R) − RV0 ),
sin ` cos b = R V (R) − V0 ≡ W (R).
Take data with
fixed value of
W (R) and
obtain the
variation in
angular width
θb (`). Determine
best fits for
R/R0 and hz /R0 .
Thereby we find
vr (R) and hz (R).
-75 km/s < W < −70 km/s
J.M. Lattimer
AST 346, Galaxies, Part 2
tange
Galactic Rotation Curve
int
nt p o
J.M. Lattimer
od
meth
Merrifield, AJ 103, 1552 (1992)
AST 346, Galaxies, Part 2
laye
r th
ickn
ess
Implications of Galactic Rotation
For a spherical system,
M(R) =
RV (R)2
RV (R)2
= 7.5 × 1010
M .
G
R0 V02
Most of this is dark matter, and no reason to believe it is not spherically
distributed. If ρ ∝ r −2 , the mass is linearly proportional to r , matching
the constant velocity rotation profile. Gauss’ Law ∇2 Φ = 4πG ρ implies
Φ(r ) = 4πGr02 ρ0 ln(r /r0 ),
ρ = ρ0 (r0 /r )2 ,
known as the singular isothermal sphere. To match the rotation of the
Milky Way, we should have ρ0 r02 ' 0.9 × 109 M kpc−1 .
A more realistic solution is one without the cusp at r = 0:
r02
,
M(r ) = 4πρ0 r02 [r − a tan−1 (r /a)].
+ a2
r
r
q
G M(r )
a
r
2
V (r ) =
= 4πG ρ0 r0 1 − tan−1 .
r
r
a
The velocity varies linearly near r → 0 and vanishes as observed.
ρ(r ) = ρ0
r2
J.M. Lattimer
AST 346, Galaxies, Part 2