The Local Milky Way Color-Magnitude Diagram · Analyzing Nearby Stars I Star Formation Rate IM=L ˘2; L D ˘1:5 1010 L =)M˘3 1010 M I˝ gal ˘10 Gyr, so if half of gas is returned

The Local Milky Way Color-Magnitude Diagram

Distances from Hipparcostrigonometric parallaxes

15,630 stars

d < 100 pc

Half of stars withMV > 10 are notdetected.

age differences��

��

metallicity differences��

��

red

gian

ts⇒

⇐re

dcl

um

p

Her

tzsp

run

gga

p

white dwarfs

B0 A0 F0 G0 K0 M0

J.M. Lattimer AST 346, Galaxies, Part 2

Local Stellar Luminosity Function

Φ(M) = number stars MV±1/2volume over which seen

Ψ(M) = ΦMS(M) Θ(

τgalτMS(M)

)Θ(x) = 1 if x ≤ 0Θ(x) = x if x > 0τgal ' 10 Gyr

• Hipparcos◦ Reid et al. (2002)

Dim stars hard to find; Brightstars are rare.

Stars not uniformly distributed.

Stars in binaries mistaken forbrighter single stars.

⇐W

iele

nd

ip

LV Φ(MV )

MΦ(MV )

total

MSA, F

K giants

B, O


Stellar Initial Mass Functionξ(M)∆(M)= number stars born with mass between M and M+ ∆M

Salpeter initial mass function ξ(M) = ξ0

M�

(MM�

)−2.35

Total number:∫Mu

M`ξ(M)dM = ξ0

1.35

(M�M`

)1.35[

1−(M`

Mu

)1.35]

Total mass:∫Mu

M`Mξ(M)dM = ξ0M�

0.35

(M�M`

)0.35[

1−(M`

Mu

)0.35]

Total luminosity:∫Mu

M`Lξ(M)dM = ξ0L�

2.15

(Mu

M�

)2.15[

1−(M`

Mu

)2.15]

Pleiades cluster Salpeter mass function

ξ(M)

Mξ(M)


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 perpindicular2a = (s/1 arcsec)(d/1 pc) AU, P2a3 = 4π2GMBH

Vp(1−e) = Va(1+e), V 2p −V 2

a = 2GMBHa−1[(1− e)−1 − (1 + e)−1

]

Galactic center�

2a=

sd

6

?


Distances From Motions

Example: Distance to supernova ring, if ring is circulart− = R(1− sin i)/c , t+ = R(1 + sin i)/c

SN 1987a@@I

∆t = tA − t0 = t−

∆t = tB − t0 = t+

∆t = 0

-


Spectroscopic and Photometric Parallax

Assume stars of same spectral type and Zhave equal L.Correct for dust reddening.Works to 10% for main sequence stars,60% for K giants.Needs lots of observation time.

Cheaper alternative is to estimatespectral type from color, otherindicators determine dwarf/giant.Works best for clusters, wherereddening is easier to determine.

Photometric distances towards South Galactic pole

V − I colors of 12,500 stars with mV < 19

5 < MV < 6

nZ ∝ e−|z|/hz

halo→

hhhhhhhhhh


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,Se−R/hR,S , IS(R) = LSΣS(R)

LD,S = 2πIS(0)h2R,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 SunM/LV ∼ 0.74 for all stars except white dwarfs and gasM/LV ∼ 5 from Galactic rotation


Analyzing Nearby Stars

I Star Formation RateI M/L ∼ 2, LD ∼ 1.5× 1010 L� =⇒ M∼ 3× 1010 M�.I τgal ∼ 10 Gyr, so if half of gas is returned to stars, star formation

rate is 3− 6 M�/yr.I Gas mass is 5− 10× 109 M�, enough for 1–3 Gyr.

I V/Vmax test – a measure of the uniformity of a spatial distributionI An average of the ratio of the volume V from which an object could

be seen divided by the volume Vmax defined by the maximumdistance at which an object in the sample is detectable.

I Assume sample is of equally luminous objects and limited by abrightness limit which corresponds to the distance zmax and thevolume Vmax . A brighter object with m < mmax originates from asmaller distance z < zmax .

IVVmax

= z3

z3max

=⇒⟨

VVmax

⟩=

R zmax0 z5dz

z3max

R zmax0 z2dz

I When n(z) is uniform, 〈V/Vmax〉 = 1/2


Stellar Velocities and Scale Heights

Stars show the cumulative effects of acquiring random gravitational pullsfrom 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


Stellar Velocities and Scale Heights


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 awaycontaining the Sun and starsyounger than 30 Myr. They lie ina layer tilted by 20◦ about a linealong the Sun’s orbit at ` = 90◦.

I Open clusters100 L� < L < 3× 104 L�M/L < 1M�/L�0.5 pc < rc < 5 pcσs < 0.5 km s−1

I Globular clusters104 L� < L < 106 L�1M�/L� <M/L < 2.5M�/L�0.5 pc < rc < 4 pc25 pc < rt < 85 pc4 km s−1 < σs < 20 km s−1

Hipparcos dataMV < 3, 100 pc < d < 500 pc

Gould’s Belt

IRAS data

M7 M22


Distances and Ages of Clusters

100 Myr isochrone100 Myr, w/o dust reddening16 Myr isochrone

•Pleiades

blue giants, ≈ 5 M�

(m −M)0 = 5.6

← binaries↓

pre-

mai

nse

qu

ence

star

s

⇓


Globular Clusters

47 Tuc

M92

SMC

redgian

ts

[Fe/H]= −0.71 [Fe/H]= −2.15

12 Gyr 13 Gyr

12± 2 Gyr 13± 2 Gyr

HorizontalBranch Horizontal

Branch


Globular Cluster Distribution

Metal-rich Metal-poor [Fe/H]< −0.8

` = 20◦

` = 90◦

b = 0◦

Metal-poor clusters have highly eccentric, random orbitsand do not share the Galactic disk rotation.

Metal-rich clusters share Galactic rotation like stars in the thick disk.

♦ disrupted dwarf galaxy remnants


Halo Properties

I Oldest stars in halo a result of Galactic cannibalism.I Gravity slows galaxies during encounters and leads to mergers.I The Sagittarius dwarf galaxy is the closest satellite galaxy, and it is

partially digested.I ω Cen is a bizarre globular cluster with non-uniform metallicity;

possibly the remnant core of a digested satellite galaxy.I Magellanic Clouds will share the fate of Sagittarius dwarf galaxy in

3–5 Gyr.I Some believe the blue horizontal branch globular clusters joined the

Milky Way after their galaxies were cannibalized.

I Metal-poor halo consists of globulars and metal-poor stars down to[Fe/H]= 10−5

I Most globulars have dissolved, forming many halo stars.I Palomer 5 has lost 90% of its stars and has tidal tails extending 10◦.I Metal-poor halo stars form moving groups, including some carbon

stars and M giants stripped from Satittarious dwarf galaxy.I Total mass of halo metal-poor stars is about 107 M�.I Halo is somewhat flattened but rounder than bulge.


Palomar 5


Sagittarius Dwarf Galaxy


The Halo as an Ensemble of Disrupted Satellites


The Bulge and NucleusI Bulge best studied with infrared light

due to dust (31 mag extinction).

I Bulge accounts for 20% of Galaxy’slight with a size of order 1 kpc.

I Bulge is not dense inner portion of halo:stars are old but have average metallicty> 0.5Z�, Zmax ∼ 3Z�.

I Average rotation speed 100 km s−1

(half disk), stars have large randommotions.

I Close to the center are dense, activestar forming regions.

I Center has a 2 pc radius torus with106M� of gas surrounding a3× 107M� star cluster within 10′′

(0.2 pc) including > 30 massive stars.

I Innermost stars < 0.05 pc from centralblack hole, 4× 106M� (Sag A∗).


Sag A∗

R0 = 8.28± 0.1± 0.29 kpc

MBH = 4.30± 0.20± 0.30× 106M�


Galactic Rotation

Differential rotation in Galaxy noticed by1900 and explained by Oort in 1927.

?GC

φ


Geometric Relations

V0 = V (R0)R0 sin ` = R sin(90◦ + α) = R cosαR0 cos ` = R sinα + d

φ+ `+ α = 90◦

Vr (R, `) = V (R) cosα− V0 sin `

= R0 sin `

(V (R)

R− V0

R0

)Vt(R, `) = V (R) sinα− V0 cos `

= R0 cos `

(V (R)

R− V0

R0

)− d

V (R)

R.

When d << R,

Vr ≈ R0 sin `

(V (R)

R

)′R0

(R − R0) = d A sin(2`)

Vt ≈ R0 cos `

(V (R)

R

)′R0

(R − R0)− dV (R)

R

= d [A cos(2`) + B]

A ≡ −R

2

(V (R)

R

)′R0

B ≡ − 1

2R[RV (R)]′R0

A + B = −V (R)′R0

A− B = V0/R0

φ


Galactic Rotation – A Simple Model

V (R) = V0, equally luminous clouds in rings of radii 2, 3 . . . 14 kmspaced every degree in φ. Dot sizes indicate relative brightnesses.


Galactic Rotation Curve

For inner Galaxy, R < R0, employ tangent point method. Vr (d , `) has amaximum 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 parallaxmeasurements.

��



Method of Merrifield (1992) for outer rotation curve. Local hydrostaticequilibrium dictates that the thickness hz of the HI layer is a function ofR but not `. A ring with constant thickness has a variable angular size:

θb = 2 tan−1 (hz/2d) , d = R0 cos `+√

R2 − R20 sin2 `. For circular

rotation, and assuming co-rotation, the radial velocity satisfiesRVr = sin ` cos b(R0V (R)−RV0), Vr

sin ` cos b = R0

R V (R)−V0 ≡W (R).

Take data withfixed value ofW (R) andobtain thevariation inangular widthθb(`). Determinebest fits forR/R0 and hz/R0.Thereby we findvr (R) and hz(R).

-75 km/s < W < −70 km/s



Merrifield, AJ 103, 1552 (1992)

�

tangentp

ointm

ethod

layer thickness


Implications of Galactic Rotation

For a spherical system,

M(R) =RV (R)2

G= 7.5× 1010

[RV (R)2

R0V 20

]M�.

Most of this is dark matter, and no reason to believe it is not sphericallydistributed. If ρ ∝ r−2, the mass is linearly proportional to r , matchingthe constant velocity rotation profile. Gauss’ Law ∇2Φ = 4πGρ implies

Φ(r) = 4πGr20 ρ0 ln(r/r0), ρ = ρ0(r0/r)2,

known as the singular isothermal sphere. To match the rotation of theMilky Way, we should have ρ0r

20 ' 0.9× 109M� kpc−1.

A more realistic solution is one without the cusp at r = 0:

ρ(r) = ρ0r20

r2 + a2, M(r) = 4πρ0r

20 [r − a tan−1(r/a)].

V (r) =

√GM(r)

r=√

4πGρ0r20

√1− a

rtan−1 r

a.

The velocity varies linearly near r → 0 and vanishes as observed.J.M. Lattimer AST 346, Galaxies, Part 2

Documents

The Local Milky Way Color-Magnitude Diagram · Analyzing Nearby Stars I Star Formation Rate IM=L ˘2; L D ˘1:5 1010 L =)M˘3 1010 M I˝ gal ˘10 Gyr, so if half of gas is returned