Download ∫ Rotation Curves

Rotation Curves
• Spherical mass shell, uniform density
Inside shell: no effect
Outside shell:
acts as if all mass at center
• Fcentripital = Fgravity
Hwk 2 due Sept 23
Hwk 3 not yet assigned, but due Sept 30
Midterm 1 Wed, Oct 2
dFgravity ρ × d (vol )
∝
dΩ
r2
2
d ( vol ) = dΩ r dr
mv 2 GM ( r )m
=
r
r2
1/ 2
 GM ( r ) 
v=

 r 
• Inside spherical mass distribution
r
4
M ( r ) = ∫ ρ ( r′)4π r 2dr′ = ρ π r 3
0
3
v∝r
• Outside spherical mass distribution
M ( r ) = const.
• Spher. Distr.
+ exponential disk
v ∝ r −1 / 2
The Milky Way’s Rotation Curve
Radio data
Tangent point method
Young star clusters
1
receeding
Expected:
approaching
Rotation curves in
other galaxies
Spec
slit
Observed:
systemic
velocity
• Vera Rubin & Kent Ford
(late 1970’s)
• Image tube spectrograph
[CO pg. 917]
Density as shown by flat rotation curves
•
dM(r)/dr ~ constant
•
•
Unbounded mass
distribution??
NFW profile
ρ (r) =
(r)
•
•
•
ρ0
(r / a )(1 + r / a )2
Predicted for Cold dark
matter (CDM)
Actual derived dark matter
profiles often slightly
different than this
What is CDM? Coming later
in course.
2
The Milky Way’s
Dark Matter
“Halo”
M31
770 kpc
Components of our Galaxy
10 kpc
(1010 M☼)
Nuclear bulge
1
Disk
7
Halo
0.3
Dark matter
190 out to r ~ 230 kpc
Milky Way
H I Spectra of
Spiral Galaxies
3
Tully-Fisher Relation
Rotation Curves
• Maximum rotation velocity-Luminosity
relation [FIG 25.10]
• Tully-Fisher relation
• MB = -9.95 log10VMax + 3.15
• MB = -10.2 log10VMax + 2.71
• MB = -11.0 log10VMax + 3.31
Semi-derivation of
Tully-Fisher Relation:
MB = -9.95 log10Vmax + 3.15
MB = -10.2 log10Vmax + 2.71
MB = -11.0 log10Vmax + 3.31
(Sa)
(Sb)
(Sc)
(Sa)
(Sb)
(Sc)
• Mass interior to outermost R where rotation curve can be measured:
Mass =
• Assume
2
Vmax
R
G
L = Mass / const.
• “Freeman Law” (observed fact ---maybe):
L
= const.
Surf .Bright. =
4πR 2
4
L = const × Vmax
Important as a
DISTANCE
calibrator!
• Convert to Absolute B-band magnitudes:
 L 
 = −10 log10 Vmax + const.
M B = M sun − 2.5 log10 
 Lsun 
4
SO FAR:
• Galaxy types
• Ancient history
• Milky Way and spiral galaxy morphology
• Nuclear bulge
• Disk
• Stellar halo
• Dark matter halo
• Star-forming regions
• Chemical enrichment
Hwk 2 due Sept 23
Hwk 3 not yet
assigned, but due
Sept 30
Midterm 1
Wed, Oct 2
Kinematics of spiral galaxies
• Rotation curves Î mass distribution
(includes sidetrack about measuring distances)
• Spiral structure [CO 25.3]
• General properties of S, E, Irr galaxies
• Midterm 1 (Wed. Oct 2)
5