Today in Astronomy 328: the Milky Way Image: wide-angle photo and overlay key of the Sagittarius region of the Milky Way, showing vividly the effect of obscuration by dust clouds. The very center of the Milky Way lies behind particularly heavy dust obscuration. (By Bill Keel, U. Alabama.)
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Today in Astronomy 328: the Milky Way Image: wide-angle photo and overlay key of the Sagittarius region of the Milky Way, showing vividly the effect of.
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Today in Astronomy 328: the Milky Way
Image: wide-angle photo and overlay key of the Sagittarius region of the Milky Way, showing vividly the effect of obscuration by dust clouds. The very center of the Milky Way lies behind particularly heavy dust obscuration. (By Bill Keel, U. Alabama.)
The Milky Way
• Summary of major visible components and structure
• The Galactic Rotation
• Dark Matter and efforts to detect it
Brief History
• Late 1700s - Herschels counted stars in 683 regions of sky, assumed all are equally luminous. Concluded that Sun at center of a flattened system.
What is the Shape of the Milky Way?
• 1920 - Kapteyn used a greater number of star counts and came to roughly the same conclusion
Star Counts:
If stars are distributed uniformly in space, then in any patch of sky, the total number of stars with flux less than a limiting flux, f is:
(Note: This formula was derived on the board in class)
3/ 20 0 .N ff Af
What is the shape of the Milky Way?
•First answer, due to Kapteyn, came from star counts. If stars were distributed uniformly in the universe, then the number counted in any patch of the sky, with flux larger than f0, is given by
•Actual star counts at low fluxes are less than predicted by this relationship and the numbers at larger fluxes.
3/ 20 0 .N ff Af
0log f
3/ 20f
0
logN
ff
Star counts in directions of the Milky Way disk (blue), and in the perpendicular directions (red).
Conclusion: stellar density not uniform but decreases with distance from Sun; faster in direction perpendicular to Milky Way and slower in the direction of the Milky Way
Milky Way is a highly flattened disk
• 1919 - Shapley studied globular clusters; used distance derived from pulsating stars to determine that Sun is not at center of Milky Way. These were found at great distances above and below the plane of the Galaxy, where extinction effects are much less than that found along the Milky Way
Figure: Chaisson and McMillan, Astronomy Today
•Definitely bound by gravity
•Contain large numbers of stars in a very small volume: 20,000-1,000,000 stars in a volume 20 pc in diameter
•very round and symmetrical in shape - very old -- among the first stellar complexes formed in the galaxy
Globular clusters
Distances from Variable Stars
Morphology of Galaxy
Disk• Young thin disk• Old thin disk• Thick disk
Thin disk
• Diameter ~ 50 kpc• Young thin disk scale height = 50 pc• Old thin disk scale height = 325 pc• Contains youngest stars, dust, and gas• Contains Sun, which is 30 pc above midplane
• M* = 6 1010 Msun
• Mdust+gas = 0.5 106 Msun (scale height 0.16)
• Average stellar mass ~ 0.7 Msun
• LB ~ 1.8 1010 Lsun
• Population I stars in the Galactic plane• Contains ~ 95% of the disk stars• [Fe/H] ~ -0.5 - +0.3• Age ~ < 12 Gyr• Spiral structure seen in neutral H, HII regions, young O and B stars
Thick disk
• Diameter ~ 50 kpc• Scale height = 1.4 kpc
•M* = 2-4 109 Msun
•LB ~ 2 108 Lsun
• [Fe/H] ~ -1.6 - -0.4 (less metal rich than thin disk)• Age ~14-17 Gyrs
Gas and Dust
Spheroidal Components• Central bulge• Stellar Halo• Dark Matter Halo
Central Bulge
• Diameter ~ 2 kpc• Scale height = 0.4 kpc
•M* = 1 1010 Msun
•LB ~ 0.3 1010 Lsun
• [Fe/H] ~ -1.0 - +1.0 (less metal rich than thick and thin disk)• Age ~10-17 Gyrs
Stellar Halo
• Diameter ~ 100 kpc• Scale height = 3 kpc• number density distribution ~ r-3.5
Rotation of Galaxy implies that there is a lot of mass in our Galaxy that we don’t see (ie, if we count up the mass from the stars that emit visible light, it’s much less than that implied by observing the motion of stars as a function of radius from the center of the Galaxy.
How do we know that the stars in the disk rotate around the center of the Galaxy? How do we know the rotational velocity of the Sun? How do we know the rotation curve?(rotational velocity as a function of radius from the Galactic center?)
Dark Matter Halo?
Rotation of Galaxy implies that there is a lot of mass in our Galaxy that we don’t see (ie, if we count up the mass from the stars that emit visible light, it’s much less than that implied by observing the motion of stars as a function of radius from the center of the Galaxy.
How do we know that the stars in the disk rotate around the center of the Galaxy? How do we know the rotational velocity of the Sun? How do we know the rotation curve?(rotational velocity as a function of radius from the Galactic center?)
Determining the rotation when we are inside the disk rotating ourselves
To determine the rotation curve of the Galaxy, we will introduce a more convenient coordinate system, called the Galactic coordinate system. Note that the plane of the solar system is not the same as the plane of the Milky Way disk, and the Earth itself is tipped with respect to the plane of the solar system. The Galactic midplane is inclined at an angle of 62.6 degrees from the celestial equator, as shown above.
23.5°
39.1°
The Galactic midplane is inclined 62.6° with the plane of the celestial equator. We will introduce the Galactic coordinate system.
l
l=0°
l=180°
l=90°
l=270°
Galactic longitute (l) is shown here
b
Galactic latitude(b) is shown here
Galactic Coordinate System:
l
b
Assumptions:1. Motion is circular constant velocity, constant radius2. Motion is in plane only (b = 0) no expansion or infall
GC
d
R
R0
l
l = 0
l = 90
l = 180
l = 2700
0
R0 Radius distance of from GCR Radius distance of from d Distance of to 0 Velocity of revolution of Velocity of revolution of 0 Angular speed of Angular speed of
R0 Radius distance of from GCR Radius distance of from d Distance of to 0 Velocity of revolution of Velocity of revolution of 0 Angular speed of Angular speed of
RT
Rv
2
(rad/s)
Keplerian Model for [l = 0, 180]:
GC
R
R0
l = 0
l = 180
0
d2
1vR = 0
vR = 0
vR = 0
2
2
R
mGM
R
vm
FF gc
R
GMv enc vR
Keplerian Model for [l = 45, 135]:
GC
d
R0
45
l = 0
l = 90
l = 180
l = 2700
2
0
2
45
R > R0
R < R0
GC
d
R0
45
l = 0
l = 180
00 45
R > R0
R < R0
Star movingtoward sun
Star moving awayfrom sun
0R-1R = vR < 01
11R
2R
0R
0R
0R-2R = vR > 0
Relative Radial Velocity, v R
0 45 90 135 180 225 270 315 360
Angle, l (o)
v R
InnerLeading
Star
OuterStar
LeadingInnerStar
(moving awayfrom Sun)
LaggingOuterStar
(movingtowards Sun)
LeadingStar
At Same Radius
InnerLeading
Star
LaggingOuterStar
(moving awayFrom Sun)
LeadingInnerStar
(movingtowards Sun)
LaggingStar
At Same Radius
Keplerian Model for [l for all angles]:
Star
Sun
Galactic Center
Star
Sun
Galactic Center
StarSun
Galactic Center
Star
Sun
Galactic Center
Star
Sun
Galactic Center
Star
Sun
Galactic Center
StarSun
Galactic Center
Star
Sun
Galactic Center
Star
Sun
Galactic Center
R < R0 R = R0 R > R0 R = R0 R < R0
At 90 and 270, vR is zero for small d since we can assume the Sun and star are on the same circle and orbit with constant velocity.
GC
d
R
R0
l
0
l
l
RT
90-
What is the angle ?
We have two equations:
+ l + = 90 (1)
+ l + = 180 (2)
If we subtract (1) from (2), i.e. (2) – (1):
- = 90 = 90 +
c
C
b
B
a
A
sinsinsin
GC
d
R
R0
l
0
l
l
90-90 +
Now let us derive the speed of s relative to the , vR (radial component).
R = cos
0R =
0 sinl
l Relative speed, vR = R – 0R
= ·cos – 0·sinl
We now can employ the Law of Sines
a b
cA
B
C
lRR
sin90sin0
lRR
sincos0
lR
Rsincos 0
Therefore,
lRRR
lR
R
llR
RvR
sin
sin
sinsin
00
0
00
00
From v = R, we may substitute the angular speeds for the star and Sun,0
00 ;
RR
lRvR sin00 lRvR sin00
GC
d
R
R0
l
0
l
l
90-90 +
Now let us derive the speed of s relative to the , vT (tangential component).
T =
si
n
0T =
0 co
sll vT = T – 0T = ·sin – 0·cosl
GC
d
R
R0
l
90 +
90 -
90 -l
Rcos
Rsin
R0 sin(90-l)=R
0 cosl
sincos0 RdlR R
dlR
cossin 0
Therefore,
lRdlR
lRR
dlRR
ldlRR
vT
coscos
coscos
coscos
000
00
00
00
dlRvT cos00 dlRvT cos00
Summarizing, we have two equations for the relative radial and tangential velocities:
dlRvT cos00 dlRvT cos00 lRvR sin00 lRvR sin00
Now we will make an approximation.
dlRvT cos00 dlRvT cos00 lRvR sin00 lRvR sin00
We can work equally with (R) or v(R) for the following approximation. Here we will work with (R).
00 RRR
Let us write R=R0+R. Then, the Taylor Expansion yields
202
2
0
02
02
2
00
02
2
2
0
00
00
00
00
00
!2
1
!2
1
!2
1
RRdR
RdRR
dR
Rd
RRRdR
RdRR
dR
RdR
RRdR
RdR
dR
RdR
RRR
RRR
RRRR
RRRR
RRRR
Here we make the approximation to retain only the first term in the expansion:
lRRRdR
d
lRRRRdR
d
Rv
R
RR
sin
sin1
00
0
0020
0
0
0
0
If we continue the analysis for speed, we would use the substitution: =R. Therefore, =/R. The derivative term on the right-hand side of the equation must be evaluated after substitution by using the Product Rule.
20
0
0 0
00
1
RdR
d
R
R
R
dR
d
dR
Rd
R
RRRR
Therefore, the radial relative speed between the Sun and neighboring stars in the galaxy is written as
00
0
RRdR
RdRR
RR
When d<<R0, then we can also make the small-angle approximation: R0=R+dcos(l).
dcos(l)
R
d
l
R0
ldRR cos0 ldRR cos0
llddR
d
R
lRRRdR
dv
R
RR
sincos
sin
0
0
0
0
00
0
Using the sine of the double angle, viz. 2sincossin 21
lddR
d
Rv
RR 2sin
00
0
We may abbreviate the relation to
ldAvR 2sin ldAvR 2sin
00
0
2
1
RdR
d
RA
00
0
2
1
RdR
d
RAwhere
If we then focus our attention to the transverse relative speed, vT, we begin with
dlRvT cos00 dlRvT cos00
dllddR
d
R
dlRRRRdR
d
R
dlRv
R
R
T
coscos
cos1
cos
0
0
0
0
0020
0
0
00
Picking up on the lessons learned from the previous analysis, we write simply
Using the cosine of the double angle, viz. 1cos22cos 2
dlddR
d
Rv
RT
12cos2
1
00
0
Because RR0, 0, which implies the last term is written as: dR
dd
0
00
Therefore,
BdlAd
ddR
d
RlAd
dR
ddR
d
Rld
dR
d
Rv
R
RRT
2cos
2
12cos
2
12cos
2
1
0
00
0
0
0
0
0
0
0
0
where BlAdvT 2cos BlAdvT 2cos
00
0
2
1
RdR
d
RB
00
0
2
1
RdR
d
RB
Summarizing,
where
BlAdvT 2cos BlAdvT 2cos
00
0
2
1
RdR
d
RB
00
0
2
1
RdR
d
RB
ldAvR 2sin ldAvR 2sin
00
0
2
1
RdR
d
RA
00
0
2
1
RdR
d
RA
The units for A and B are
pcs
km
kpcs
kmor
We can define a new quantity that is unit-dependent.
So that the transverse relative speed becomes
dv lT 74.4
74.4
2cos BlAl
74.4
2cos BlAl
The angular speed of the Sun around the Galactic Center is found algebraically
when [d] = parsec, [vT] = km/s.
BAR
0
00
Likewise, the gradient of the rotation curve at the Sun’s distance from the Galactic Center is
BAdR
Rd
R
0
The quantities used can all be measured or calculated if the following order is obeyed.
1-1- kpcskm 2.14.14
2sin Measure 1.
ld
vAv Rcalculate
R
1-1- kpcskm 8.20.122cos Measure 2. lAd
vBv Tcalculate
T
BA 0 Calculate 3.
)(
get weB, andA of definition theFrom 4.
0
BAdR
d
R
So, summarizing, for stars in the local neighborhood (d<<R0), Oort came up with the following approximations:
00
0
00
0
dRdΘ
RΘ
21
- B
dRdΘ
- RΘ
21
A
Vr=Adsin2l
Vt= =d(Acos2l+B)
Where the Oort Constants A, B are:
0=A-B
d/dR |R0 = -(A+B)
Keplarian Rotation curve
Dark Matter Halo
• M = 55 1010 Msun
• L=0• Diameter = 200 kpc• Composition = unknown!
90% of the mass of our Galaxy is in an unknown form