The Milky Way Galaxy

1
The Milky Way Galaxy

• 02/03/2005

2
The Milky Way

• Summary of major visible components and structure
• The Galactic Rotation
• Dark Matter and efforts to detect it

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Morphology of Galaxy
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Dark Matter Halo?
Rotation of Galaxy implies that there is a lot of
mass in our Galaxy that we dont see (ie, if we
count up the mass from the stars that emit
visible light, its 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?)
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Determining the rotation when we are inside the
disk rotating ourselves
23.5
39.1
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.
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The Galactic midplane is inclined 62.6 with the
plane of the celestial equator. We will introduce
the Galactic coordinate system.
l0
Galactic longitute (l) is shown here
l
l90
l270
l180
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Galactic latitude(b) is shown here
b
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Galactic Coordinate System
b
l
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Let us introduce the following coordinate system
l
b
d
R0
R
GC
GC
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• Assumptions
• Motion is circular ? constant velocity, constant
  radius
• Motion is in plane only (b 0?) ? no expansion
  or infall

l 180?
w0
Q0
l 90?
l 270?
l
d
w
Q
R0
R
R0 Radius distance of ? from GC R Radius distance
of ? from ? d Distance of ? to ? Q0 Velocity of
revolution of ? Q Velocity of revolution of
? w0 Angular speed of ? w Angular speed of ?
GC
l 0?
w (rad/s)
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Keplerian Model for l 0?, 180?
?
l 180?
vR 0
Q1
vR 0
Q0
R0
d
vR 0
Q2
R
GC
l 0?
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Keplerian Model for l 45?, 135?
Q1R
Q0R-Q1R vR lt 0
w1
Q1
Star moving toward sun
l 180?
l 180?
R gt R0
R gt R0
Q0R
w0
w0
Q0
Q0
45?
45?
l 90?
l 270?
Q0R
45?
45?
d
d
w2
R0
R0
Q2
Q2R
Q0R-Q2R vR gt 0
R lt R0
R lt R0
Star moving away from sun
GC
GC
l 0?
l 0?
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Keplerian Model for l for all angles
Outer Star
Leading Inner Star (moving away from Sun)
Leading Star At Same Radius
Inner Leading Star
Inner Leading Star
Lagging Outer Star (moving towards Sun)
Lagging Outer Star (moving away From Sun)
Leading Inner Star (moving towards Sun)
Lagging Star At Same Radius
R lt R0
R R0
R gt R0
R R0
R lt 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.
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Keplerian Model for l 0? and 180?
l 180?
t 0
t gt 0
Falling backward
vR 0
Q1
l 180?
l 180?
1
R gt R0
1
Q1
ma
vR 0
CCW Rotation of INNER OUTER local ?s relative
to ?
Q0
0
0
R0
R0
Q0
d
d
mb
vR 0
Q2
2
2
Racing forward
l 0?
l 0?
Q2
R lt R0
R
R
GC
GC
l 0?
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What is the angle g?
Q0
We have two equations
d l a 90? (1)
l
d l g 180? (2)
d
d
l
If we subtract (1) from (2), i.e. (2) (1)
90-a
g
?
g - a 90?
g 90? a
a
l
R0
Q
d
QR
QT
R
d
GC
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Now let us derive the speed of ?s relative to the
?, vR (radial component).
Q0
Relative speed, vR QR Q0R Qcosa Q0sinl
l
Q0R Q0 sinl
l
d
We now can employ the Law of Sines
d
l
90-a
90? a
a
l
R0
Q
d
QR Q cosa
R
GC
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Therefore,
From v Rw, we may substitute the angular speeds
for the star and Sun,
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Now let us derive the speed of ?s relative to the
?, vR (tangential component).
Q0
vT QT Q0T Qsina Q0cosl
l
Q0T Q0 cosl
We will use trigonometry similar to that used
when looking at the energy conservation of a
pendulum.
l
d
d
l
90-a
90? a
a
l
R0
Q
d
QT Q sina
R
GC
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?
Therefore,
l
R0sin(90-l)R0cosl
d
90? a
Rsina
R0
90? - a
R
Rcosa
a
90? - l
GC
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Summarizing, we have two equations for the
relative radial and tangential velocities
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Let us study w(R)
l 180?
2
1
l 90?
l 270?
3
4
R0
GC
l 0?
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Quadrant ? R gt R0, 90? lt l lt 180?
l 180?
Conclusion Star is moving towards the Sun vR lt
0 Always!
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R gt R0
d
l 90?
l 270?
l fixed
vR
R
R0
d
w0R0sinl
GC
l 0?
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Quadrant ? R gt R0, 180? lt l lt 270?
l 180?
Conclusion Star is moving away from the Sun vR
gt 0 Always!
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R gt R0
d
l fixed
l 90?
l 270?
vR
w0R0sinl
R
R0
d
GC
l 0?
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Quadrant ? R lt R0 and R gt R0, 0? lt l lt 90?
l 180?
l 90?
l 270?
l fixed
d
I
R lt R0
vR
Star moving away!
When Rlt R0, then vRgt0 and wgtw0.
R0
R
d
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Case I RltR0 small Case II RgtR0 large
GC
II
R gt R0
l 0?
Star moving towards!
When Rgt R0, then vRlt0 and wltw0.
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Quadrant ? R lt R0 and R gt R0, 270? lt l lt 360?
l 180?
l fixed
l 90?
l 270?
Quadrant ? is the negative of Quadrant ?!
vR
d
4
GC
l 0?
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Now we will make an approximation.
We can work equally with w(R) or v(R) for the
following approximation. Here we will work with
w(R).
Let us write RR0DR. Then, the Taylor Expansion
yields
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Here we make the approximation to retain only the
first term in the expansion
If we continue the analysis for speed, we would
use the substitution QRw. Therefore, wQ/R. The
derivative term on the right-hand side of the
equation must be evaluated after substitution by
using the Product Rule.
Therefore, the radial relative speed between the
Sun and neighboring stars in the galaxy is
written as
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When dltltR0, then we can also make the small-angle
approximation R0Rdcos(l).
?
Using the sine of the double angle, viz.
We may abbreviate the relation to
where
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If we then focus our attention to the transverse
relative speed, vT, we begin with
Picking up on the lessons learned from the
previous analysis, we write simply
Using the cosine of the double angle, viz.
Because R?R0, w?w0, which implies the last term
is written as
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Therefore,
where
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Summarizing,
where
The units for A and B are
or
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We can define a new quantity that is
unit-dependent.
So that the transverse relative speed becomes
when d parsec, vT km/s.
The angular speed of the Sun around the Galactic
Center is found algebraically
Likewise, the gradient of the rotation curve at
the Suns distance from the Galactic Center is
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The quantities used can all be measured or
calculated if the following order is obeyed.
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So, summarizing, for stars in the local
neighborhood (dltltR0), Oort came up with the
following approximations
VrAdsin2l
Vt d(Acos2lB)
Where the Oort Constants A, B are
w0A-B
dQ/dR R0 -(AB)
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Keplarian Rotation curve
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(No Transcript)
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Dark Matter Halo

• M 55 ? 1010 Msun
• L0
• Diameter 200 kpc
• Composition unknown!

90 of the mass of our Galaxy is in an unknown
form
This could be a topic for your final project