Download - Rotation of the Galaxy
![Page 1: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/1.jpg)
Rotation of the Galaxy
![Page 2: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/2.jpg)
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°
![Page 3: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/3.jpg)
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
![Page 4: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/4.jpg)
b
Galactic latitude(b) is shown here
![Page 5: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/5.jpg)
Galactic Coordinate System:
l
b
![Page 6: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/6.jpg)
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)
![Page 7: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/7.jpg)
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
![Page 8: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/8.jpg)
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
![Page 9: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/9.jpg)
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.
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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 +
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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
![Page 12: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/12.jpg)
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
![Page 13: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/13.jpg)
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
![Page 14: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/14.jpg)
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
![Page 15: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/15.jpg)
Summarizing, we have two equations for the relative radial and tangential velocities:
dlRvT cos00 dlRvT cos00 lRvR sin00 lRvR sin00
![Page 16: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/16.jpg)
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
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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
![Page 18: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/18.jpg)
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
![Page 19: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/19.jpg)
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
![Page 20: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/20.jpg)
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
![Page 21: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/21.jpg)
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
![Page 22: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/22.jpg)
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
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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
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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)
![Page 25: Rotation of the Galaxy](https://reader035.vdocuments.us/reader035/viewer/2022062806/56814f0a550346895dbc9da7/html5/thumbnails/25.jpg)
Keplarian Rotation curve
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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