Lab7 2
pdf
keyboard_arrow_up
School
University of California, Los Angeles *
*We aren’t endorsed by this school
Course
3
Subject
Astronomy
Date
Apr 3, 2024
Type
Pages
10
Uploaded by ColonelCaribouMaster938
Lab 7
Name
ID#
Section
Lab Partners
Structure and Motion of Spiral Galaxies
The Sun and 200 billion of its closest stellar neighbors make up what we call the Milky
Way galaxy, named for the white lane of light these stars create in Earth’s night sky.
Learning Objectives
At the completion of this lab, you should be able to:
•
Identify and calculate the important
quantities
that
describe
circular
motion.
•
Plot and interpret rotation curves for
solid bodies,
planetary systems,
and
spiral galaxies.
•
Describe
the
mass
distribution
of
a
system based on its rotation curve.
•
Explain
how
spiral
galaxies
provide
evidence
for
the
existence
of
dark
matter.
•
Describe how the appearance of a spiral
galaxy changes in di
↵
erent filters, and
explain why.
Solid Body Rotation Curves
Imagine four ants, each stuck to the surface of a Frisbee as shown. When you spin the
Frisbee, the ants ride around in circles whose radii are given in the table below. If the Frisbee
rotates once per second, how fast is each ant traveling? Fill in the table below. You may
leave the factors of
⇡
in your answers.
Ant
Radius
Speed
A
2 cm
B
6 cm
C
11 cm
D
17 cm
Continues on next page
!
1
12
.
566cm/s
37
.
699cm/s
69
.
15
cm/s
106
.
814CM/s
1)
On the axes below plot a
rotation curve
for the system, showing speed as a function of
distance from the center of the Frisbee. Label each ant. Draw a single line through all
the points.
2)
The rotation curve produced by ants on a spinning Frisbee is representative of
solid body
rotation
. Write a proportionality relation for speed (call it
v
) and radius (call it
R
) in
solid body rotation. You can assume period (P) is constant. Hint: Use the plot or the
formula you used to complete the table on page 1.
3)
What is an example of solid body rotation in our solar system (i.e.
the Sun, planets,
moons, etc )? Be specific as there might be more than one type of rotation that applies
to any given object.
Continues on next page
!
2
110
=
·
Cou
90
-
80
-
70
-
⑳
00-
C
50
-
40
-
30-
·
B
20
n
-
10
·
A
I
1
11
I
I
1
I
2
46
81012
14
16
18
V
=
R
Earth
can
be
an
example
of
a
solid
body
rotation
in
our
solar
system
because
it
Rotates
around
Its
axis
For non-solid systems, the force of gravity determines orbital motion. We know that the
strength of gravity depends on mass and distance. Therefore, we can learn about the
mass
distribution
(how much mass is located in a given area) in astronomical systems by studying
rotation curves. We will start with a simple case: our solar system.
Keplerian Orbits
4)
Let’s consider the planets orbiting the Sun.
They actually travel in ellipses, but their
orbits are nearly circular.
As you did for the ants, write a formula for the speed of a
planet around the Sun in terms of its distance from the Sun,
R
. Use the period,
P
, for
the time required for one orbit. Rearrange your speed formula to solve for
P
.
Recall the proportionality relation form of Kepler’s Law of planetary motion,
P
2
/
a
3
.
Note that
a
is the same as
R
when we’re approximating the planetary orbits as circles, so
use
R
for the distance.
5)
Use the expression you wrote down in question 4) relating speed (
v
), period (
P
), and
distance from the Sun (
R
), to eliminate
P
from the proportionality form of Kepler’s Law
(
P
2
/
a
3
). You should be left with an expression only in terms of speed (
v
) and distance
(
R
). Simplify your answer.
6)
Rearrange this proportionality to solve for speed (
v
) in terms of distance (
R
).
Based on this expression, you now know how the speed of a planet varies with distance
from the Sun. We will be taking a closer look at the actual rotation curve of the solar system
and its relation with the solar system’s mass distribution.
Continues on next page
!
3
-
#
-
V
=
A
V
= I
P
=
①
-
O
e
Par
3
2
π
KC
R
(
-)2xn
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Below is a plot of the measured speed of each planet versus its distance from the Sun.
7)
Draw a smooth curve to connect the data points in this plot. Compare this curve to the
solid body rotation curve you drew earlier (page 2). Does the solar system rotate like a
solid body? Describe the di
↵
erences.
8)
Based on the plot, which planets move more slowly, inner or outer planets? Explain why
this is so. (Hint: Why are the planets in orbit at all?)
This type of rotation curve is called
Keplerian
because the relationship between speed
and distance is determined by Kepler’s Law, which is a consequence of universal gravitation.
Now let’s consider how the Keplerian rotation curve is related to the mass distribution in
the solar system.
9)
(a) Where is most of the mass in the solar system?
(b) If we were to draw a circle
centered on the Sun and gradually increase the size of this circle along the ecliptic, is
there significantly more mass inside this circle as we increase the radius of the circle?
Continues on next page
!
4
(
--
the
solar
system
does
not
rotate
as
a
solid
body
because
the
speed
does
not
increase
as
a
linear
function
to
the
distance
of
the
planet
from
the
Sun
.
Outer
planets
move
move
slowly
because
they
are
farther
from
the
sun
a
.
most
of
the
mass
b
.
as
We
Increase
the
radius
of
the
circle
,
the
mass
would
not
significantly
increase
In
the
solar
system
is
locat
a
because
there
Is
a
lot
of
mass
In
the
Sun's
Center
.
Relation between Enclosed Mass and Rotation Curves
One way to describe the location of mass in a system is
enclosed mass
, or
M
R
, defined as
the amount of mass enclosed in a circle of radius,
R
. Beyond the radius where enclosed mass
stops increasing, a system will exhibit a Keplerian rotation curve. However, while
enclosed
mass
is still increasing a rotation curve can increase, be flat, or decrease, depending on how
much matter is present.
Make sure you have read the above paragraph before proceeding
Galactic Rotation Curves
Now you will apply these ideas of rotation curves and enclosed mass to an example spiral
galaxy, NGC 3198, which is similar to our own Milky Way. You can find images of NGC
3198 both on the lab table and at the end of Lab 7. You will use these images to answer
several questions throughout the remainder of this lab.
10)
Consider the color image of the galaxy NGC 3198. As far as you can tell from this image,
how does enclosed mass change as you (a) go from the center of the galaxy to the edge
of the galaxy, and then (b) from the edge of the galaxy to the edge of the picture?
11)
(a) At what distance from the center (in kiloparsecs, not inches) does it look like the
galaxy “ends” (measure along the longest direction)?
(b) Do you expect the enclosed
mass to change significantly after this point? Explain your reasoning.
Continues on next page
!
5
as
you
go
from
the
center
of
the
galaxy
to
the
edge
,
the
enclosed
mass
increases
.
as
you
go
from
the
edge
of
the
galaxy
to
the
edge
,
the
enclosed
mass
stays
the
same
.
a
.
at
10
.
5
KDC
from
the
center
b
.
Since
there
is
not
a
lot
of
mass
surrounding
the
galaxy
,
the
enclosed
mass
will
not
significantly
change
.
12)
Based on your answers to questions 10) and 11), draw a diagram with enclosed mass on
the y-axis and distance from the center of the galaxy on the x-axis. Draw and label a
vertical line on your graph where the galaxy ends.
13)
Represent the rotation curve from the center to the edge of galaxy as a increasing curve.
Based on your description of the galaxy’s mass distribution in the questions above (and
the paragraph on top of page 5), what do you expect the galaxy’s rotation curve to look
like beyond where the galaxy appears to end?
Describe and sketch the rotation curve
(speed vs. distance from the center) you expect. Draw and label a vertical line on your
graph where the galaxy ends.
Continues on next page
!
6
enclosed
mass
!
I
l
'
is
is
distance
from
the
center
the
speed
will
Increase
until
it
gets
to
the
edge
of
the
galaxy
.
speed
-
1
!
↓
i
is
is
distance
from
the
center
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Below is the measured rotation curve for the stars in NGC 3198.
14)
Draw and label a vertical line at the radius where you observed the galaxy to end.
15)
How does this rotation curve di
↵
er from your prediction (question 13)) beyond where the
galaxy appears to end?
16)
What does the shape of the rotation curve imply about the enclosed mass beyond where
the galaxy appears to end in the image?
17)
Think back to the last page. What assumptions about light and mass were you making
to answer those questions? In other words, how can you tell there is mass?
Many spiral galaxies, including the Milky Way, have rotation curves similar to that of
NGC 3198. The enclosed mass (
M
R
) continues to increase beyond the radius at which the
light of the galaxy stops. (In fact, the enclosed mass curve increases proportional to distance:
M
R
/
R
.)
18)
Based on your analysis of the image and rotation curve of NGC 3198, why is the flatness
of the rotation curve beyond the visible edge of the galaxy evidence for dark matter?
Continues on next page
!
7
from
question
13
,
I
predicted
the
galaxy
wil
ed
he
ending
at
10
.
5
KPc
but
the
graph
at
around
7
KPC
.
-
-
Implies
that
the
enclosed
mass
will
Increase
the
assumption
is
that
all
objects
that
project
light
have
mass
flatness
explains
that
the
mass
must
Increase
.
Since
beyond
the
galaxy
there
Is
very
little
light
,
the
mass
is
coming
from
dark
matter
.
Galaxy Spiral Patterns
The flat rotation curves observed for stars and gas clouds of spiral galaxies also present a
problem with the gracefully curving spiral arms we observe: If the spirals are physical objects,
they will end up curling on to themselves! We’ll examine this problem first qualitatively using
rotation curves then more quantitatively using stars in NGC 3198.
19)
Compare the solid body rotation curve (page 2) to the rotation curve of NGC 3198 over
the range of 10–30 kpc. (a) Do the stars and gas clouds in this spiral galaxy rotate like
a solid body? (b) Does NGC 3198 exhibit Keplerian rotation (page 4)?
20)
Look at the three images of NGC 3198 taken with three di
↵
erent filters: red, blue, and
infrared. These are called negatives: the darker it is in the image for a given filter, the
more light that is being emitted in that filter. In which image are the spiral arms most
pronounced? In which image does the galaxy look the smoothest (i.e., the spiral arms do
not stand out prominently from other parts of the galaxy)?
21)
Recall what you have learned about di
↵
erent types of stars from Lab 6. Describe a main
sequence star that would be brighter in a blue filter than in an infrared filter (think about
mass, temperature, lifetime, etc.).
22)
Based on your answers to questions 20) and 21), what type of stars are found in spiral
arms – old stars or young stars?
23)
What does the galaxy look like between the spiral arms in the colored picture? In which
filter is this part of the galaxy brightest, relative to the arms? What type of main sequence
stars must be located between the spiral arms?
Continues on next page
!
8
a
the
stars
&
gas
clouds
In
the
spiral
galaxy
do
NOT
rotate
like
a
solid
body
b
NGC
3198
does
NOT
exhibit
Keplerian
rotation
most
pronounced
In
Blue
I
infrared
most
smooth
In
I
a
star
that
would
be
brighter
in
a
blue
filter
would
have
a
large
mass
,
high
temperature
,
&
a
short
lifetime
.
young
stars
O
After completing question 23), you may be wondering why old stars and young stars are
segregated in NGC 3198 (and in spiral galaxies in general). In the last part of this lab, we
will be tackling this problem and its connection to star formation in galaxies.
Astronomers have determined that the pattern of arms of a spiral galaxy rotates like
a solid body, even though the stars themselves do not. This fact, along with the galaxy’s
rotation curve (page 7), can be used to determine where star formation occurs in a spiral
galaxy.
24)
Plot two rotation curves: (a) one for the spiral pattern of a galaxy and (b) one for the
stars in that galaxy. The curves should both begin at the origin and intersect at one other
point. Label that point
“corotation.”
Check your curves with your TA.
25)
Describe what is happening to the stars relative to the spiral arms at radii smaller than
corotation, the point where the stars and spiral arms have the same speed.
26)
Describe what is happening to the stars relative to the spiral arms at radii beyond
corotation.
Continues on next page
!
9
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
27)
Based on questions 25) and 26), do stars stay in the spiral arms forever?
28)
How does this answer explain your observations from page 8?
Where must stars be
formed, in or between the spiral arms?
End of Lab 7
10