Wk12-GalaxyRotationActivity
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Course
1101
Subject
Astronomy
Date
Dec 6, 2023
Type
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6
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Astronomy 1101
Galaxy Rotation and Dark Matter
Lab Activity Worksheet
Galaxy Rotation
Figure 1, below, shows a visible-light image of the galaxy NGC3198, a galaxy that is roughly
half the size and half the total luminosity of the Milky Way:
Figure 1
: Visible-light image of galaxy NGC 3198 – a galaxy is located in the constellation Ursa
Major.
This image was taken as part of Advanced Observing Program (AOP) program at Kitt Peak Visitor Center
during 2014. Credit: KPNO/ NOIRLab/ NSF/ AURA/ John Vickery and Jim Matthes/ Adam Block
1.
How would
you
classify NGC 3198 using the Hubble’s Galaxy Classification scheme from
the book?
S
piral galaxy
2.
Based just on this
image
, do you think the galaxy is rotating clockwise (as seen from
our vantage point) or counter-clockwise? Explain the basis of your answer.
Counter clockwise based on the how the arms are bent
3.
Based on the colors you see in the image, what can you say about the ages of stars
you would find in this galaxy? Think back to what you know about stellar evolution
and what you know about galaxies.
They are young hot. Stars on the arms and in the center they are older stars because they are
more yellow.
1
Figure 2 (below) shows the luminosity profile of NGC 3198: the total luminosity of all stars
inside radius R, where R is the distance from the center of the galaxy. The vertical axis is in units
of billions of L
sun
. The radius R on the horizontal axis is in units of kiloparsecs (kpc).
Figure 2:
Luminosity Profile of NGC 3198
4.
What are the units of the vertical axis of the graph?
Billions times the luminosity of the sun
As you can see, the total luminosity of ALL the stars in NCG 3198 is about
9 Billion L
sun
. About
half of that light comes from inside the radius R = 5 kpc and about half of comes from between
5 and 15 kpc. The curve stops increasing at 15 kpc because there are almost no stars beyond that
radius in NGC 3198, as you can see from Figures 1 & 2.
If all of the stars in NGC 3198 were exactly like the Sun, we could just multiply the total
luminosity by 1 M
sun
/ L
sun
(1 solar mass per unit of solar luminosity) to infer that the total mass
of all the stars in the galaxy is about 9 billion M
sun
. However, the light actually comes from a mix
of main sequence stars, red giant stars, white dwarfs, etc. with a wide range of luminosities and
masses. By the time you average over all the stars and their numbers, it turns out that you need to
multiply by a mass-to-light ratio of about 4 M
sun
/ L
sun
to infer the stellar mass.
Stellar Mass = Interior Luminosity
×
(4 M
sun
/ L
sun
)
2
The stellar mass of NGC 3198 is: (9 billion L
sun
)
×
(4 M
sun
/ L
sun
) = 36 billion M
sun
Figure 3 (below) shows the rotation curve of NGC 3198, a plot of the disk’s rotation speed
(measured from Doppler shifts) versus distance from the center of the galaxy. The rotation curve
is measured from radio observations of hydrogen gas, and it extends to 30 kpc, roughly twice the
radius of the galaxy in the visible image.
Figure 3:
Rotation Curve of NGC 3198
We can also use Newton’s laws of gravity to infer the mass of the galaxy from the velocities of
the gas, just as we have used the orbital speeds of stars to infer the mass of the Galactic Center
black hole in a previous lab. Here we use our familiar formula M = v
2
×
R/G, but to make your
lives easier, we have converted the units for you to express it in a scaled form:
Total Mass = (2.3
×
10
5
M
sun
)
×
(v in km/s)
2
×
(R in kpc).
When we apply this formula to the measured rotation speed at radius R, it tells us the total mass
of the galaxy inside a circle of radius R. For example, if you measured a velocity of 110 km/s at
a radius of 2 kpc, you would infer that the mass inside 2 kpc is
Total Mass inside 2 kpc = (2.3
×
10
5
M
sun
)
×
(110 km/s)
2
×
(2 kpc) = 5.6
×
10
9
M
sun
3
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5.
Using
the
Stellar Mass
formula and the luminosity profile in Figure 2:
Stellar Mass = Interior Luminosity
×
(4 M
sun
/ L
sun
)
along with the
Total Mass
formula and the observed rotation curve shown in Figure 3:
Total Mass = (2.3
×
10
5
M
sun
)
×
(v in km/s)
2
×
(R in kpc)
Compute the “stellar mass” and “total mass” of NGC 3198 interior to these 5 radii on the
next page: R = 2 kpc, 5 kpc, 15 kpc, 20 kpc, and 30 kpc.
Computed Masses
R = 2 kpc
Stellar mass interior to R, from luminosity profile:
6 x 10^9 Msun
Total mass interior to R, from rotation curve:
5.6 x 10^9 Msun
R = 5 kpc
Stellar mass interior to R, from luminosity profile:
19 x 10^9 Msun
Total mass interior to R, from rotation curve:
2.45 x 10^10 Msun
R =15 kpc
Stellar mass interior to R, from luminosity profile:
34.8 x 10^9 Msun
Total mass interior to R, from rotation curve:
7.87x 10^10 Msun
R = 20 kpc
Stellar mass interior to R, from luminosity profile:
36 x 10^9 Msun
Total mass interior to R, from rotation curve:
7.46 x 10^10 Msun
4
R = 30 kpc
Stellar mass interior to R, from luminosity profile:
36 x 10^9 Msun
Total mass interior to R, from rotation curve:
1.55 x 10^11
5
6.
Is the stellar mass interior to 2 kpc sufficient to explain the observed rotation speed measured
at 2 kpc? In other words, do the stars themselves have enough mass to approximately
produce the gravitational force necessary to explain the speed of the gas at 2 kpc?
The stellar mass would be sufficient because the stellar mass and total mass are about the same.
7.
Is the stellar mass interior to 30 kpc sufficient to explain the rotation speed measured at
30 kpc?
The stellar mass interior of 30 is not sufficient to explain the rotational speed because the total
mass is larger than the stellar mass.
8.
Between 2 kpc and 5 kpc, by what ratio does the stellar mass increase (M
5kpc
/ M
2kpc
)?
19/6 =3.17
9.
Between 2 kpc and 5 kpc, by what ratio does the total mass increase (M
5kpc
/ M
2kpc
)?
(4.45 x 10^10)(5.6 x 10^9)=4.375
10. Between 15 kpc and 30 kpc, by what ratio does the stellar mass increase (M
30kpc
/ M
15kpc
)?
(36 x 10^11)(34.8 x 10^9) = 1.034
11. Between 15 kpc and 30 kpc, by what ratio does the total mass increase (M
30kpc
/ M
15kpc
)?
(1.55 x 10^11)(7.87 x 10^10) = 1.97
12. Name 2-3 types of astronomical objects that you know about have a lot of mass but emit little
or no light?
Neutron stars, white dwarfs, black holes
13. Propose a hypothesis to explain the observed rotation curve of NGC 3198 – specifically, to
explain why the change of rotation with radius is different from what is predicted based on
the star light alone.
There is dark matter that we cannot observe which can explain the change of rotation and why
the star light alone cannot predict rotational speed.
6
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