Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Dark Matter
I. Purpose
To evaluate the hypothesis that the luminous matter in a spiral galaxy accounts
for only a fraction of its total mass.
II. Background
Simply put, Dark Matter is matter which may only be detected by its gravitational
influence. In common use, “dark matter” might describe anything that does not
emit detectable radiation or reflect light from some other source. Many “ordinary”
objects in interstellar space may be counted as dark matter (rogue planets,
extinct stars, cold clouds of gas and dust, etc.).
If sufficiently heated, these things will emit some radiation. If illuminated, these
things will absorb, reflect, or scatter light from other sources. We know that such
ordinary matter accounts for only a tiny fraction of all of the gravitational mass in
the universe.
The Dark Matter in which cosmologist are most interested does not absorb or
radiate light.
Figure 1: Radial Velocity
Ordinary matter is sometimes referred to baryonic matter since most of its mass
is it is made up of baryons (protons and neutrons). Matter made of protons and
neutrons has mass, but must also interact with electromagnetic radiation (light).
Many researchers have hypothesized that much of the gravitational mass of the
universe, Dark Matter, is non-baryonic; that it is made of some as yet unobserved
particle that has mass, but does not interact with electromagnetic radiation.
1/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
In this exercise, we will compare the mass distribution of the galaxy NGC 2742
as measured by the orbital speed of stars at some distance from its center, to its
mass distribution as determined from an estimate of the number of stars with
orbital radii smaller than that distance.
We will treat this difference as the contribution of dark matter to the gravitational
mass of the galaxy.
III.Weighing the Galaxy
Figure 2 is the rotation curve for the Galaxy NGC 2742 (NGC stands for “New
General Catalog”, a list of “nebulae” compiled in the latter part of the 19th
century.) It plots the radial velocity (how fast objects are moving in the line of
sight direction, as shown in Figure 1) against the radial distance from the center
of the galaxy.
Figure 2: Rotation Curve
After Courteau, S.
1997, Astronomical
Journal, 114, 2402. .
One side of the galaxy is moving with a positive velocity (spinning away from us),
while the other side has a negative velocity (spinning towards us). These data
come from the measurement of spectral lines that have been either blueshifted
(stars moving towards us) or redshifted (stars moving away from us).
Since the plane of the galaxy is tipped at an angle to the line of sight of observers
on the earth, the velocity that we detect is not the total velocity of objects orbiting
around the center of NGC 2742. A correction has been made in the curve to take
2/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
into account the inclination angle. Using Newton’s gravity equation, we will be
able to determine the gravitational mass profile of the galaxy.
IV.Rotation Curves
Optical rotation curves, like the one in figure 2, are compiled using long slit
spectroscopy.
As we discussed in lecture, spectroscopic examination of the light emitted by a
distant source can be used to extract information about the temperature,
chemical composition, and motion of that object. In this lab, we are interested in
the motion of stars around the center of a galaxy, and so will use the Doppler
Shift of spectral lines in the light from stars at different positions along the visible
profile of the galaxy.
Since the light from a galaxy is actually the light from many sources, and since
the galaxy as a whole may be moving with respect to our point of observation, we
use long slit spectroscopy to collect the spectrographic and spatial information in
one image, comparing the motion on one part of the galaxy to the motion in a
different part of the
galaxy.
An optical slit is a very
narrow aperture cut
into an opaque screen.
To collect a long slit
spectrum, such an
aperture is interposed
between the collection
optics (telescope
lenses) and the
diffraction grating or
prism used to disperse
the light from the
source into a spectrum.
From Wikipedia
Figure 3
The slit is oriented so that it spans some dimension of the object under
examination. The spectrum so collected shows how the motion of that object
changes along that dimension.
By examining the position of an individual line, or a narrow group of lines, as we
move along the slit we can compare the radial velocity of objects at the edges of
the source to the radial velocity of objects close to the center of the source.
In this way, we can measure the motion of the stars around the center of the
galaxy separately from the motion of the galaxy as a whole
3/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Figure 4
The Doppler Shift only contains information about motion along the line of sight
(towards, or away from, the observer). That means we are only seeing the
component of motion towards or away from us. If we can work out the angle by
which the the plane of the galaxy is inclined to our line of sight, it is a fairly
straightforward exercise in trigonometry to determine the orbital speed of objects
moving about the center of mass of the galaxy.
The angle of inclination is measured with respect to the plane perpendicular our
line of sight. Thus, a galaxy with a large angle of inclination would be nearly
parallel to our line of sight and the radial velocity would be very close to the
orbital velocity. A galaxy with a small angle of inclination would be oriented more
nearly perpendicular to the line of sight to the observer, and the measured radial
velocity would be a small fraction of the orbital velocity.
We would expect, therefore that a highly inclined galaxy would be an ideal choice
for our observations.
There is another consideration, however. Notice the white lane in the image of
the galaxy above. This is gas and dust in the plane of the galaxy. This material
blocks the light from the inner regions of the galaxy, and so interferes with both
the spectroscopy and photometry used to determine the mass of the galaxy.
A better choice, therefore, would be a galaxy with a well characterized angle of
inclination; where stars across the width of the galaxy are clearly visible.
At a distance of about 65 million light years, and an inclination of about 45°, NGC
2742 is such an object.
4/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Figure 5
We want to use this velocity profile to extract a gravitational mass distribution for
the galaxy.
As we discussed in the unit on celestial mechanics, we can combine Newton’s
Law of Gravitation and Kepler’s Harmonic Law in a general law for the period of
orbit of any body around some more massive body.
4π 2
T=
GM
kpc ⋅ km2
G = 4.31× 10
M☉s 2
−6
Where M is the mass inside the radial distance R.
For convenience, G is here expressed in terms of kiloparsecs and
solar masses (M☉ )
Noting that the average linear (tangential) speed of the orbiting body is perimeter
of the orbit (the linear distance traveled in on revolution) divided by the period
(the time it takes to complete one circuit), we can write an expression for the
central mass of the system (the mass of thing about which the orbiting body
revolves).
M=
v 2R
G
G = 4.31× 10−6
kpc ⋅ km2
M☉s 2
Where M is the mass inside the radial distance R from galactic center,
and M☉ is one solar mass (the mass of our sun).
5/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
The linear velocity is the speed we compute from the doppler shift, these are the
values in Figure 1.
To simplify our calculations, we may write this as
( )
1 2
M=
v R
G
Where v is measured in
km
s
M☉s 2
1
5
= 2.32 × 10
G
kpc ⋅ km2
(Equation 1)
, the distance R from galactic center is measured in
kiloparsecs (kpc), and M☉ is one solar mass (the mass of our sun).
You will recall that we developed these formulae for two bodies which were small
compared to the distance separating their centers.
For stars orbiting the center of a galaxy, force pulling the motion of the star into
an orbit is due ALL of the mass (stars, gas clouds, planets, and anything else)
inside the radius of its orbit.
With a little mathematics (which is beyond the scope of the class) we can
demonstrate that star orbiting a radially symmetric distribution of mass will
experience exactly the same force of attraction as it would if all of the mass
inside its orbital radius was concentrated in a single point at the center of the
orbit. That is to say, that if the mass per unit volume does not change along any
circumference around the center of the galaxy (over appropriate distance scales),
then we can treat all of the mass inside of the orbit of a particular radial distance
from the center of the galaxy as if it were a single, small, body at the center of
circle described by that radius.
So we may use Equation 1 to compute the total mass enclosed by the orbit of
any body for which we know the orbital speed and the radius of its orbit.
V. Luminosity Profile
Figure 5 is a photograph of NGC2742 taken with the Schmidt Telescope at the
Palomar Observatory. It is a spiral galaxy. Its elliptical appearance in this image
is due to the inclination of the plane of the galaxy to or line of sight. If we could
see the galaxy from “above”, it would look something like the cartoon galaxy in
figure 1; a circular object with several spiral arms of bright stars, separated by
regions of dimmer stars and clouds of cold gas and dust.
You will note that the brightest part of the galaxy is a small region close to its
center, and that it gets gradually dimmer as you look farther away from the
center .
Since we want to compare the total gravitational mass enclosed by some radius
to the amount of luminous mass enclosed by that same boundary, we need to
measure the total light emitted by all of the glowing stuff (stars and hot gas)
inside of some region of the galaxy.
6/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Figure 6
This quantity is called the “Surface Brightness” of the galaxy. We can make
estimate of the way that the surface brightness varies with radial distance from
the center of the galaxy by taking an image like the one in figure 2 and
calculating the average brightness (“whiteness”) of all of the pixels with some
elliptical region (representing an orbital radius), then multiplying that average by
the total area enclosed by that ellipse.
Figure 6 shows the results of such an analysis. Note that the the average
brightness decreases (the averaged area gets darker) as the elliptical area
increases.
If we collect all of the light from the region of the sky spanned by the galaxy, and
account for things like scattering by gas and dust in the interstellar medium, and
the light of nearby stars which lie along our line of sight to the galaxy, we can
compute a total surface luminosity of about 7.7 × 109 times the luminosity of the
sun.
7/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Figure 7
Using this value for the total light emitted by the stars enclosed by the largest
ellipse (in the upper left of figure 6), we can construct a plot of surface luminosity
as a function of radial distance from the center of the galaxy (figure 7).
Note that the curve on that plot is included merely as a guide to the eye. It is not
the prediction of any theory of galactic structure (although it agrees well with
theory).
Remember that luminosity is dependent on the mass and age of a star. By
examining the spectrographic types evident in large sample of galaxies like NGC
2742 we know that most of the light we see is due to relatively small number of
young, hot, and bright stars. There are also many more dim, cool, low mass
stars than than there are bright massive stars in any region of a spiral galaxy.
From an examination the numbers of stars of different masses and luminosities
of stars in our own galaxy, we can estimate a ratio of about 1.23 solar mass
units per solar luminosity unit over the entire volume spanned by the galaxy.
You will use this ratio to estimate the mass of luminous material as a function of
radial distance from the center of the galaxy.
8/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
VI.Procedure
A. Select 7 radii, all of which are to one side or the other of 0 kpc and are evenly
spaced to the edge of the galaxy. Record those radii in Table 1 on the
worksheet.
B. Looking at figure 1, determine the radial velocity at each of your radii. Record
those values in the space provided in Table 1.
C. Use your values for Radius and Velocity to determine the mass of the galaxy.
( )
1 2
M=
v R
G
M☉s 2
1
5
= 2.32 × 10
G
kpc ⋅ km2
(Equation 1)
Record your results in the “Gravitational Mass” column in Table 1.
NOTE: The galaxy is comprised of billions of stars. Your numbers should be
very large.
Now that we’ve found the total mass of the galaxy, we will investigate how much
of that mass comes from luminous matter like stars and gas.
The gravitational mass we calculated in step C included all the mass inside of a
spherical shell of some radius. The curve in figure 3 represents the light from all
of the stars inside a particular radius.
For the purposes of calculating the mass of the luminous material in the galaxy
we estimate 1.23 solar masses for each unit of solar luminosity.
D. Look at Figure 7. At the radii that you have chosen, determine a value for the
integrated luminosity of the galaxy and record it in Table 1
E. For every solar luminosity, count 1.23 solar masses and record the number
in the “Radiation Mass” column of Table 1.
F. Plot each of the masses (gravitational and “radiation”) versus radius on the
grid provided. Clearly label each plot.
G. Answer the questions on the worksheet.
References:
Stinson, Greg. "Dark Matter." University of Washington Astronomy Department. Web. Retrieved
17 Nov. 2011.
LeDrew, Glenn. "The Real Starry Sky." Journal of the Royal Astronomical Society of Canada 95
(2001): 32-33. Print.
This work is licensed under a Creative Commons License.
9/13
Astronomy 101 !Assesment Lab !!
10/13
!
!
!
!
!
DARK MATTER
Astronomy 101 !Assesment Lab !!
!
!
To receive credit for this
assignment, complete the data
table, plot, and answer the
!
!
!
DARK MATTER
Name ____________________
Partners____________________
____________________
____________________
Table 1
A. Radius
B. Rotational
C. Gravitational Mass
D. Luminosity
E. Luminous Mass
(kPc)
Velocity (km/s)
(solar masses)
(solar luminosities)
(solar masses)
F. Plot
11/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
G. Questions
1. Determine the mass to light ratio for NGC 2742 at your largest radius by
dividing your gravitational mass by your radiation mass.
a) Put the mass to light ratio another way: What percentage of the total mass
is luminous?
b) What percentage of the total mass cannot be accounted for in the light
that we see (we call this the dark matter)?
2. Figure 7 plots how much luminosity is produced inside of some radius. It does
not decline at large radii even though the galaxy gives off less light out there.
Explain why.
3. We assumed that for every 1 solar luminosity that we see, there are 1.23
solar masses of matter.
a) How would our assumption for the amount of luminous matter change if
NGC 2742 has more low mass stars than we thought, but remains just as
luminous? Remember that a star’s luminosity depends sensitively on its
mass.
b) Following from part a, how would the mass to light ratio (total mass
divided by luminous mass) change if NGC 2742 contains more low mass
stars than we thought?
12/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
4. Why would these galaxies be inappropriate for our search for dark matter?
Answer for each of the pictured galaxies.
M101 The Pinwheel Galaxy
M104 The Sombrero
a. M101:
b. M104:
5. Comment on the existence of Dark Matter. Are you convinced that it exists?
13/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
Notes on rewrite of the Dark Matter Lab
The original version of this lab is available from the UW Astronomy Department
Laboratory Clearinghouse website and seems to be pretty widely use (by
evidence of its appearance on other college astronomy websites).
The lab required significant revision to bring it up to standard for Astronomy 101
at Clark. The original lab exists as a series of linked webpages. The graphics
are confusing and poorly made. The Translation from HTML to regular text
introduced typographic problems. The sarcastic tone probably worked when
Stinson was presenting the class himself, to students familiar with his style, but
the “humor” is not portable.
Also, as originally written, the lab compares the mass derived from the radial
velocity curve (actual data) to a theoretical model for the surface luminosity which
is well beyond the math expectations of a 100-level astronomy class, multiplied
by the application of a somewhat whimsical mass:luminosity correlation.
While rewriting the description of this model, to remove the smirking
condescension, I became concerned that the abstruse mathematics and
insufficiently supported correlation might make it difficult for students to achieve
the pedagogical goals of the lab.
Inspired by Stinsonʼs comment
that “There might be some
simple trick that aleviates [sic]
the problem [of producing an
actual surface luminosity plot
for NGC 2724]” I grabbed an
image from the HST Phase II
sky survey and did a quick test
to see if we could generate a
surface luminosity curve for
NGC2742.
I brought the image into
Photoshop, where I duplicated
it 40 times.
I selected an elliptical region of
the same aspect ratio as the
galaxy, and scaled about
1/36th of the apparent area spanned by the galaxy.
14/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
I centered this selection on the brightest region in the galaxy, and executed an
“average blur”, which computes the arithmetic mean of grayscale values of the
pixels in the region of interest and then fills the selected region with that color.
I repeated this for each of the images, increasing the dimensions of the blurred
region in uniform increments, and recording the grayscale value of the averages
region. I then multiplied the grayscale by the area of the enclosing ellipse for
each image, and plotted the resulting number as a function of the major axis of
the ellipse.
Proxy Luminosity
Grayscale*Selected Area (GS•px2)
pseudo light curve
80000
y = -0.0013x3 + 0.8347x2 + 32.717x - 859.18
R² = 0.998
64000
48000
32000
16000
0
0
78
157
235
313
392
470
Proxy Radius
Length of Major Axis of Selection (px)
I fitted a 3rd order polynomial as a guide to the eye, to help the students pull
values from the graph. I might as well have fitted the exponential function
Stinson used, but I didnʼt want to plug in the formula.
The contour of the curve is a close match for the curve provided by Stinson. I
think the outliers between 390 and 470 px are due to a nearby star in the field.
The apparent magnitude of NG 2742 is variously reported as 11.4 to 12.3.
Picking an intermediate value of 11.7, we compute an absolute visual magnitude
of -19.90 which, according to the Wolfram Alpha calculator, corresponds to a
luminosity of about 7.77 x 109 Lsun.
15/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
To get from solar luminosities to solar masses I tried to find a reference for the
prevalence of stars by mass and luminosity in the Milky Way. This turns out to
not to be a simple question.
I did, however, find one reference:
Title: The Real Starry Sky
Authors: Ledrew, G.
Journal: Journal of the Royal Astronomical Society of Canada, Vol. 95, p.32
Bibliographic Code: 2001JRASC..95...32L
PDF here.
The author estimates a mass to light ratio of 1.23 solar masses/solar luminosity
for a 1000 psc3 volume centered on the Sun.
I do not expect that this is a particularly meaningful number, but it is derived from
analysis of actual data, so I thought it might lend the exercise some
verisimilitude. Also, multiplied by luminosity provided by Wolfram Alpha, it is to
the same order of magnitude the values provided by Stinsonʼs 2× correlation to
the modeled curve.
1.1E+10
Solar Masses of Luminous Matter
To verify that
students would
get the same
qualitative result.
I pulled a few
points from the
modeled curve,
multiplied by the
2× correlation
per Stinson, and
plotted them on
the pseudo light
curve, scaled to
the luminosity
per Wolfram and
mass ratio per
LeDrew.
8.3E+09
5.5E+09
2.8E+09
0E+00
0
2.75
5.50
8.25
11.00
Distance from Center (Kpc)
The result was an unreasonably close match. In the figure below, the green
circles were pulled from Stinsonʼs curve at 1 Kpsc intervals. The blue crosses
are the pseudo light curve, normalized to mass of 9.56 × 109 solar masses (7.77
× 109 solar luminosities by 1.23 solar masses per) and scaled to the optical
diameter of NGC 2742.
16/13
Astronomy 101 !Assesment Lab !!
!
!
!
!
!
DARK MATTER
I honestly do not know what to make of this. Certainly, this suggests that we can
use the curve generated from analysis of the HST image of NGC 2742 for the
lab. However, there are a mess of assumptions that must be explained to the
students; including, but not limited to:
• The luminosity of the image in the range of wavelengths sampled by the HST
camera is representative of the total radiation from sources in the galaxy
across all visible wavelengths.
•
The distribution of stars by mass and luminosity in the small volume of the
Milky Way surveyed by LeDrew is representative of the distribution of masses
and luminosity's for all sources across the diameter of NGC 2742.
I donʼt know if this is that much of an improvement over Stinsonʼs theoretical
curve.
Since we are evaluating this lab for the standards assessment I did not rewrite
the questions (“...depends sensitively...”), except to reflect changes in the figure
numbers and to clean up the graphics.
--john
17/13