Name:
Lesson 6 Lab - The Cosmic Distance Ladder
Exercises
The Cosmic Distance Ladder Module consists of material on seven different distance
determination techniques. Four of the techniques have external simulators in addition to
the background pages. You are encouraged to work through the material for each technique
before moving on to the next technique.
Radar Ranging
Question 1: (2 points) Over the last 10 years, a large number of iceballs have been found
in the outer solar system out beyond Pluto. These objects are collectively known as the
Kuiper Belt. An amateur astronomer suggests using the radar ranging technique to learn
the rotation periods of Kuiper Belt Objects. Do you think that this plan would be
successful? Explain why or why not?
Radar ranging cannot be used to determine the rotational periods of Kuiper belt objects.
The sound wave reflection method is a technique that is similar to this radar method. If
you send a sound wave (speak or shout) into a cave, you will hear an echo because of
sound wave reflection. Similar to this, radar technology requires sending radar signals to
spacecraft and returning those signals to Earth.
Parallax
In addition to astronomical applications, parallax is used for measuring distances in many
other disciplines such as surveying. Open the Parallax Explorer where techniques very
similar to those used by surveyors are applied to the problem of finding the distance to a
boat out in the middle of a large lake by finding its position on a small scale drawing of the
real world. The simulator consists of a map providing a scaled overhead view of the lake
and a road along the bottom edge where our surveyor represented by a red X may be
located. The surveyor is equipped with a theodolite (a combination of a small telescope and
a large protractor so that the angle of the telescope orientation can be precisely measured)
mounted on a tripod that can be moved along the road to establish a baseline. An
Observer’s View panel shows the appearance of the boat relative to trees on the far shore
through the theodolite.
Configure the simulator to preset A which allows us to see the location of the boat on the
map. (This is a helpful simplification to help us get started with this technique – normally
the main goal of the process is to learn the position of the boat on the scaled map.) Drag
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the position of the surveyor around and note how the apparent position of the boat relative
to background objects changes. Position the surveyor to the far left of the road and click
take measurement which causes the surveyor to sight the boat through the theodolite and
measure the angle between the line of sight to the boat and the road. Now position the
surveyor to the far right of the road and click take measurement again. The distance
between these two positions defines the baseline of our observations and the intersection
of the two red lines of sight indicates the position of the boat.
We now need to make a measurement on our scaled map and convert it back to a
distance in the real world. Check show ruler and use this ruler to measure the distance
from the baseline to the boat in an arbitrary unit. Then use the map scale factor to calculate
the perpendicular distance from the baseline to the boat.
Question 2: (2 points) Enter your perpendicular distance to the boat in map units.
______________ Show your calculation of the distance to the boat in meters in the box
below.
1au = 3 metres
7.5au = 20m/au x 7.5au = 150m
Configure the simulator to preset B. The parallax explorer now assumes that our surveyor
can make angular observations with a typical error of 3°. Due to this error we will now
describe an area where the boat must be located as the overlap of two cones as opposed to
a definite location that was the intersection of two lines. This preset is more realistic in
that it does not illustrate the position of the boat on the map.
Question 3: (2 points) Repeat the process of applying triangulation to determine the
distance to the boat and then answer the following:
What is your best estimate for the The perpendicular distance =
perpendicular distance to the boat?
6.6au=6.6 multiplied by 20m = 132m.
What is the greatest distance to the boat that The greatest perpendicular distance =
is still consistent with your observations?
7.5au = 7.5 multiplied by 20m = 150m.
What is the smallest distance to the boat that The perpendicular shortest distance =
is still consistent with your observations?
5.9au = 5.9 multiplied by 20m = 118m.
Configure the simulator to preset C which limits the size of the baseline and has an error
of 5° in each angular measurement.
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Question 4: (2 points) Repeat the process of applying triangulation to determine the
distance to the boat and then explain how accurately you can determine this distance and
the factors contributing to that accuracy.
Because the baseline is so short, it is not possible to obtain an accurate distance
measurement for the boat.
Distance Modulus
Question 5: (2 points) Complete the following table concerning the distance modulus for
several objects.
Apparent
Absolute
Distance
Magnitude m
Magnitude M
Modulus mObject
Distance (pc)
M
Star A
2.4
2.4
0
10
Star B
6
5
1
16
Star C
10
0
2
25
Star D
8.5
0.5
0
3.98
Question 6: (2 points) Could one of the stars listed in the table above be an RR Lyrae star?
Explain why or why not.
A RRLyrae star exhibits an apparent magnitude between 7 to 8.5 and an absolute
magnitude from 0.6 to 0.1.
Spectroscopic Parallax
Open up the Spectroscopic Parallax Simulator. There is a panel in the upper left entitled
Absorption Line Intensities – this is where we can use information on the types of lines
in a star’s spectrum to determine its spectral type. There is a panel in the lower right entitled
Star Attributes where one can enter the luminosity class based upon information on the
thickness of line in a star’s spectrum. This is enough information to position the star on the
HR Diagram in the upper right and read off its absolute magnitude.
Let’s work through an example. Imagine that an astronomer observes a star to have an
apparent magnitude of 4.2 and collects a spectrum that has very strong helium and
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moderately strong ionized helium lines – all very thick. Find the distance to the star using
spectroscopic parallax.
Let’s first find the
spectral type. We can see in the
Absorption Line Intensities panel
that for the star to have any
helium lines it must be a very hot
blue star. By dragging the
vertical cursor we can see that for
the star to have very strong
helium and moderate ionized
helium lines it must either be O6
or O7. Since the spectral lines
are all very thick, we can assume
that it is a main sequence star.
Setting the star to luminosity
class V in the Star Attributes panel
then determines its position on the HR Diagram and identifies its absolute magnitude as 4.1. We can complete the distance modulus calculation by setting the apparent magnitude
slider to 4.2 in the Star Attributes panel. The distance modulus is 8.3 corresponding to a
distance of 449 pc. Students should keep in mind that spectroscopic parallax is not a
particularly precise technique even for professional astronomers. In reality, the luminosity
classes are much wider than they are shown in this simulation and distances determined by
this technique are probably have uncertainties of about 20%.
Question 7: (2 points)Complete the table below by applying the technique of spectroscopic
parallax.
Observational Data
Analysis
m
Description of spectral lines
Description of line
thickness
M
m-M
d
(pc)
6.2
strong hydrogen lines
moderate helium lines
very thin
-7.4
13.6
5,240
13.1
strong molecular lines
very thick
16
-2.9
2.6
7.2
strong ionized metal lines
moderate hydrogen lines
very thick
3.6
3.6
52.6
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Main Sequence Fitting
Open up the Cluster Fitting Explorer. Note that the main sequence data for nearby stars
whose distances are known are plotted by absolute magnitude in red on the HR Diagram.
In the Cluster Selection Panel, choose the
Pleiades cluster. The Pleiades data are then
added in apparent magnitude in blue. Note that
the two y-axes are aligned, but the two main
sequences don’t overlap due to the distance of
the Pleiades (since it is not 10 parsecs away).
If you move the cursor into the HR diagram, the
cursor will change to a handle, and you can shift
the apparent magnitude scale by clicking and
dragging. Grab the cluster data and drag it until
the two main sequences are best overlapped as
shown to the right.
We can now relate the two y-axes. Check show
horizontal bar which will automate the process
of determining the offset between the m and M
axes. Note that it doesn’t matter where you
compare the m and M values, at all points they
will give the proper distance modulus. One set
of values gives m –
M = 1.6 – (-4.0) = 5.6 which corresponds to a distance of 132 pc.
Question 8: (2 points) Note that there are several stars that are above the main sequence in
the upper left. Can you explain why these stars are not on the main sequence?
The main sequence stage stars are the stars that burn hydrogen to produce helium in their
core through the nuclear fusion process. Our Sun is also one such star. Once, the star runs
out of hydrogen fuel, they move away from the Main Sequence Stage i.e., above the main
sequence stage and turns into a red giant.
Question 9: (2 points) Note that there are several stars below the main sequence especially
near temperatures of about 5000K. Can you explain why these stars are not on the main
sequence? When a main sequence star burns through the hydrogen in its core, reaching the
end of its life cycle. At this point, it leaves the main sequence. Stars smaller than a 1/4 the
mass of the sun collapse directly into white dwarfs.
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Question 10: (2 points) Determine the distance to the Hyades cluster.
Apparent magnitude m
0.5
Absolute Magnitude M
Distance (pc)
1.3197
42
Question 11: (2 points) Determine the distance to the M67
cluster.
Apparent magnitude m
14.3
Absolute Magnitude M
Distance (pc)
7.6
219
Cepheids
Question 12: (1 points) A type II Cepheid has an apparent magnitude of 12 and a pulsation
period of 3 days. Determine the distance to the Cepheid variable and explain your method
in the box below?
12-(-1)=-5+5log10d
18/5=log10d
d=1018/5
d= 3981pc
d = 3981 pc
This is correct as taking 12-(-1) and using the pc distance chart it gives me an answer of
4000pc.
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Supernovae
Open up the Supernovae Light Curve Explorer. It functions similarly to the Cluster
Fitting Explorer. The red line illustrates the expected profile for a Type I supernovae in
terms of Absolute Magnitude. Data from various supernovae can be graphed in terms of
apparent magnitude. If the data represents a Type I Supernovae it should be possible to fit
the data to the Type I profile with the appropriate shifts in time and magnitude. Once the
data fit the profile, then the difference between apparent and absolute magnitude again
gives the distance modulus.
As an example load the data for 1995D. Grab and drag the data until it best matches the
Type I profile as shown. One can then use the show horizontal bar option to help calculate
the distance modulus. One pair of values is m – M = 13-(-20) = 33 which corresponds to
a distance of 40 Mpc.
Question 13: (1 points) Determine the distance to Supernovae 1994ae and explain your
method in the box below?
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I made the Type (I) profile match the nova. At day 0 it's apparent magnitude was 13
and it's absolute magnitude was -19.5. Therefore, 13-(-19.5) = 32.5. Taking this to the
distance modulus chart, I come up with roughly 3.2x107 pc or 32Mpc
Question 14: (1 points) Load the data for Supernova 1987A. Explain why it is not possible
to determine the distance to this supernova?
Supernova 1987A is not a Type (I) supernova, meaning this method will not work to
determine the distance. This is a Type II supernova which will result in a neutron star or
black hole
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