ICW 14 - Cosmic Distances
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School
University of Washington, Bothell *
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Course
101
Subject
Astronomy
Date
Dec 6, 2023
Type
Pages
6
Uploaded by JudgeRoseKoala29
Worksheet
—
Cosmic Distances
Page 1 of 6
Astronomy
BPHYS 101
Worksheet
—
Cosmic Distances
Identify Yourself
Radar Ranging
Read the information page on
Radar Ranging
.
Question 1:
Over the last 10 years, many
“
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?
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 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.
Name 1:
Name 2:
Name 3:
Worksheet
—
Cosmic Distances
Page 2 of 6
Question 2:
Enter your perpendicular distance to the boat in
map units
. __________________
Show your calculation converting the distance to the boat from
map units
to
meters
in the box below.
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:
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
perpendicular distance to the boat?
What is the greatest distance to the boat that is
still consistent with your observations?
What is the smallest distance to the boat that is
still consistent with your observations?
Configure the simulator to
[preset C]
, which limits the size of the baseline and has an error of 5° in each
angular measurement.
Question 4:
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.
Worksheet
—
Cosmic Distances
Page 3 of 6
Distance Modulus
Read the information page on
Distance Modulus
.
Question 5:
Complete the following table concerning the distance modulus for several objects.
𝑚
𝑀
𝑚 − 𝑀
𝑑
Object
Apparent Magnitude
Absolute Magnitude
Distance Modulus
Distance
[pc]
Star A
2.4
10
Star B
5
16
Star C
10
25
Star D
8.5
0.5
Question 6:
Could one of the stars listed in the table above be an RR Lyrae star? Explain why or why not.
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 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 be either 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
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Worksheet
—
Cosmic Distances
Page 4 of 6
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:
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
13.1
strong molecular lines
very thick
7.2
strong ionized metal lines
moderate hydrogen lines
very thick
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 vertical axes are aligned, but the two main sequences do not 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.
We can now relate the two vertical 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:
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?
Question 9:
Note that there are several stars below the main sequence, especially near temperatures of
about 5000 K. Can you explain why these stars are not on the main sequence?
Worksheet
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Cosmic Distances
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Question 10:
Complete the table and determine the distances to the listed star clusters.
𝑚
𝑀
𝑚 − 𝑀
𝑑
Object
Apparent Magnitude
Absolute Magnitude
Distance Modulus
Distance
[pc]
chi Persei
h Persei
Praesepe
NGC 3293
Messier 67
NGC 188
Hyades
Cepheid Variable Stars
Read the information page on
Cepheid Variable Stars
.
Question 11:
Suppose an observed Type II Cepheid has an apparent magnitude of 12 with a pulsation
period of 3 days. Determine the distance to the Cepheid variable and explain your method in the box
below?
Worksheet
—
Cosmic Distances
Page 6 of 6
Supernovae
Open up the
Supernovae Light Curve Fitting Explorer
.
It functions similarly to the Cluster Fitting
Explorer. The red line illustrates the expected profile fora 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 40Mpc.
Question 12:
Complete the table and determine the distances to the supernovae.
𝑚
𝑀
𝑚 − 𝑀
𝑑
Supernova
Apparent Magnitude
Absolute Magnitude
Distance Modulus
Distance
[pc]
1993J
1994I
1994ae
1999dq
1990N
Question 13:
Load the data for Supernova 1987A. Explain why it is not possible to determine the distance
to this supernova?
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