HRDiagram_Worksheet
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Ohio State University *
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Course
3500
Subject
Astronomy
Date
Dec 6, 2023
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13
Uploaded by ConstableSeaUrchin2285
Name: Alexander Chen
1
Astronomy 1102
Hertzsprung-Russell Diagram
Assignment Worksheet
A fundamental
physical
property of a star is its
luminosity
, the rate it radiates energy into space. The
corresponding
observable
property of a star is its
apparent brightness
(or
brightness
for short), a
measurement of how bright it
appears
to us as seen from a distance here on the Earth.
The brightness of a star depends on its luminosity and its distance: if two stars have the same
luminosity, the more distant one appears fainter. This is the same effect as seeing two 100-watt light
bulbs (which have the same luminosity) at different distances – the one across the street appears fainter
than the one on the table next to you because light from the distant bulb gets spread out over a much
larger area before it reaches you. To estimate the luminosity of a star we must measure its brightness
and its distance. The distances to nearby stars are measured using their parallaxes.
Another basic observable property of a star is its
color
. Astronomers typically describe color
quantitatively as a
color index
, the ratio of the star’s brightness seen at red and blue wavelengths. If a
star is red (it emits more red light than blue light) the color index is positive; more positive is redder.
If instead the star is blue (emits more blue light than red light), the color index is negative; more
negative is blue. For purposes of this assignment, you just need to know that the color indexes of stars
lie between
0.5 and
2.5; stars with color index color index of
0.5 are the bluest stars, while stars
with color index of +2.5 are the reddest stars.
The color of a star depends on its
surface temperature
. Hotter stars have bluer colors and therefore
more negative color indexes, while cooler stars have redder colors and more positive color indexes.
This makes color index a proxy for surface temperature.
For reference, our Sun has a surface temperature of 5772 Kelvin and so emits primarily pale-yellow
light. Its color index is positive (about +0.66).
In this assignment, you will first compute the luminosities of a few stars from their observed brightness
and distance. You will then make and examine a Hertzsprung-Russell (or H-R) Diagram, a plot of
luminosity vs. color index, and then apply what you learned to H-R diagrams for nearby stars.
The H-R diagram is one of the primary tools that astronomers use to understand the properties of stars.
The first versions of the such diagrams were made by the Danish astronomer Ejnar Hertzsprung and
the American astronomer Henry Norris Russell around 1910. Using H-R diagrams, astronomers have
been able to piece together the life cycle of stars and how stars relate to each other.
The assignment has 4 parts and is meant to be able to be done entirely on this worksheet. You can
write directly on the PDF file of the worksheet using a tablet app like Notability or OneNote, or you
can write on a paper copy and scan it into digital form to submit as your homework.
2
Part 1: From Distance and Brightness to Luminosity
Brightness, luminosity, and distance of an object are related in the following way :
Brightness = Luminosity ÷ (4π d
2
)
where
d
is the distance. If you move a star 2 times farther away from you it will appear to be 2
2
= 4
times fainter than before, but its total energy output (luminosity) is the same. If we measure a star’s
brightness and its distance, we can estimate its luminosity by:
Luminosity = Brightness × (4π d
2
)
The star Alpha Centauri (α Cen) is one of the closest stars to the Sun, at a distance of
d
= 4.37 light years = 4.13 × 10
16
meters. The apparent brightness of α Cen is
Brightness = 2.71 × 10
8
watts/m
2
(watts per square meter).
1.
From the above equation, what is the luminosity of α Cen, in watts?
Luminosity of α Cen = _____5.80870416*10^26__________
2.
What is the ratio of the luminosity of α Cen to the luminosity of the Sun, 3.828 × 10
26
watts?
(Luminosity of α Cen / Luminosity of Sun) = ______1.517*10^52_________
If you did everything right, your answer to the last question should be about 1.5.
Now repeat this calculation for:
3.
Betelgeuse
, the red star that is the left shoulder of the constellation Orion:
Brightness = 9.90 × 10
8
watts/m
2
Distance = 6.08 × 10
18
m
(Luminosity of Betelgeuse / Luminosity of Sun) = _____1.201*10^57__________
4.
Rigel
, the blue star that is the right knee of the constellation Orion:
Brightness = 5.68 × 10
8
watts/m
2
Distance = 8.02 × 10
18
m
(Luminosity of Rigel / Luminosity of Sun) = ______1.199*10^57_________
5.
Sirius B
, a faint blue star that is a binary companion to the bright star Sirius:
Brightness = 1.20 × 10
10
watts/m
2
Distance = 8.14 × 10
16
m
(Luminosity of Sirius b / Luminosity of Sun) = ______2.610*10^50_________
3
Part 2: Plotting an H-R diagram
The table below lists the color indices and luminosities of 50 stars whose distances were determined
via trigonometric parallax measurements. Luminosity is given in units of the Sun’s luminosity (e.g., a
Luminosity of 0.01 is 1/100
th
the luminosity of the Sun).
1.
Plot the positions of these 50 stars on the blank H-R diagram on the next page.
The stars with the * next to them in the table have been plotted for you; check that you understand
their locations on the plot. Plotting these points will take some time but watch for patterns as they
emerge.
Color Index
Luminosity
Color Index
Luminosity
0.89
0.2*
0.08
77
1.02
8*
0.45
5
0.55
0.0002*
1.04
34
1.04
0.1*
1.36
0.05
1.02
0.15
0.80
18
0.74
0.3
0.61
4.6
1.10
0.1
1.14
0.1
0.50
8
0.89
27
1.42
0.08
1.62
0.0009
0.32
7.6
1.12
0.07
0.20
14
1.08
21
0.39
8.7
0.96
0.1
0.24
8.8
0.52
3.7
1.41
0.03
1.47
0.004
0.72
2.4
0.97
0.2
0.31
13
0.77
0.3
1.08
29
0.65
1
1.02
0.2
0.69
3
0.08
58
0.86
0.6
0.07
58
1.07
0.2
0.50
2
0.93
29
1.03
31
0.53
9.4
0.72
0.7
0.13
27
0.36
0.0003
0.96
0.2
0.18
0.0006
0.26
0.0006
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2.
The color index of α Cen is 0.69. Plot it on your graph and label it.
Red dot
3.
The color index of Sirius b is
0.03. Plot it on your graph and label it.
Purple dot
4.
Where (roughly) would Rigel and Betelgeuse appear on this diagram? Explain in words.
Luminosity of Rigel is 4.590*10^31 and Luminosity of Betelgeuse 4.599*10^31, the luminosity of
them is very close, they will appear at luminosity approximate 4.6, and the color index also
approximately close to each other.
5
5.
Write down two things you notice about your handmade graph. Each of your observations should
be written as a complete sentence:
A.
On the H-R diagram, stars are not uniformly distributed. The main-sequence stars appear in a
downward sloping line.
B.
There are only a few white dwarfs.
6
Part 3: Making sense of the H-R diagram
As discussed in the lectures, the luminosity (L) of a star depends on its temperature (T) and its radius
(R) as given by this equation:
L = 4
R
2
T
4
,
where
is the Stefan-Boltzmann constant which has the same value for all stars (and can hence be
ignored for the following discussion). In words, what this equation tells us is:
If two stars have the same temperature, the larger radius star is more luminous.
If two stars have the same radius, the hotter star is more luminous.
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1.
With this information in mind, label the four corners of your H-R diagram to indicate where you
find stars that are
hot and big [where big means large radius]
hot and small
cool and big
cool and small.
2.
Based
only
on the H-R diagram, what do you think is the approximate color index of the Sun?
Explain your answer.
a Luminosity of 0.01 is 1/100
th
the luminosity of the Sun is 1 on y axis the approximate color index of
the Sun is between 0.5 to 1
3.
On your H-R diagram, mark and label (A, B, C) the location of three hypothetical stars that have
the following properties:
Star A has the same luminosity as the Sun but a cooler surface temperature.
Star B has the same temperature as the Sun but a larger radius.
Star C has the same luminosity as the Sun but a smaller radius
4.
Write down (in complete sentences) two hypotheses about things you have noticed in your H-R
diagram and/or two conclusions that you can draw about stars based upon the way the H-R diagram
looks.
A.
The stars is a line from hotter, more massive blue stars to cooler, less massive red stars.
8
B. The sun seems to be at the middle of the graph.
9
Part 4: The H-R Diagram of Nearby Stars
Figure 1 shows an H-R diagram of 1900 stars drawn from the Hipparcos parallax catalog.
Figure 1: H-R Diagram of Nearby Stars
1.
Write down four qualitative observations about things you notice in this H-R diagram. Each of
these observations should be expressed as a complete sentence. Some repetition with the
observations already made in Part 3 is OK
A.The stars is a line from hotter, more massive blue stars to cooler, less massive red stars.
B.The sun seems to be at the middle of the graph.
C. The H-R Diagram distribute like a big letter Y.
D. There is very few point on the button left, which there should be white dwarfs.
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Figure 2 shows the same 1900 stars as Figure 1 but now it plots the
apparent brightness
of the star as
seen from the Earth instead of
luminosity
. The vertical axis is scaled so that 1.0 represents the
brightness of α Centauri, one of the nearest stars to the Sun. Although there are stars brighter than α
Centauri in the sky, this graph shows only a fraction of the stars in the Hipparcos parallax catalog,
which is the reason that there are no stars above 1.0 in the graph.
Figure 2: Brightness versus Color Index of Nearby Stars
2.
Suppose that you used a digital camera and filters to measure the brightness and color index of
many stars. You therefore have the information needed to plot a graph like Figure 2. What
additional measurements would you need to make before you could plot a graph like Figure 1?
Would making these additional measurements be easy or hard? Explain your answer.
We need additional measurement of its luminosity and distance, therefore we can calculate the
luminosity of planets through Luminosity = Brightness × (4π d
2
), not know the distance would making
these measurement to be hard.
11
3.
Explain why Figures 1 and 2 look different. (Don’t just say how they look different, give a 1-3
sentence explanation of why the points have such different distributions in the two diagrams.)
They have different measurement on their Y axis. The distribution is different because the luminosity
is no equal to brightness. The figure 2vertical axis is scaled so that 1.0 represents the brightness of α
Centauri but not for figure one
4.
If you want to learn how stars work, which of these diagrams, Figure 1 or Figure 2, would be more
useful? Explain your answer.
If you want to learn how stars work the figure 1 would be more helpful, because the
Luminosity is an
absolute measure of radiated
electromagnetic power , the radiant power emitted by a light-emitting
object. We can learn how stars generate the heat and what form of stars it is.
12
Figure 3 shows an H-R diagram for the
nearest
200 stars to Earth.
Figure 3: H-R Diagram of the 200 nearest stars
Figure 4 shows an H-R diagram for the 200
brightest
stars as seen from Earth.
Figure 4: H-R Diagram of the 200 brightest stars
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5.
Explain why Figure 3 and Figure 4 look so different. (Hint: Think about how the distance and
luminosity of a star affect its apparent brightness.)
The luminosity of a star, is the amount of light it emits from its surface. The difference between
luminosity and apparent brightness depends on distance. Apparent brightness is not an intrinsic
property of the star it depends on your location. So, everyone will measure a different apparent
brightness for the same star if they are all different distances away from that star.
6.
Based on thinking carefully about Figures 3 and 4, which kind of star do you think is most
common throughout the Milky Way galaxy:
A. Stars with luminosity less than 0.001 solar luminosities.
B. Stars with luminosity greater than 1000 solar luminosities.
Explain the reasoning behind your answer
Stars with luminosity less than 0.001 solar luminosities is most common throughout the Milky Way
galaxy, because most of the star is less than solar luminosity, and on the graph most points is less than
0.001, therefore option A is most likely to be common.