hr diagram homework

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The traditional Hertzsprung-Russell (H-R) diagram activity is a plot of the nearest and brightest stars. This activity is an extension of the traditional plotting activity, and begins with an H-R diagram with enough bright and nearby stars plotted to define the shape and location of the main sequence, and the location of the giant and white dwarf branches. The activity is part of the Variable Star Astronomy (formerly Hands-On Astrophysics) curriculum developed by the American Association of Variable Star Observers (AAVSO) in Cambridge, MA. There are other versions of the activity and they will be described in the “Using the Variable Star/H-R Diagram Activity Materials in the Classroom” section below. The traditional H-R diagram plotting activity shows the different branches of the diagram and the location of giants, supergiants and white dwarfs – all part of the evolutionary track involving main sequence stars. However, it presents a rather static view of stellar evolution.

The emphasis of this activity is the plotting of four types of intrinsic pulsating variable stars – Cepheids, RR Lyraes, Miras and Semiregulars. These variable stars can be thought of as representing transition stages for some stars as they evolve from the main sequence and “move” to other branches of the H-R diagram. These variable stars occupy regions on the H-R diagram known as instability strips, and plotting their variability as they transition from one evolutionary stage to another gives a better perspective of stellar evolution as a continuously changing process. This is further emphasized by plotting these pulsating variables at both maximum and minimum brightness to show how much they vary in brightness and temperature as they transition through the instability strips.

The student handout includes all information necessary for completing the H-R diagram plotting activity. Teachers may use the background information for their own edification, or download all or part of the information for student use depending on individual classroom needs. The first 2 pages of the background information provides a description of the H-R diagram – including absolute magnitude, temperature, spectra, stellar classification, luminosity, and the major branches. The remaining 6 pages describe variable stars and light curves, cataclysmic and intrinsically pulsating variable stars, and H-R diagram instability strips. The background information includes links for additional in-depth information on stellar evolution and Type Ia and Type II supernovas on the Chandra website at http://chandra.harvard.edu/edu/formal/index.html .

The 7-page Student Variable Star H-R Diagram Activity handout provides all the information necessary for completing the activity – including the H-R diagram worksheet. The three sets of questions on page 4 are for the purpose of determining if students have a basic understanding of the H-R diagram, and the two sets of questions on page 6 are a discussion of their results. Answers will vary; however, students should have answers comparable to the following:

Before the activity:

1.) Plotting both the brightest and the nearest stars are necessary to see a normal or typical distribution of the stellar population, and the same distribution and percentage of stars on the individual branches would be seen from any position within the Milky Way Galaxy as there is no preferred view of the galaxy. This is the basis of the cosmological principle – that the universe is homogeneous and isotropic over large scale distances – the assumption that observers on Earth do not occupy a unique location within the universe.

2.) Main sequence stars have a specific relationship based on mass – the most massive stars have the brightest absolute magnitudes (luminosities) and highest surface temperatures, and the least massive stars have the dimmest absolute magnitudes and lowest surface temperatures. That relationship only holds for the main sequence – other branches have no specific relationship. As stars transition from the main sequence to other regions on the diagram, there are various tracks that they follow, and there are areas for which specific combinations of stellar luminosities and temperatures does not exist. Other stellar evolution stages and/or objects, such as supernovas, neutron stars and black holes are too extreme to be plotted on the H-R diagram.

3.) The answers will vary – from the introduction students will know that the pulsating variables occupy specific regions of instability so they will be able to reasonably predict the locations.

NOTE: A completed H-R diagram answer key with the plotted variables and a separate H-R diagram answer key with plotted variables and the branches labeled are provided.

After the activity:

1.) Various answers depending on the responses to question 3 above.

2.) The Cepheids, RR Lyaes and the Miras are grouped fairly closely together; however, the two Semiregulars are not. Students should understand that variable stars are classified from their light curves and think about why the two Semiregulars are not together. Semiregular stars are giants and supergiants so there is a large range in mass, which can lead to different evolutionary tracks along the H-R diagram.

This activity focuses on plotting pulsating variable stars. The student H-R diagram worksheet has bright and nearby stars already plotted. If you would prefer your students to plot the bright and nearby stars themselves before plotting the variable stars, a blank H-R diagram and the star data tables are available separately to download. The variable star data tables list the stars, spectral class, absolute magnitude and distance in parsecs. The distances are included because all the absolute magnitudes were calculated from parallax and apparent magnitude measurements by the HIPPARCOS mission. This allows for consistency for the absolute magnitude values. Another version of this activity on the AAVSO website lists the distance in parsecs but not the absolute magnitudes – the students have to calculate the absolute magnitudes using the parallax measurement and the distance modulus. All versions of the H-R diagram activity, including information for teachers, are part of the Variable Star Astronomy (VSA) curriculum posted on the AAVSO website: chapter 9 entitled The Life of a Star. Variable Star Astronomy (VSA) can be accessed at http://www.aavso.org/education/vsa .

The Chandra E/PO office has developed an activity with a scoring rubric that is useful as a post assessment for the Pulsating Variable Stars and the H-R Diagram activity. It is designed to be used as either a pre or a post assessment activity to determine student understanding of stellar evolution. The image set for the ac¬tivity includes images of the different stages of stellar evolution, light curves and H-R diagrams. (HTML, PDF and PowerPoint (PPT) versions) Educators can request as many classroom sets of the Stellar Life Cycle cards as necessary.

Stellar Life Cycles Page: http://chandra.harvard.edu/edu/formal/stellar_cycle/ The Stellar Life Cycles Card Sets Request Form: http://chandra.harvard.edu/edu/request_special.html Stellar Cycles Assessment Activity: http://chandra.harvard.edu/edu/formal/stellar_cycle/task_desc.html Teacher Guide and Answer Key: http://chandra.harvard.edu/edu/formal/stellar_cycle/guide.html Scoring Rubric: http://chandra.harvard.edu/edu/formal/stellar_cycle/rubric.html

Chandra is designed to observe X-rays from high-energy regions of the universe – including cataclysmic variables (supernovas, novas), and X-rays from binary systems such as the pulsating red giant Mira A and its white dwarf companion Mira B. As a result, the American Association of Variable Star Observers (AAVSO) and the Chandra X-Ray mission have collaborated with variable star observations and educational materials in their mutual quest to understand stellar processes and evolution. Two other activities and investigations from the Variable Star Astronomy materials, enhanced with extensions and flash versions, are posted on the Chandra website.

Stellar Heartbeats is an introductory activity designed to familiarize students with the magnitude scale and the Julian Day by estimating the changing magnitude of a variable star using comparison stars, plotting a light curve and determining the period. There are HTML, Flash, PDF, and PowerPoint versions. http://chandra.harvard.edu/edu/formal/variable_stars/activity1a.html

A Variable Star in Cygnus uses a set of photos of the variable star W Cyg. By using actual images of W Cyg students learn how to estimate the changing magnitudes of a variable star with actual comparison stars against a background of the real sky. Students then plot a light curve and determine the period. There are HTML, Flash, PDF, and PowerPoint versions. http://chandra.harvard.edu/edu/formal/variable_stars/activity2a.html

Chandra Chronicles Articles describing how the AAVSO amateur observers assisted the Chandra X-Ray Observatory for two observing campaigns of the variable star SS Cygni: Backyard Astronomers Trigger Multi-satellite Observing Campaign on SS Cygni Astronomers Team Up for Chandra Observations of SS Cygni

NATIONAL SCIENCE EDUCATION STANDARDS (Grades 9-12) http://www.nap.edu/openbook.php?record_id=4962&page=173

Formulate and revise scientific Explanations and Models Using Logic and Evidence: Student inquiries should culminate in formulating an explanation or model. Models should be physical, conceptual, and mathematical. In the process of answering the questions, the students should engage in discussions and arguments that result in the revision of their explanations. These discussions should be based on scientific knowledge, the use of logic, and evidence from their investigation.

Understandings about Scientific Inquiry: 5. Scientific explanations must adhere to criteria such as: a proposed explanation must be logically consistent; it must abide by the rules of evidence; it must be open to questions and possible modification; and it must be based on historical and current scientific knowledge.

The Origin and Evolution of the Universe 3. Stars produce energy from nuclear reactions, primarily the fusion of hydrogen to form helium. These and other processes in stars have led to the formation of all the other elements.

BENCHMARKS FOR SCIENCE LITERACY PROJECT 2061 (Grades 9-12) http://www.project2061.org/publications/bsl/online/index.php?home=true

1. THE NATURE OF SCIENCE

• Science is based on the assumption that the universe is a vast single system in which the basic rules are everywhere the same and that the things and events in the universe occur in consistent patterns that are comprehensible through careful, systematic study. 1A/H1*
• In science, the testing, revising, and occasional discarding of theories, new and old, never ends. This ongoing process leads to a better understanding of how things work in the world but not to absolute truth. 1A/H3bc*
• Sometimes, scientists can control conditions in order to obtain evidence. When that is not possible, practical, or ethical, they try to observe as wide a range of natural occurrences as possible to discern patterns. 1B/H3*
• Scientists often cannot bring definitive answers to matters of public debate. There may be little reliable data available, or there may not yet be adequate theories to understand the phenomena involved, or the answer may involve the comparison of values that lie outside of science. 1C/H9** (SFAA)

4. THE PHYSICAL SETTING THE UNIVERSE

• The stars differ from each other in size, temperature, and age, but they appear to be made up of the same elements found on earth and behave according to the same physical principles. 4A/H1a
• Eventually, some stars exploded, producing clouds containing heavy elements from which other stars and planets orbiting them could later condense. The process of star formation and destruction continues. 4A/H2ef
• Increasingly sophisticated technology is used to learn about the universe. Visual, radio, and X-ray telescopes collect information from across the entire spectrum of electromagnetic waves; computers handle data and complicated computations to interpret them; space probes send back data and materials from remote parts of the solar system; and accelerators give subatomic particles energies that simulate conditions in the stars and in the early history of the universe before stars formed. 4A/H3

11. COMMON THEMES B. Models

• A mathematical model uses rules and relationships to describe and predict objects and events in the real world. 11B/H1a*
• A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning. 11B/H1b The behavior of a physical model cannot ever be expected to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. The inappropriateness of a model may be related to differences between the model and what is being modeled. 11B/H5** (SFAA)

C. Constancy and Change:

• Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change. 11C/H4
• The present arises from the conditions of the past and, in turn, affects what is possible in the future. 11C/H6*
• It is not always easy to recognize meaningful patterns of change in a set of data. Data that appear to be completely irregular may be shown by statistical analysis to have underlying trends or cycles. On the other hand, trends or cycles that appear in data may sometimes be shown by statistical analysis to be easily explainable as being attributable only to randomness or coincidence. 11C/H9** (SFAA)

THE NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS STANDARDS (Grades 9-12) http://www.nctm.org/resources/content.aspx?id=12630

STANDARD 2: Algebra Use mathematical models to represent and understand quantitative relationships.

• Draw reasonable conclusions about a situation being modeled.

STANDARD 5: Data Analysis and Probability Standard Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them:

• Understand histograms, parallel box plots, and scatterplots and use them to display data.

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Mr Toogood's Physics

A Level Physics notes to support my lessons at LCS

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3.9.2.5 The Hertzsprung-Russell (HR) diagram

General shape: main sequence, dwarfs and giants.

Axis scales range from –10 to +15 (absolute magnitude) and $\quantity{50 000}{K}$ to $\quantity{2 500}{K}$ (temperature) or OBAFGKM (spectral class).

Students should be familiar with the position of the Sun on the HR diagram.

Stellar evolution: path of a star similar to our Sun on the HR diagram from formation to white dwarf.

Life cycle of stars - What you need to know

The life cycle of stars is quite complicated, and involves a lot of different branches of physics, from the gas laws to nuclear physics. A lot of the details are beyond the scope of an A level course, but I will include them later on this page as they are, nonetheless, very interesting.

What is important for you to know are the following:

• • How to use the Hertzsprung-Russell diagram.
• • The path the a star like the sun will take through it.
• • How the mass of a star will determine how it will end its life.

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The Hertzsprung-Russell diagram

The HR diagram is a very useful tool for helping understand the differences between the different types of stars and how a star changes throughout its life. It plots spectral class, or temperature on the x-axis and absolute magnitude or luminosity on the y-axis. It is not a graph, in the true sense on the word, and neither axis starts at zero.

Stars tend to fall into one of four distinct areas on the diagram depending on their size and the stage of their lives.

The main stripe down the middle of the diagram is called the main sequence, and this is where stars will stay throughout the majority of their lives. The hottest stars are on the left of the diagram, and will, typically be much brighter. The cooler stars are usually dimmer and appear on the right of the main sequence.

There are two ‘clouds’ in the top right hand corner of the diagram of giant stars and supergiant stars. These are large stars which are approaching the end of their lives. Most typical stars (such as the Sun) will expand towards the end of their lives and move into the giant cloud. These stars and much brighter, so appear higher up on the y-axis, but are also much cooler, class class K or M, due to their surface area increasing, and following from from Stefan’s law. When such a star finishes fusing all of its fuel it will slowly cool to form a stellar remnant called a white dwarf. These are very small dim objects, but are also very hot, class B. The sun was formed from a cloud of dust and gas called a nebula. This was quite cool and very dim, so would not appear on the diagram. So when the Sun was formed it just appears on the HR diagram in its current position. The Sun will follow a path during its life similar to the one on the diagram below, and you are required to know this path.

You may be expected to draw an HR diagram in your exam, and you may be asked to label the x-axis with either spectral class or temperature, so it worth practising it. Some key points to remember when drawing the diagram are:

• The absolute magnitude on the y-axis should start at 15 at the bottom and go to -10. Make sure that you get these numbers the correct way around, from largest to smallest, not the other way.
• If plotting temperature, the x-axis should go from $\quantity{50\,000}{K}$ on the left to $\quantity{2500}{K}$ on the right.
• The main sequence should be drawn as a band, not a line and must have some curvature.
• Neither the giant cloud nor the white dwarf cloud can touch the main sequence.
• The giants should have an absolute magnitude less than 0.
• The dwarfs should have an absolute magnitude greater than 10.
• You do not need to draw the supergiants.

The Hertzsprung-Russell diagram is also useful for comparing two different stars in terms of their size or temperature. For example if two G class stars are observed to have different absolute magnitudes they can be plotted on the HR as below.

If they are both the same spectral class, they must have the same temperature, but Star A has a brighter absolute magnitude, so has a higher power output. We can no compare these two stars using Stefan’s law, $P=σAT^{4}$ :

So star A must have a greater surface area, and therefore a greater diameter.

Worked example

A red giant and a main sequence star have the same absolute magnitude, 0, and surface temperatures of $\quantity{3000}{K}$ $\quantity{15\,000}{K}$ respectively.

• What spectral classes are the two stars?

You are expected to remember the associated temperatures for the different spectral classes, especially at either end of the scale. An HR diagram can help with this. If a star has a surface temperature of $\quantity{3000}{K}$ it is one of the coolest stars, class M. The main sequence star is much hotter, and its temperature of $\quantity{15\,000}{K}$ puts its in class B.

• Show that the radius of the red giant is 25 times that of the main sequence star.

As both of the stars have the same absolute magnitude, they must both have the same power output.

• The red giant - $P_{RG}=σA_{RG}{T_{RG}}^{4}$
• The main sequence star - $P_{MS}=σA_{MS}{T_{MS}}^{4}$

Rearranging and cancelling gives:

So the red giant has a surface area, 625 times that of the main sequence star. Since $A=4πr^{2}$ , the red giant is $625^{\frac{1}{2}}=25$ times larger.

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• Homework - HR diagrams

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H-R Diagram

A collection of stars visible from Earth can be arranged and classified based on their color, temperature, luminosity, radius, and mass. This can be done using one or two-dimensional plots, including a Hertzsprung-Russell diagram of luminosity vs. temperature.

Lesson Materials

Student Exploration Sheet

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