Homework_7

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University of Michigan *

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102

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Astronomy

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Feb 20, 2024

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HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes INSTRUCTIONS: - SHOW ALL YOUR WORK - When submitting PLEASE, select the pages and match them to the correct problem Introduction Everything about a star, including temperature and brightness or luminosity , its lifetime and what will eventually happen to it depends on the star’s mass . This project focuses on the relationship between a star’s mass and its luminosity and between the mass, luminosity, and lifetime of a star. Background For main sequence stars there is a mass- l uminosity relationship that is estimated to be the equation below (the mass raised to the power α ). For stars up to twenty (20) times the mass of the Sun, is estimated to be about 3.5. This can be tested by plotting the luminosity of a number of stars as a function of their mass and doing a power-curve fit of the plotted data. L ~ M α Mass-Luminosity Relationship Taking Data Useful data for this experiment can be found at AstroPixels.com ( http://astropixels.com/stars/brightstars.html ). From their list of the 50 Brightest Stars ; find all stars with masses between 1 and 20 solar masses that have a diameter of ten (10) times that of the Sun or less. Also exclude stars that have a luminosity that is more than 10,000 times that of the Sun. These much larger and brighter stars are NOT main-sequence stars. You may use data from double stars (stars for which two spectral types are listed). In some cases, individual luminosities, masses, and diameters are given for both stars; if not, the data is all for the brighter component. You should be able to find data you can use for about twenty (20) stars. Record your data in TABLE 1 . For now, ignore the Lifetime (last) column. 1
HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Problem 1: Table and Graph (2 POINT) REMEMBER! TABLE 1 should ONLY include stars that are : between 1 and 20 Solar Masses less than 10 Solar Diameters less than 10,000 Solar Luminosities. TABLE 1 Main-Sequence Stars Star Name Spectral Type Luminosity (Sun=1) Mass (Sun=1) Lifetime t=10*(M/L) (Billions of Years) 2
HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Plotting Your Data Transfer your data from TABLE 1 to an Excel spreadsheet. Under Insert select a scatter plot graph of the mass vs. luminosity . Make a rough sketch of your plot on the axes provided. Draw a curve on your graph that best fits the data points you plotted. 3
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HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Problem 2: Curve-fit (2 POINT) Now do a power curve-fit of a graph of your data. Do this by clicking on Add Chart Element and selecting More Trendline Options under the same menu will bring up a Format Trendline dialog box. Click on the button that looks like a bar-graph, this will allow you to again select Power and Display Equation on chart . The equation should be of the form; L ~ M α Write your equation; L = ______ Show work here (i.e. explain why your equation is what it is. A short one-sentence explanation is enough): And record your exponent; α = ______ Is your value for α similar to what was expected ( α~ 3.5) ? YES / NO (circle one) Show work here (i.e. why or why not you think your exponent value is close or not. Again a one sentence explanation is enough): 4
HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Problem 3: Stellar Lifetimes (2 POINT) The main sequence life of a star lasts as long as hydrogen fusion is occurring in its core. A more massive star will have more hydrogen to burn, so we would expect the lifetime of a star, t, to be proportional to its mass, M, t~M. However, more luminous main sequence stars, which are also hotter, will burn their hydrogen faster, so the lifetime of a star, t, should be inversely proportional to its luminosity, L, t~ (1/L). Combining the two proportionalities gives t~ (M/L). The main sequence lifetime of a star should be proportional to the ratio of its mass divided by its luminosity. Since masses and luminosities of all the stars in your data are given in terms of the mass and luminosity of the Sun, for the Sun, t~ (1/1) =1. The Sun is expected to have a main sequence life about 10 billion (10,000,000,000) years, so t=10*(M/L) will give the lifetime of a star in billions of years. Use the mass, M, and luminosity, L, for each star in TABLE 1 and the formula t=10*(M/L) to calculate the lifetime in billions of years of each star in your data. Now, plot a graph in Excel of the mass vs. lifetime of the stars from TABLE 1. Sketch the graph on the axes provided. Draw a curve on your graph that best fits the data points you plotted. 5
HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Since t~ (M/L), and, as you proved with your first plot that L ~ M α , that also means that, substituting for L, t~(M/M α ) = M 1 α . Now use your value of α from your first plot to calculate 1- α = _____ = β . Now write your equation for stellar lifetimes t~ M β ; t= ______. Do a power curve-fit for your graph of mass vs. lifetime. Record the equation and exponent; t=_____ β=_____ Are your equation and value for β similar from you curve-fit similar to what you expected from the equation you wrote? YES / NO (circle one) Problem 4: Follow-up Questions (1.5 POINTS, 0.5 POINTS EACH) 1. Based on your data and graphs; how does the luminosity of stars vary with their mass? 2. Also based on your data and graphs; how do the lifetimes of stars vary with their mass? 3. Based on your answers to the previous questions; how do the lifetimes stars vary with their luminosity? Explain your answer. 6
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HOMEWORK #7 The Mass Luminosity Relationship and Stellar Lifetimes Problem 5: Multiple Choice Practice (2.5 POINTS, 0.5 POINTS EACH) 1. More massive main sequence stars are _____ than less massive ones. A. more luminous (brighter). B. less luminous (dimmer). C. [there is no correlation] 2. More massive main sequence stars have ______ hydrogen fuel to burn than less massive ones. A. more B. less C. [there is no correlation] 3. More massive main sequence stars live _____ lives than less massive ones. A. longer B. shorter C. [there is no correlation] 4. More luminous (brighter) main sequence stars live _____ lives than less luminous (dimmer) ones. A. longer B. shorter C. [there is no correlation] 5. The reason(s) for the answers to questions 3 and 4 is (are) A. that more massive stars last longer because they have more hydrogen fuel to burn. B. that more massive stars have more fuel, but because they are hotter and brighter they burn it more quickly. C. [neither of the above-there is no correlation] 7