Homework_7
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School
University of Michigan *
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Course
102
Subject
Astronomy
Date
Feb 20, 2024
Type
Pages
7
Uploaded by CommodoreGuanacoPerson959
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