Lab-04_Physical-Properties-of-a-Star1
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Feb 20, 2024
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Lab 4: Physical Properties of a Star
OBJECTIVES You will be able to:
1.
Plot the H-R diagram for stars and use it to estimate the temperature and luminosity of a
star, given its spectral class. 2.
Calculate the mass, radius, and lifetime of a star, using the appropriate equations and
graphs. Marketable Skills:
This course assesses the following Core Objectives. In this assignment, you will develop the following marketable skills:
Critical Thinking
Analyze Issues
Anticipate problems, solutions, and consequences.
Apply knowledge to make decisions
Detect patterns/themes/underlying principles
Interpret data and synthesize information
Communication
Summarize information
Use proper technical writing skills
Personal
Responsibility
Accept responsibility
Exhibit Time Management
Show attention to detail
Learn and grow from mistakes
Empirical Quantitative
Communicate results using tables, charts, graphs
Contextualize numeric information/data
Demonstrate logical thinking
Draw inferences from data, use data to formulate conclusions
EQUIPMENT
Calculator and semi-logarithmic graph paper (supplied at the end of this lab, page-12).
INTRODUCTION There are five physical quantities, which are used to define a star: 1.
Temperature
2.
Luminosity
1
3.
Mass
4.
Radius and 5.
Lifetime. Let us examine how each of these quantities can be deduced. Temperature The temperature (T) of the photosphere is measured in degrees K. This can be calculated by
direct observation from Earth using Wien’s Law. But an easier method uses the spectral type.
Astronomers have correlated the spectral lines seen with the degree of ionization present in the
star’s photosphere. Since temperature determines the degree of ionization, once the spectral
type of a star is identified, it is possible to use a table like the one below to determine a star’s
temperature. Remember the spectral sequence is O, B, A, F, G, K, M, with the O stars being the
hottest. Each letter category is in turn divided into 10 sub-categories, ranging from zero to nine.
A star with the classification B9 is therefore slightly cooler than B8, but hotter than A0.
Table 1
Spectral Type
Temperature O5
30,000 K
B0
25,000 K
A0
10,000 K
F0
8,000 K
G0
6,000 K
K0
5,000 K
M0
4,000 K
M7
2,000 K
Luminosity Luminosity is the energy emitted by the star's photosphere each second and over all
wavelengths of the electromagnetic spectrum. If the distance to the star is known, the luminosity
can be calculated by using the familiar equation:
Apparent Brightness
=
Luminosity
4
π r
2
where r = distance. Unfortunately, distances further than 200 pc (parsec) cannot be measured using parallax, and
therefore the luminosity of more distant stars has to be estimated using other techniques. The HR diagram is a great tool for estimating luminosity. It is a plot between a star’s spectral
type and luminosity, and it was drawn for stars whose distance and luminosity were known.
Thus, if we only know a star’s spectral type, we can use the HR graph to estimate its luminosity.
Then we can use the equation above to find its distance. Estimating the distance of a star in this
manner is called the spectroscopic parallax method
.
Mass The mass of a star is a measure of how many and what types of atoms it contains. Astronomers
first measured the mass of stars in binary systems (i.e., systems that contain two stars
2
gravitationally bound to each other). Approximately 50% of the stars are members of binary
systems. For nearby systems with a measured parallax and known distance, Newton's Law of
Gravity and Kepler's Third Law of Planetary Motion can be used to calculate the total mass of
the stars in these systems. But not all stars are in binary systems, and not all binary systems have a measurable
parallax. When astronomers compared the mass and luminosity of hydrogen-burning, main
sequence stars, they discovered a direct relation between these two quantities. The Mass-
Luminosity relation is also a graph shown on page 5. Although it is only valid for main sequence
stars, it is an easy way to estimate a star’s mass. Thus, if a star’s luminosity is calculated to be
1000, the graph can be used to estimate that its mass will be 7 solar masses, or 7 times the
mass of the Sun.
Radius The luminosity represents the total energy output of the star per second. This is related to the
star’s temperature and s
ize of the star. A larger star will naturally have a higher energy output
than a smaller one at the same temperature. Since stars are assumed to be spherical, it is
possible to relate the luminosity (L) and temperature (T) of a star to its radius (R), through the
equation, Luminosity L
=
4
πσ T
2
R
2
where π and σ are constants.
However, since it is more meaningful to compare stellar data to that of our Sun, we can simplify
the above equation as L
=
T
4
R
2
where L, T, and R, are all expressed in solar units and the constants 4πσ have been removed
as they would be the same for the Sun and another star. Rearranging this equation gives
R
2
=
L
T
4
∨
R
=
(
√ L
)
T
2
It is important to remember that in the equation above, the luminosity and temperature must be
expressed in solar units. This means if you determine the real temperature of the star to be
8000 K, its value is (8000/5800) = 1.38 times that of the Sun. The number 1.38 rather than 8000
K will be used in the equation above.
Lifetime The most abundant element in stars is hydrogen. Although helium and other elements are also
found, their mass is a very small percentage of the mass of a star. How long a star will burn will
depend on how much mass it has to begin with. The more mass it has, the longer it can remain
"alive". But how fast it burns its fuel will also play a role. If its luminosity is high, it will be using
up large amounts of its fuel very fast. In that case, it will not last very long, like the journey time
of a "gas-guzzling" automobile. The star’s life is thus inversely related to its luminosity and
directly related to its mass. To calculate the star’s time on the main sequence, use lifetimet
=
Stellar Mass
Luminosity
∨
t
=
M
L
3
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where M = stellar mass and L = stellar luminosity. Once again, since M and L are in solar units,
the star’s lifetime t will also be in comparison to the Sun.
Summary For a hydrogen-burning, main sequence star, the following procedure can be used to determine
its physical quantities: 1.
Read the spectral classification of the star and estimate its temperature (T) in degrees
Kelvin using the chart on page 1. 2.
From the H-R diagram, use the spectral class to estimate the luminosity (L). To do this
you must draw a line vertically upwards from the x-axis until it meets the H-R graph and
then draw a line horizontally until it meets the y-axis. 3.
Use the Mass-Luminosity Relationship graph on page 4 to estimate the mass (M).
Follow the same steps noted in 2 above, namely draw a horizontal line from the
luminosity axis until it meets the plotted line and then draw a vertical line until it meets
the mass axis. 4.
Calculate the relative radius using the R
=
(
√ L
)
T
2
5.
The estimated duration of the hydrogen-burning phase or lifetime t
=
M
L
Each of the quantities T, L, M, R, and t will be expressed in solar units, m
eaning in comparison
to the Sun. For the Sun these quantities are:
T
Sun
= 5,800 K L
Sun = 4 x 10
26
Watts M
Sun
= 2 x 10
30
kg R
Sun = 7 x 10
8
m t
Sun
= 10
10
years
4
5
PRE-LABORATORY QUESTIONS 1.
Which of the following stars will be the hottest? a)
O9 b)
A5 c)
A7 d)
K4 2.
Which of the following stars will be the coolest? a)
G2 b)
G8 c)
K5 d)
K2 3.
The temperature of a G5 star will be approximately a)
7000 K b)
6000 K c)
5500 K d)
10,000 K 4.
Spectroscopic parallax is a method to determine a)
the spectral classification of the star. b)
the parallax angle for the star c)
the temperature of the star. d)
the distance of the star. 5.
The graph between the mass and luminosity of stars shows that a)
as the mass of a star increases, its luminosity decreases. a)
as the mass of a star increases, its luminosity varies b)
as the mass of a star increases its luminosity increases. c)
none of the above are correct as there is no correlation between mass and luminosity. 6.
A star with a large radius will have a)
high luminosity and high temperature. b)
low luminosity and low temperature. c)
low luminosity and high temperature. d)
high luminosity and low temperature. 7.
The most abundant element present in all stars is a)
oxygen. b)
metals like iron. c)
hydrogen. d)
helium. 8.
The lifetime of a star depends on its a.
luminosity. 6
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b.
mass. c.
density
d.
both a & c 9.
A star has a volume of 5 solar units. This means its volume is a)
the size of Jupiter. b)
one-fifth the size of the Sun. c)
five times bigger than the Sun. d)
25 times bigger than the Sun. 10.
In the H-R diagram the two quantities plotted are a)
mass and luminosity. b)
distance and temperature. c)
volume and distance. d)
luminosity and spectral type. LAB EXERCISE A. Recording the Luminosity Look at Table 2 below which lists the spectral type and luminosity for 11 main sequence stars.
You will be using this table to graph the spectral type against luminosity. Table 2: Stellar Data Star Name
Spectral Type
Luminosity
Acrux
B3
7500
Achernar
B5
3600
Vega
A0
60
Sirius
A1
26
Fomalhaut
A3
18
Procyon A
F5
8.6
Alpha Centauri
G2
1.6
Sun
G2
1.0
Epsilon-Eridani
K2
0.37
61-Cygni A
K5
0.17
Lacaille
M1
0.05
B. Drawing the H-R Diagram 11. Print out the semi-log graph paper at the end of this exercise. The spectral types will be marked along the horizontal (x) axis and the luminosity scale
along the y axis must be chosen as shown. Notice that the luminosity values in the table
range from 7500 for Acrux to about 0.05 for Lacaille. To enable us to plot this large
range of numbers, we allow each large square to increase by a factor of 10. Notice that
the lines on the graph paper along the vertical axis are not evenly spaced. This scale is
called "logarithmic". Since the numbers change evenly along the x-axis, the graph paper
is called "semi-logarithmic". Choose the numbers along the y-axis as given on the graph
paper.
7
12. Plot the luminosity versus spectral type for the 11 stars given above. These are all
main sequence stars. To plot the first point for Acrux go along the x-axis (to the right) to
B, and move two more squares to the right to get to B3. Then move up along the vertical
(y-axis) to the line for 7000 and use your judgement to go up a little more to get to 7500.
Plot a point there and write Acrux next to it. Continue till you have plotted the points for
all the stars listed in Table-2:
Stellar Data above. Acrux and Sun have been plotted on
the graph paper on page 12. 13. Draw a smooth curve through the middle of the points. You do NOT have to join all
the dots but draw a wide line with a highlighter pen representing the "average" position.
This is the main sequence line.
C. Using The H-R Diagram Let us use all the information accumulated to calculate the physical properties of Denebola
(beta Leo). Its spectral classification is A3. IMPORTANT! The steps (i) – (vii) below show you how the stellar properties will be determined.
All the calculations for Denebola are shown. Please work your way through this example. It will
be worth the effort! The calculations are summarized in the second column of the table. i.
Denebola is A3 and from the H-R diagram x-axis, you can estimate that Denebola’s
temperature is between 9,000 and 10,000 K. Let’s choose it to be 9,400 K. Temperature (T
1
in Kelvin) = 9,400 K
ii.
Temperature as compared to the Sun = T
1
/ 5800 = 9,400 /5800 = 1.6 T = 1.6 (this value will be used in step vi) iii.
Let’s check luminosity of Denebola from H-R graph. If you draw a line from A3 up to the main sequence line you plotted, and then read the corresponding luminosity, it turns out to be about 20 times that of the Sun. Therefore, L = 20 (Note this value may be different since it depends on where you drew your H-R line. Since we are doing approximate calculations, it is more important for you to understand the process than get the exact answer). iv.
From the mass-luminosity graph on page 4, a luminosity of 20 corresponds to 2 solar masses. M = 2 (solar mass) M = 2(2 x 10
30
kg) = 4 x 10
30
kg
v.
Radius R ¿
(
√ L
)
T
2
=
√
(
20
)
(
1.6
)
2
=
1.75
(
solar radius
)
Radius = 1.75 (7 x 10
8
m) = 1.23 x 10
9 m vi.
Lifetime t = Mass/Luminosity = 2/20 = 0.1 solar lifetime Lifetime = 0.1 (10 billion years) = 1 billion years.
D.
You will be doing calculations for two stars, 61-Cygnus and 78-Ursa Major. The table
on the next page shows all the calculations for Denebola and the equations used. Follow the example carefully.
The table below shows all the calculations for Denebola and the equations used. Follow the example carefully. 8
E.
Estimate the temperature of each star using its given spectral type and the table on page 1. Also calculate the temperature as compared to the Sun.
F.
Use your HR graph and on it draw a vertical line from the spectral type to the line you
have drawn, and then a horizontal line to read the luminosity of these stars. Please draw
these lines! Record the luminosity value in the table.
G.
Next use the mass-luminosity graph to read the mass using the luminosity value you have determined in the step above. Do the calculations for the radius and lifetime of the star.
Star Name
Denebola 61-Cygnus
78-Ursa Major
Spectral Type
A3
K5
F2
Temperature T
1
(see chart on pg 1)
9400 K
Temperature T
(compared to Sun)
T
1
/5800 9400 /5800 = 1.6
Luminosity L
(from HR graph)
20
Mass M
(from mass-
luminosity graph on pg 5)
2Msun
Real Mass
M x (2 x 10
30
) kg
2 x (2 x 10
30
) kg = 4 x 10
30
kg
Radius R = √L/T
2
√20/[1.6]
2
= 1.75 solar radius
Real Radius
1.75 x (7 x 10
8
) = 1.23 x 10
9
m
9
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Star Name
Denebola 61-Cygnus
78-Ursa Major
R x (7 x 10
8
) m
Lifetime t = M/L
2/20 = 0.1 solar lifetime
Real Lifetime
t x (10 billion years) 0.1 x (10 billion years) = 1 billion years
Input all the data you determined for 61-Cygnus which has a spectral classification of K5. (You do not need to rewrite it here, but you will be inputting the following items when you submit the lab.)
Temperature (T
1
in Kelvin) =
Temperature as compared to the Sun = T
1
/5800 = T =
Luminosity from HR graph (L) =
Mass from Mass-luminosity graph (M) =
Real Mass =
Radius R = [√L] / T
2 =
Real Radius =
Lifetime (t) = M/L =
Real lifetime
13.
Input all the data you determined for 78-Ursa Major which has a spectral classification of F2.
(You do not need to rewrite it here, but you will be inputting the following items when you submit the lab)
Temperature (T
1
in Kelvin) =
Temperature as compared to the Sun = T
1
/5800 = T =
Luminosity from HR graph (L) =
Mass from Mass-luminosity graph (M) =
Real Mass =
Radius R = [√L] / T
2 =
10
Real Radius =
Lifetime (t) = M/L =
Real lifetime
14.Summarize what you have learned from this lab.
Grading Rubric Question 1- 10: 0.5 points each
Question 11 – Graph: 8 points
Question 12 -13: 5 points each
Question 15: 2 points
Total = 25 points
11
12
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