Physics, Astronomy, &
Astrophysics
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Chapter 1 : Models of Light
INTRODUCTION
What is the subject of astrophysics? Astrophysics is interested in internal composition
and evolution of stars and galaxies. But how can we learn about their structure if they
are so far away? What kind of information can we obtain to devise models of these
objects? This information must be produced by these objects and reach us. Some
celestial objects radiate visible light. Some are invisible for our eyes but visible for radio
telescopes or X-ray telescopes. Some radiate cosmic particles. The only way to learn
about celestial objects is to study what they send us.
Astronomy : A Different Way of Making Observations
Basic Observations
1.
What do you think are the differences between astronomy and other
sciences?
2.
Why do astronomers want to study radio waves, UV rays and X-rays from
celestial objects? Isn’t light enough?
3.
Why are X-ray telescopes placed on satellites?
Let us start with light. We call it a phenomenon because we can observe it. We study its
properties in experiments. But how do we explain these properties? In physics,
explanations that we give to the phenomena are called models. The word model is
based on the idea that whatever explanation we devise, it will not comprise all the
complexity of the phenomenon. It will explain some features of it under some
circumstances, and for other circumstances a different model will be called for. The
validity of models is tested in experiments. There are 2 major models that explain
observable light phenomena, such as reflection, refraction, interference, diffraction, and
the radiation and absorption of light.
What is a meaning of the word model? In science people use this word for different
things. What follows is the description of our understanding of this word.
1.1 Models in Physics
The word “model” is used for many purposes. It usually stands for a “simple version”
of something. In physics one can say that this something can be:
(a) an object, (b) a change, (c) a phenomenon, or (d) a technical device.
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For any model, one should separate the its definition (what are the main features of the
model) and the criteria for its application (when can we apply this model explaining the
behavior of real systems).
Models of Objects
Objects with localized properties
• A particle (dimensionless object) - an object that has no size or volume but can be
located in space, possesses mass, has velocity and acceleration and can be acted
upon by forces. A real object can be described as a particle when its size is much
smaller than the distances in the problem or all its points follow the same path with
the same velocity.
• Ideal gas - an object that is made up of particles that do not interact with each other
than in elastic collisions and obey Newton’s laws. Real gas can be described as ideal
when the distances between the particles are much greater than their dimensions
and the temperatures are high.
• Point charge - a charged particle. Can be a small charged object, or a spherical
dielectric object, or a spherical metal object if we are very far away from it.
• Electron gas - an object that is made up of the ideal gas of electrons in metals which
collide with ions and move with constant velocity from a collision to a collision. Real
electrons can be described as ideal at temperatures above superconductivity. This
model explains Ohm’s and Joules law but does not explain linear dependence of
metals resistivity on the temperature.
Objects with continuously distributed properties:
• Model of a regular wave - all particles of the medium or space participate in wave
motion. Real wave can be described by this model when the size of the container is
much larger that the wavelength.
• Model of a field (gravitational, electric, magnetic, electromagnetic).
• Model of a uniform field.
Models of Change
• Linear change (motion with constant speed, Ohm’s law, thermal expansion).
• Quadratic change (motion with constant acceleration).
• Sinusoidal change (circular motion with constant speed, simple harmonic motion,
sinusoidal wave).
• Exponential change (amplitude for damped motion, radioactive decay).
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Models of Phenomena
Here we include models of phenomena we study in the physics course and models of
scientific phenomena (the first person to do this was Galileo, in the same way that he
was the first to start thinking about the model of change for the position of a falling
object).
• Free fall (phenomenon in which we neglect all forces acting on an object except the
force of gravity). Can be used when the force of friction is much smaller that the
force of gravity.
• Existence of an inertial reference frame. Can be used when the acceleration of the
reference frame is much smaller than the acceleration of gravity.
• Phenomenon of an isolated system.
• The motion of a particle attached to a massless spring which obeys Hooke’s law
(mass on a spring). Can be used when the size of the mass is much smaller that the
spring and the spring stretches much less that its own size.
• Motion of a simple pendulum (particle on a string). Can be used when the length of
the string is much greater than the size of the bob.
• Phenomenon of wiring with no resistance.
To note our explanations (ideas, hypotheses) of observed real phenomena (explaining why a
particular phenomenon happens we devise a mechanism). It reflects our preset
understanding and, of course, is a simplified version of what is going on. There can be
multiple models of the same phenomenon, a model can never be proven correct, it can
only be proven wrong.
•
•
•
•
•
Phenomenon of resistivity (scattering of phonons).
Phenomenon of superconductivity (interaction of electron pairs - Cuper pairs).
Phenomenon of a white dwarf.
Phenomenon of a neutron star.
Phenomenon of a black hole.
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Models of Technical Devices
• Internal combustion engine.
• Model of a bridge.
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Developing & Testing Models
Task 1 (You have 10 minutes to do this)
1. Observation: You observe every day the sun appearing above the horizon,
moving in an arc and then disappearing again.
2. Explanation: Use the light and the Styrofoam ball to devise two different models
that explain these observations. A model is an answer to the question “Why do
we observe this phenomenon?”)
3. Testing: Design testing experiments to reject the models. A testing experiment is
an experiment whose outcome is predicted before it is performed. Is there a
model that you could not rule out?
4. Presentation: Prepare to share your findings with the rest of the class.
Task 2
1. Observation: Use alcohol and a sponge to make a wet streak on a piece of paper.
Observe the streak. Record your observations.
2. Explanation: Devise three different explanations that account for your
observations (why the alcohol behaved the way it did).
3. Testing: Design a testing experiment to rule out each explanation. If you need any
additional equipment, ask Eugenia. Is there any explanation that you could not
rule out?
4. Presentation: Prepare to share your findings with the rest of the class.
5. Let us reflect on the processes that you used to reject different explanations. Do
you think that scientists use similar methods? Could you give me any examples
of how scientists came to believe in something?
6. Now let us try to practice this method. Each group will now receive a different
task. You will have 45 min to work on it and then you will present your findings
to the group.
Task 3
The main part of any oscilloscope is a cathode ray tube. CRT is a device that creates
a beam makes a beam of electrons that hit a fluorescent screen. The screen shines in
green light when an electron hits it.
1. Using the CRT and a magnet devise a rule for the direction of the force exerted
by the magnet on the beam of electrons. Make sure that it is a general rule and
that can be used to predict the deflection of the beam for any situation.
2. Prepare to share your findings with the rest of the class.
Task 4
1. Using materials on your table to create electric current in a circuit that does not
have a battery (this circuit is built). Devise a general rule that will account for all
possible situations.
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2. Prepare to share your findings with the rest of the class.
Task 5
1. Using the rotating stool ands the barbells to find out what happens to the
rotational motion of the person on the stool when she/he brings the barbells
close to her/his body.
2. Decide what physical quantities are needed to describe the motion
quantitatively.
3. Use the videotaped experiment to record data. Decide if there are any patterns in
the data that you collected.
4. Devise an experiment to test the patterns.
5. Prepare to share your findings with the rest of the class.
Summary
1. Think of why we made you go through these exercises.
How can they be related to the research done by physicists?
2. How is astronomical research different from the research in physics?
3. What kind of data should astronomers collect to explain the models of celestial
objects? What should we study next?
1.2 The Mechanical Wave Model
In order to understand the wave model of light, you need to refresh your memory about
mechanical waves by doing the exercises below.
(a) Mechanical Waves
Observations
1.
Produce a longitudinal pulse on a slinky. Place it on the floor and let two people
hold it. One person should shake the end of the slinky once.
2.
Produce a transverse pulse on a slinky.
3.
What happens when each pulse reaches the other end of the slinky?
4.
Observe what happens when two pulses oriented in similar directions meet.
5.
Observe what happens when two pulses oriented in opposite directions meet.
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Modeling
1.
Explain how different pulses propagate on a slinky (devise a mechanism).
2.
Can longitudinal pulses propagate in air? Give examples. Can transverse pulses
propagate in air? Give examples.
3.
Explain how two pulses can reinforce each other. Explain how two pulses can
cancel each other.
Observations
1.
What do you need to have if you want to create wave motion?
Observations & Models
2.
What is the difference between phenomena and models? Provide an
example from every day life.
3.
Is wave motion a phenomenon or a model?
4.
What is called a sinusoidal wave? Is it a phenomenon or a model? Explain.
Physical Quantities (pq’s) & Their Relationships
5.
What are physical quantities that characterize a model of a sinusoidal wave?
Demonstrate them using a slinky.
6.
For a wave of a known frequency and amplitude, draw graphs of
(a) the displacement of a single particle in the medium versus time
(b) the displacement of all of the particles in the medium versus position at a
given instant (in other words, a “snapshot” of the wave).
7.
What does it mean if the amplitude of a wave in a slinky is 20 cm?
8.
What does it mean if the frequency of a wave in a slinky is 2 Hz?
9.
What does it mean if the speed of a wave in a slinky is 2 m/s?
10.
For the previous conditions, determine the wavelength of the wave and explain
what the number you calculated means.
11.
Draw a graph amplitude versus frequency for this wave (it is called a power
spectrum). What can you learn from examining this graph?
Prediction & Testing
By changing the tension in a slinky we change the medium. Predict & Test what effect
this change has on the speed of the pulse in a given medium.
(b) Standing Waves
Observations
Use a slinky to produce a steady pattern. Place it on the floor and let two people hold it.
One person should continuously shake the end of the slinky. Vary the frequency of the
source (your hand) and observe different waves. Describe your observations using the
word “energy”.
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Modeling
1.
How does it happen that some coils vibrate with large amplitudes and some coils
do not vibrate at all? (Hint: add two sin functions.)
2.
Suggest an explanation of standing waves, in other words how do you think they
occur?
3.
Draw diagrams that represent the shape of the slinky for t=0, t=T/4; t=T/2;
t=3T/4 and t=T.
Testing
Design an experiment to test your explanation.
(c) Interference and Diffraction of Mechanical Waves
Interference
Observation
Observe the laser disc Ripple Tank scenes of vibrating water produced by 2
sources. Describe in words what you see.
Model
Propose a model for the pattern that you’ve observed.
Predicting & Testing
Using the model, predict what will happen if you change (1) the separation
between the sources, or (2) the frequency with which the sources vibrate. Test
your prediction. If your prediction was wrong, account for why.
Diffraction
Observations
1.
Observe what happens to waves that pass through two narrow openings.
2.
Observe what happens to waves that pass through multiple narrow openings.
3.
Observe what happens to waves that pass through a single narrow opening.
Model
Propose a model that explains both the 2nd and 3rd observations.
Predicting & Testing
Use the model to predict what will happen to the waves as the opening that they pass
through gets wider. Test your prediction.
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(d) Polarization of Waves
Observations
1.
Generate a longitudinal pulse on a slinky and observe its motion. Use a
bottomless box to observe its effects on pulse propagation.
2.
Generate a transverse pulse on a slinky and observe its motion. Use a bottomless
box to observe its effects on pulse propagation.
Modeling
Explain the difference between the effects of the box on different pulses. What waves
can be polarized and what waves can’t?
Testing
Use equipment provided to test whether sound is a longitudinal or transverse wave.
Question: In what materials can transverse waves propagate: gases, liquids, solids, or
vacuum? Explain your answer.
1.3 The Wave Model of Light
Now we can start discussing why a wave model can be used to explain the behavior of
light. Think about light phenomena that resemble different phenomena that occur with
mechanical waves. What does it mean for light to reflect? Refract? Interfere? Diffract?
(a) Reflection & Refraction of Light
Observations & Models
1.
2.
Use ActivPhysics2 simulations (Geometrical Optics: Telescopes) to explain
( a) how a telescope allows us to see faint objects.
( b) how a telescope allows us to separate close objects in the sky.
Using a prism and a flashlight observe what happens when white light goes
through a prism. Record your observations. Why do you see a rainbow if white
light goes through a prism?
(b) Interference & Diffraction of Light
Design an Experiment
If light is a wave, we should be able to use it to exhibit the characteristics of
interference and diffraction. Therefore, given
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(1) an assortment of single and double slits, and
(2) sources of different colors of light, design and perform an experiment (or
experiments) that show that light can both interfere and diffract.
What aspect of the testing situation corresponds to color? Defend your answer.
Developing a Mathematical Model
1.
Using a laser and double slit system provided, observe the pattern formed on the
screen.
2.
Using the information on the diagram of this situation that is provided on the
following page, develop a mathematical expression that represents the pattern in
terms of y, L, n, d, and 
3.
Would your expression be any different if many slits (instead of 2) were used?
Explain why or why not.

Qualitative Predicting & Testing
4.
Predict what will happen to the double slit pattern if you have instead a system
of very many narrow slits (known as a diffraction grating) for the laser light to
go through? (Think about what the number of slits/mm, shown on the
diffraction grating, means.)
5.
Test your prediction. If your prediction was not correct, you’ll need to revise
you model so that it can explain your observations for both the double slit
system and the diffraction grating.
6.
Predict and test how different diffraction gratings (slits/mm) would affect the
pattern.
Testing the Mathematical Model
7.
You will be provided with a source of white light and a special apparatus for
measuring the wavelength of different colors.
Decide how this apparatus can be used to measure the wavelength of different
colors.
8.
Test the relationship you’ve developed in #2 by determining the wavelengths of
red, yellow, green, and blue light. Compare your results to the values provided
on the board.
Applying The Model
9.
What happens if white light goes through a system of two very narrow slits?
Why? Test your answer (if possible).
10.
What happens if white light goes through a system of many narrow slits? Why?
Test your answer (if possible).
11.
An experiment in which white light is projected through a diffraction
grating onto a screen shows that the red part of the visible spectrum is
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12.
13.
farther from the center of the screen than the blue part of the spectrum.
Explain how this proves that red light has a larger wavelength than blue light.
Why should a diffraction grating be called an interference grating?
An important application of diffraction is the resolving power of a telescope. Many
of the sources that we study involve a system of 2 stars. These stars are so far
away that their angular separation () is very small. Using the Active Physics 2
simulation for resolving power (click on Chapter 16, Physical Optics), determine
the effect of the aperture opening size (D), wavelength (), and angular
separation () on the resolving power of a telescope.
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Diagram of 2 Slit Interference
Use ActivPhysics2 simulation (Physical optics, Double slit interference, qualitative) to
help you to visualize the situation described in the figure below.
P
y
S1 

d
S2 

screen
L
S1 and S2 are the slits through which the light is passing.
P is a point on the pattern at which constructive interference occurs.
d is the distance between the slits.
L is the distance from the slits to the screen on which the pattern appears.
y is the distance from the center of the pattern to point P.
The segment PS2 must be nlonger than PS1. Explain why.
Indicate on the diagram where the value n should be placed.
Since (in practice) d is so small compared to L, the angles  and  shown above
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are so small (<<10) that they can be considered to be equal. Therefore, we can say
that tan  = sin Explain why. (You’ll need to use this idea in developing the formula!)
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1.4 The Electromagnetic Wave Model of Light
The phenomena of interference and diffraction prove that the wave model can be
applied to study the propagation of light. But what kind of wave is the light wave? Is it a
mechanical wave (like a sound wave) or a different kind? We know that light
propagates in a vacuum and sound does not. This means that light waves are not
transmitted by vibrating particles. If this is true, then what is oscillating in a light wave?
What is it that’s waving?
Question:
Is light a transverse or longitudinal wave? Explain your answer. How can a transverse
wave travel in a vacuum?
J. C. Maxwell suggested that light is a wave that (a) travels at 3 x 108 m/sec in a
vacuum and (b) consists of electric and magnetic fields that oscillate perpendicular to
each other while at the same time producing each other. The condition necessary to produce
an electromagnetic wave is simply that an electric charge must be accelerated. Experiments
carried out by Heinrich Hertz (you can read about these at the appropriate Web sites)
tested these ideas and confirmed them.
Polarization experiments prove that light behaves like a transverse wave in which
electric and magnetic fields (which is why it is called an electromagnetic wave) are
perpendicular to each other and produce each other. In reading about polarizers at the
related Web site, keep in mind that a polarizer functions by using strings of polymers
that absorb the electric field. If the strings are vertical, the vertical electric field is
absorbed, thereby allowing only the horizontal electric field to pass through.
Consequently, if the polymers form vertical strings, the polarizer acts like a horizontal
“picket fence”.
From the geometry of experiments done with diffraction gratings we know that the
color of light that we perceive is representative of its wavelength (and depends on its
frequency). Frequencies of visible light range from 4.3 x 1014 Hz to 7.5 x 1014 Hz.
Electromagnetic waves with frequencies smaller than visible light (known as infrared,
microwaves and radio waves) as well as those with higher frequencies (ultraviolet,
X-rays, gamma rays) possess the same properties.
Observations
Use polarizing filters to observe their effect on light.
Model
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Use the Active Physics 2 polarization simulation (in Chapter 16) to develop a
model for how polarizing filters affect light.
Test
Design and perform an experiment to show that reflected light is polarized.
Models and Testing
1.
Describe the experiment that proved that light is an electromagnetic wave.
2.
State whether the following forms of electromagnetic radiation (radio, TV,
infrared, visible, ultraviolet, X, gamma) are listed in order of
(i) increasing or decreasing wavelength
(ii) increasing or decreasing frequency
3.
For any form of electromagnetic radiation, what is it that’s “waving” as the wave
propagates through space? What is happening at the source of the wave to
produce the radiation?
4.
What is known about the speed of different forms of radiation in a vacuum?
5.
What are the approximate wavelengths of red light, yellow light, blue light, radio
waves, television waves, X-rays?
6.
What are two characteristics of sound waves that prove that they are not forms
of electromagnetic radiation?
7.
How can you test the wave model of light using the polarization of light?
8.
A student believes that two polarizing filters, aligned perpendicular to each
other, will eliminate any unpolarized light that is passed through both the filters
because one filter removes the electric field and the other filter removes the
magnetic field. Explain how you would correct
this student’s ideas about
polarized light.
1.5 The Photon Model of Light & The Radiation Laws
Many physical phenomena involving light cannot be explained if we apply a wave
model to light. These are radiation experiments and photo-electric effect experiments.
Radiation experiments involve radiation of light by hot objects (solids or gases).
Photoelectric effect can be observed when light is absorbed by metals.
Two Radiation Laws concerning blackbody radiation were developed as a result of
experimental observations. (In this way, they are very much like Kepler’s Laws of
planetary motion : observational facts which could not be explained at the time they
were formulated.) The first of these, Wien’s Law, relates the wavelength of the
maximum energy that is radiated to the temperature of the blackbody. The second, the
Stefan-Boltzmann Law, determines the total energy radiated by the blackbody. (Read
about these laws at the appropriate Web sites.) Neither of these laws can explain the
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experimental curve of intensity versus wavelength, in which all frequencies of light are
present but have different intensities at different wavelengths.
The model for the radiation mechanism for the continuous blackbody spectrum was
provided by Max Planck. One of the basic assumptions of this model is that radiation
exists as discrete amounts of energy such that E = hf. A few years later, Einstein used this
expression to explain the empirical results of the photoelectric effect. (He later won the
Nobel Prize for this.) Based on Planck’s model, the two Radiation Laws can be derived
as well as the empirical laws of photoelectric effect.
Light Bulb Activity
Observations & Models
1.
Use a spectrometer to observe the spectrum of the light bulb. Develop a model
that explains this observation.
Predictions & Testing
2.
Predict what will happen to the spectrum of the light bulb if we change the
current (smaller, greater)? Record your predictions.
3.
Using the rheostat to change the current, test your prediction by observing what
actually happens. Record your observations. Revise your predictions, if
necessary.
Solar Spectrometer Activity
Observations & Models
4.
Familiarize yourself with a solar spectrometer. Study the spectrograms shown
on page 16 that correspond to different currents in the light bulb. Try to
interpret these data. Think about the relationship, if any, between the
temperature of the filament and the wavelength at which they observe a
maximum energy radiated. Is there any relationship between the temperature
and total amount of energy radiated?
Observations
5.
Describe blackbody radiation experiments and empirical laws consistent with the
observations.
Models
6.
Explain why a wave model of light led to the “UV catastrophe” of the universe.
7.
Describe Plank’s solution of the problem. What is a “black body” and how does it
work?
Observations
8.
Describe photoelectric effect experiments observed in class.
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Models & Testing
9.
How would you explain the photoelectric effect experiment that you observed
using the electromagnetic wave model of light?
10.
Use the Active Physics2 simulation for photoelectric effect (click on Modern
Physics) to collect more information about the intensity of light, magnitude of
current and “stopping potential”.
11.
Build a model of this phenomenon that explains the existence of the effect and
the observations in 3.
12.
How can you test this model?
13.
How would you explain the “stopping potential” observation and “shortest
wavelength” observation using the electromagnetic wave model of light?
14.
Prepare a short 3-5 sentences long explanation of the photon model of
photoelectric effect.
15.
What are the main features of the photon model of light?
16.
How do you know that the photon model works?
17.
What phenomena can be explained using either the wave model or the photon
model? What are the explanations? (provide at least two examples)
18.
Based on the photon model, what should we observe happen in the double slit
experiment? What can we conclude from this?
Physical Quantities & Their Relationships
19.
What is a photon – describe how you understand this entity. What is the
difference between red and blue photons? How can you explain the difference in
light radiated by a 25 W bulb and 60 W light bulb using the photon model of
light?
20.
Using a spectroscope, look at the Sun and estimate its surface temperature (to do
this, use the spectra of the light bulb at different temperatures).
21.
Using the datum for the surface temperature from #20, determine the total
energy radiated by the Sun every second. This rate of energy production is called
the luminosity.
22.
Compare this value for the luminosity of the Sun in #21 with the value that you
obtain for the luminosity of the Sun based on the amount of energy that 1/2 of
the Earth’s surface receives from the Sun. (Note : the Solar constant, which is the
rate at which each square meter of the Earth receives energy from the Sun, is 1.37
x 103 Watts/m2)
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Cool bulb vs hot bulb blackbody spectrum goes here
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Chapter 2 : The Structure of Atoms
As we understand now, our knowledge of stars come from the studies of
electromagnetic waves that they radiate. But how do they radiate light? How do stars
radiate UV radiation, or X-rays? This section will help you to understand how stars
produce visible and UV radiation.
History
By the middle of the 19th century scientists learned how to analyze light radiated by hot
gases. For example, the studies of hydrogen by Johann Balmer showed that atomic
hydrogen does not radiate white light, but light of only certain wavelengths. The same
phenomenon was observed for all gases at low density. Scientists were curious about
the mechanisms that could lead to this kind of radiation. To explain the mechanism they
needed to devise a model of the inner structure of atoms that would account for their
stability, lack of total electric charge and their radiation spectra.
Models
The first step in this work was the discovery of the electron in the studies of cathode
rays. Subsequently, J.J.Thomson proposed his “plum-pudding” model of the atom,
and E. Rutherford proposed the planetary model. Unfortunately, neither of these
models accounted for all of the observed properties. This prompted N. Bohr to devise a
new model which was later called the “Bohr hydrogen atom”.
Building a Model of the Atom
Observations
1.
Describe the experiment in which cathode rays were discovered (homework)
2.
What are cathode rays and what properties they have? How can you observe
these properties? (homework)
3.
Using a diffraction grating, describe your observations of the emission spectra
produced by hydrogen gas.
4.
Describe all observational evidence that can be used to build a model of atom.
Observations & Models
5.
Describe the main features of Thomson model of atoms and the experimental
facts that it accounts for.
6.
What experimental facts cannot be explained with Thomson’s model?
7.
Describe the main features of Rutherford’s planetary model of atoms and the
experimental facts that it accounts for.
8.
What experimental facts cannot be explained with Rutherford’s planetary
model? What does this model predict about the emission of light?
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Bohr’s Hydrogen Atom
In his model, Bohr modified Rutherford’s planetary model for hydrogen to account for
all experimental observations. The electron was still orbiting the nucleus but Bohr
postulated that:
• There are certain stationary states (radii) in which the electron even moving in a
circular orbit does not radiate electromagnetic waves. In these states the angular
momentum of the electron is quantized: the magnitude of the orbital angular
momentum (mvr) equals a positive integer multiple of Planck’s constant (h) divided
by 2 mvr = nh/(2)
(n=1,2,3,...).
• Radiation is emitted by the atom when the electron undergoes a transition from one
stationary state to another.
• This energy is emitted in a form of a photon, which frequency is determined by the
difference in energies of the states: E = hf
Models & Physical Quantities
1.
What assumptions were made by Bohr to explain your observations in #3 of the
previous question set?
2.
What are the most important physical quantities that describe the hydrogen
atom?
3.
How are these quantities related? Which relationships are based on classical
physics and which relationships are based on Bohr’s assumptions?
4.
Using Bohr’s postulates and the physical relationships that are assumed to be
present, determine the radius of the hydrogen atom when the electron is in the
ground state.
Predictions & Testing
5.
How can you test that the value you determined in #4 is valid?
6.
Using Bohr’s postulates and the classical physical relationships that are assumed
to be present in the hydrogen atom, develop an expression for energy as a
function of n.
7.
Use your expression from #6 to determine the transitions that are occurring to
produce the wavelengths observed in #3 of the previous page.
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The Solar Spectrum
Observations & Models
1.
Observe the spectra of sodium, and helium light sources. Record your
observations. Explain them qualitatively.
2.
Observe the main parts of the solar spectrometer. Answer the following
questions:
(a) What do we need the fiber optics part for?
(b) What are other important parts of a spectrometer?
(c) Why is a computer added to the spectrometer?
(d) What is improved due to the computer?
3.
Observe the spectrum of sodium gas using a solar spectrometer (see page 20 for a
computer-generated spectrum of sodium). Explain your observations. Where is
the energy you observe coming from?
4.
Using a hand held spectrometer observe the spectrum of the sun. Record (draw)
your observations. Repeat the same observations using a solar spectrometer.
Refer also to the computer-generated spectra (page 20). Answer the following
questions:
(a) What can this spectrum mean?
(b) How can it be formed?
(c) Can we determine the chemical structure of the Sun using this spectrum?
(d) Using the chart on page 21 that lists the spectral lines of different elements,
find out why the Sun was called a “metal star” right after people obtained its first
spectra.
22
Sodium & solar spectrum go here
23
Chart of solar spectrum lines (Appendix E) goes here
24
Reference : Physical Quantities & Relationships in the Bohr Atom
1) Circular Motion
Again, you’ve probably had this in your physics class, but according to Newton’s
second law if an object has an acceleration, then the net force of all forces acting on it
must be equal to the product of the mass of the object and the acceleration. If an object is
moving in a circle with constant speed, its acceleration is pointed towards the center of
the circle and is equal to v2/r, it is called a centripetal acceleration. Consequently, the net
force of all forces acting on the object is equal to
Fc = mv2/r
This is not a special force, it is just a net force of all forces acting on an object that is
moving in a circle with constant speed. In this case the net force has a special name, it is
called a centripetal force. If you analyze any circular motion, an amusement park ride,
whirling a ball on a string in a circle, planetary orbits, or the electron orbits in Bohr’s
atom, you will always find that a net force is pointed towards the center of the circle or
has a component that is pointed towards the center!
2) Angular Momentum (mvr)
This is a weird expression. Where does it come from? Well, remember linear
momentum, mv? As it turns out, there is an analogous type of momentum for rotating
masses, and it’s called angular momentum. It is found by the product IYou’re
probably wondering how that relates to mvr. Here’s how:


I is known as the moment of inertia, the analog of mass in a linear situation. But I
depends on the product mr2. In other words, if 2 particle masses are revolving at the
same rate around the same axis, but one mass is farther from the axis of rotation, then
that mass will be harder to stop (think about why)…therefore, it has a bigger
moment of inertia. The fact that this moment of inertia is represented by mr2 is
found from using calculus (which we’re not going to do here).
 is the rotational speed (measured in radians/sec). It is found by:
 = 2/T =v/r.

Now, simply multiply (mr2) x (v/r) and…voila! We get mvr!!!
25
Note : The quantity of angular momentum, like linear momentum, is conserved in
interactions in which there are no forces that are “external” to the system. The
Conservation of Angular Momentum will be very useful to us in the near future.
3) The Coulomb Force
You’ve probably had this in your physics and chemistry classes, but just in
case…Charles Coulomb, in 1785, experimentally determined that the force between two
electric charges q1 and q2 can be determined from the relationship
Fel = kq1q2/d2
(k = 9 x 109 Nm2/Coul2)
This expression bears a curious resemblance to the gravitational force, but it is much
stronger…1040 times stronger!!! It’s interesting that nature has evolved in such a way
that the electric force is needed to hold atoms together (gravity is useless on that level),
but gravity holds the universe together on a cosmic level (if the large masses in our
universe were charged, the electric force would cause so much havoc that we wouldn’t
be around to talk about it!!!)
4) Electric Potential Energy ( Uel = - kq1q2/r)
You probably haven’t seen this expression, and you really need to know calculus to
derive it, but here’s the general argument:
Work is done when a force is applied through a distance. In the case of charges, the
force is the Coulomb force. So the electric work done by an external force in separating
two unlike charges until they are infinitely far apart (which will increase their potential
energy…you’re moving them the way they don’t want to go; therefore, they have the
potential to acquire more KE when they are released) is found by
W = (kq1q2/r2)(r) + [(kq1q2/(r +r)2](r) + [kq1q2/(r + 2r)2](r) + … = kq1q2/r
The problem here, which can only be resolved by calculus, is that as the charges are
moved further and further apart by increments of r, the Coulomb force gets weaker
and weaker. Since the potential energy at infinity is conveniently set equal to 0, we then
have
26
W = Uel = Uf – U0 = 0 – U0

kq1q2/r = -U0


U0 = -kq1q2/r
5) Total Mechanical Energy
Etotal = K + U
Chapter 3 : The Ideal Gas Model
Many phenomena occurring to celestial objects can be explained with the help of
another model - the model of ideal gas. This model is connected to the Molecular
Kinetic Theory. The Web sites will help you to review the major parts of this theory.
Observations
1.
What is gas? What do we know about its properties? How do we know what we
know (What experimental observations give us the evidence for these
statements? What theoretical assumptions can we use and why?) (homework)
Models
2.
What additional characteristic does an ideal gas have?
What evidence do we have for this?
Physical Quantities & Their Relationships
3.
What are physical quantities that describe the behavior of ideal gas? What are
their units? What is the relationship between them? How do we know that these
laws are true? (homework)
4.
When is a real gas not an ideal gas? Provide possible situations involving
pressure & volume.
5.
Derive an expression for the average force exerted by a particle of ideal gas
against a wall of its container.
6.
Using the answer to #5, derive the expression for the average kinetic energy of a
particle of ideal gas in terms of the temperature of the gas.
Applying the Model
7.
Approximately how much energy does the air in this room possess?
8.
Calculate the speed of air molecules in this room.
9.
Find the speed of H2 molecules at room temperature.
10.
Why is there so little hydrogen in the earth’s atmosphere?
27
11.
12.
13.
14.
15.
16.
17.
(Hint: consider, among other things, what the escape velocity (see next page)
would be for hydrogen in the earth’s atmosphere.)
Can we conclude that the air in a typical classroom is an ideal gas? How do you
know? Test your conclusion using the fact (observation) that the density of air at
room temperature is known to be 1.21 kg/m3.
Use the definition of a mole of an ideal gas at STP to show that the particles are
indeed weakly interacting (that is, far apart).
Calculate the energy of a photon that can ionize the hydrogen atom.
Calculate the temperature at which the hydrogen atom will be ionized.
Imagine that you have white light produced by a hot filament (remember, it
produces a continuous spectrum). What will happen if this light goes through a
cloud of cold hydrogen? Hot hydrogen? Ionized hydrogen?
How often do molecules collide? (Calculate the number of collisions per second).
The best vacuum that can be produced in the laboratory is about 10-15 atm. How
many molecules still exist in one cubic centimeter of space at this pressure and
temperature 0°C?
28
Reference : Escape Velocity & Gravitational Potential Energy
In order to “escape” the gravitational field of a star, planet, moon, etc., a body of mass
m needs enough velocity (and therefore Kinetic Energy) so that it can reach,
theoretically, infinity. This velocity is known as the escape velocity. It’s sort of like the
velocity with which you must throw a ball up in the air so that it keeps going and
going, gradually getting slower and slower, higher and higher, until it reaches
“infinty”, where it stops. At this point, the ball (or body of mass m) has no kinetic
energy and no gravitational potential energy (Ug). In other words, it has a total
mechanical energy of 0.
1) Gravitational Potential Energy
“How can there be no gravitational potential energy at infinity?”, you may ask.
Mathematically, Ug is very similar to Uel. Arbitrarily, the value for Ug when two masses
are infinitely far apart is set at zero. At a given separation r, the value for gravitational
potential energy is given by
Ug = -Gm1m2/r
The mathematical argument for this expression is very similar to the one for Uel. (And,
also like Uel, one really needs to know calculus to fully understand where this
expression comes from.) Notice that the expression for Ug is (-). Why is this? We know
that if two masses get farther apart, they gain Ug because (+) work has to be done to
separate them. That is, Uf – U0 must be (+). Our expression for Ug satisfies this condition
because as r gets larger, Ug becomes less (-) until finally it is 0 when the separation is ∞.
In other words, Ug increases, which is what we expect.
You may be familiar with the expression mgh as representing gravitational potential
energy. Well, the truth is, it represents the change in Ug in lifting a mass m a distance h
near the surface of the earth…a very specific situation. By letting m1 = m, m2 =Mearth,
r0 = Rearth, and rf = Rearth + h, and remembering that
g = GMe/Re2, it’s easy to show that
Ug = Uf – U0 = mgh
2) Escape Velocity
Let’s return to our condition for our mass m to “escape” from the gravitational field of a
mass M. This situation can be represented with the Conservation of Energy :
(1/2)mv2
+ (-Gm1m2/r) = 0
Using this idea we can calculate the escape velocity.
29
Chapter 4 : The Doppler Effect
Observations, Models & Testing
1.
The instructor will whirl a ball on a string around in a circle. Inserted in the ball
is a high frequency buzzer. Record your observations. State a qualitative model
of what is happening to the sound you hear.
2.
On the laser disc, a source will dip into the ripple tank at a steady frequency
while moving at a constant speed from left to right. Use your model to predict
what will happen to the wavelength received by an observer for whom the
source is (a) approaching, or (b) receding. Test your prediction.
3.
Try to develop another model of the Doppler Effect that you could demonstrate
to a friend. Use the rotating buzzer to test your model.
The Doppler effect is a phenomenon that is usually associated with sound. In
astrophysics, however, the wave model of light can be used to develop an expression
for  (called the Doppler shift), which is the difference between the wavelength
emitted by a source () and the wavelength received by us (’ due to the relative speed
v between the source and us. We can measure the Doppler shift between an energy
spectrum that we receive from a source and a “laboratory” energy spectrum. We can
then use this result to measure the relative speed v. The use of the Doppler Effect is a
fundamental tool in the analysis of stellar phenomena.
Physical Quantities & Their Relationships
4.
Develop an expression for ’ in terms of , v, and c. It may be helpful to develop a
visual model of the situation. Consult the related Web site.
5.
Suppose the observed wavelength of the sodium line in a star is 5891 A, whereas in
the laboratory the wavelength is 5890 A.
(a) At what speed is the star moving relative to us?
(b) Is the star approaching or receding?
6.
Light from a galaxy in the Constellation Virgo is observed to be 0.4% longer than
corresponding light that has a wavelength of 656 nm when measured in the
laboratory. What is the radial speed of this galaxy with respect to the earth? Is it
approaching or receding?
30
7.
The period of rotation of the sun at its equator is about 24.7 days; its radius is 7 x
105 km. What Doppler wavelength shift is expected for light with wavelength 550
nm emitted from the edge of the sun’s disk?
31
Predicting & Testing
8.
Use the attached read-outs of a frequency spectrograph (page 29) to predict the
speed at which a rotating sound source
(a) approaches us
(b) recedes from us.
Test your prediction.
9.
Use Laboratory Exercises in Astronomy – The Earth’s Orbital Velocity to predict
the earth’s orbital velocity using the Doppler Effect.
Test the predicted value(s).
Applying the Model
10.
Why does the spectrograph at the bottom of page 29 display a “continuous”
spectrum of frequencies?
11.
At what point(s) in the rotational motion of the buzzer used to test the prediction
of #9 is there no Doppler shift? Explain.
12.
Describe in words how the sun’s rotation can be determined from an analysis of
the spectra of light coming from various parts of the disk.
13.
Explain how the earth’s orbital speed can be determined from observations of the
spectrum of a star.
14.
Suppose a source is monochromatic yellow light. How does the appearance of
the source change as the source
(a) approaches the observer, or
(b) recedes from the observer.
32
33
stationary source
:
rotating source :
34
Chapter 5 : Stellar Parameters
It is time to start learning about stars and their parameters, that is, the basic physical
characteristics of a star that astronomers use to classify stars and study their evolution.
It’s important not only to know what these characteristics are, but also how to
determine them.
5.1 Stellar Distances & Stellar Luminosity
If we want to compare the properties of stars we need to know the distances to them. But
how can we do it? How do we usually measure distances on Earth? Take a measuring
instrument, put simultaneously to the two ends of the object and see the reading. Or travel
from one object to another, and then multiply average speed by time. But it is impossible to
do this with the stars. They are too far away. Then how do we measure distances to them?
Observations
1.
Let’s start with a simple question: what do stars that we see in the skies look like?
Make a list of characteristics that might be important for our understanding the
nature of stars.
Models
2.
Design a model that can determine the distance to an object that is “far” away using
only protractors and meter sticks. (But, the distance can’t be easily measured using a
meter stick.)
Read the Web sites related to parallax to find out more about the method that you just
devised.
Predicting & Testing
3.
Use your method to predict, say, the distance to a tree (outside, if it’s sunny) or the
length of the hallway (inside, if it’s raining). Test your prediction.
Physical Quantities & Their Relationships
4.
At what distance would one half of the Earth’s orbit produce a parallax of 1”?
5.
If a star’s parallax is 0.04”, what is its distance in parsecs, in light years, in
astronomical units, in kilometers?
6.
The smallest parallaxes that can be measured are about 0.01” what is the
maximum distance that can be measured by the method of stellar parallax?
7.
The brightness of a star represents its luminosity per unit of surface area. Does the
distance to a star affect the brightness of a star?
8.
Recall our expression for luminosity from Chapter 1 (section 1.5). What part of
that expression represents brightness?
35
Applying the Models
11.
What would be the advantages of measuring stellar parallaxes from Mars rather
than from Earth?
12.
Hold your thumb at arm’s length, and observe it first with your left eye and then
with your right eye. Analyze how the parallax model can be applied in order to
measure how far your thumb is from your eyes.
13.
You have two stars. They have the same luminosity and are located at the same
distance from earth. Can they have different brightness? Explain your answer.
14.
How will the luminosity of the sun change if the observer moves away from it to
twice the distance between the Earth and the Sun? How will the Solar constant
change it this happens? Explain.
15.
Distinguish in words between brightness, luminosity, and apparent brightness.
Write expressions that show the mathematical relationship between these
quantities.
5.2 Stellar Magnitudes
A third type of stellar parameter is stellar magnitude. There are two kinds of stellar
magnitudes…apparent and absolute. Apparent magnitude (m) represents the observed
brightness of a star. It is given by an integer ; the more negative the integer is, the brighter the
star appears to be. Absolute magnitude (M) represents the apparent magnitude that a star
would have if it were placed at a distance of 10 parsec (pc) away from the earth.
(1 parsec is about 200,000 AU, or about 3.26 light years.) As we shall see in the exercises
below, astronomers prefer to use absolute stellar magnitudes because they can be directly
related to luminosity.
Physical Quantities & Their Relationships
1.
How do we measure stellar brightness? Use the following data to invent/derive
the relationship between stellar apparent magnitudes (m1 and m2) and the
apparent brightness (l1 and l2) that we measure here on earth.
Difference in magnitude
(m2 – m1)
Apparent brightness ratio
(l1 / l2)
0.0
0.5
0.75
1.0
1.5
1:1
1.6 : 1
2:1
2.5 : 1
4:1
36
2.0
2.5
3.0
4.0
5.0
6.0
10.0
15.0
20.0
25.0
2.
3.
4.
6.3 : 1
10 : 1
16 : 1
40 : 1
100 : 1
251 : 1
10,000 : 1
1,000,000 : 1
100,000,000 : 1
10,000,000,000 : 1
Using the expression derived in #1, derive an expression for the relationship
between a star’s apparent magnitude (m), absolute magnitude (M), and its
distance (d) from the earth.
Determine the absolute magnitude of the Sun if its apparent magnitude is -26.5.
Using the expression derived in #1, derive a relationship involving the absolute
magnitude (M) and luminosity (L) of a star and the absolute magnitude and
luminosity of the Sun.
Applying the Model
5.
Can two stars have the same apparent magnitudes but different absolute
magnitudes? Give an example. Explain.
6.
Can two stars have the same absolute magnitudes but different apparent
magnitudes? Give an example. Explain.
7.
Why is the star’s apparent magnitude not a good indication of the star’s energy
output?
8.
Why is the star’s absolute visual magnitude not an accurate measure of a star’s
total energy output?
5.3 Stellar Spectra
The studies of stellar spectra in the 19th century demonstrated that stars’ spectra look
different. The studies of stellar spectra allow astronomers to determine chemical
composition of stars, their surface temperatures, density of the atmospheres, their
motion, whether they belong to binary systems, and their age.
Please take some time now to study the Web sites related to (a) the sun’s spectrum, (b)
stellar spectra, and (c) the Harvard classification scheme (O B A F G K M … Oh! Be A
Fine Girl, Kiss Me!).
The Harvard classification scheme is the result of the efforts of a group of women
astronomers at Harvard in the early 1900’s. Read about them at the web sites.
Observations
37
1.
2.
3.
Using Laboratory Exercises in Astronomy - Spectral Classification, try to
classify as many stellar spectra (numbered 1 – 30) as you can.
Which characteristics of a star can be determined by one night’s observing with a
naked eye?
Which characteristics of a star can be determined by one night of telescopic
observations, including auxiliary instruments?
Applying the Model
4.
Which kinds of stars are expected to emit large amounts of ultraviolet radiation?
5.
Which kinds of stars would be expected to emit large amounts of infrared
radiation?
6.
What characteristic determines the color of the star?
7.
The spectral class of the star depends on which two properties of the star?
8.
List the following stars in order of increasing surface temperature: A0, B3, F2,
M3, G2, O8.
9.
Determine the color and approximate surface temperature of the following stars:
Sirius A1 ; Centauri G2 ; Arcturus K2 ; Rigel B8 ; Betelgeuse M2 ; Crucis B0
10.
11.
12.
13.
Three stars are observed to put out their maximum light at the following
wavelength. Estimate the spectral class of each.
2.6 x 10-7 m
5.0 x 10-7 m
9.7 x 10-7 m
In which stellar spectral class is each of the following most likely to be found in
the spectrum?
(a) Strong hydrogen lines
(b) Lines of neutral metals
(c) lines of neutral helium.
Stellar spectra exhibit absorption lines that can be used to identify chemical
elements whose atoms could absorb recorded frequencies. At the same time this
information can be deceiving because the conditions for absorption vary with
temperature. Most of the stars are made of the same chemical elements: 70% H
and about 25% He. Chemical composition changes only slightly as the star
evolves.
How do astronomers estimate the chemical compositions of stars?
What different kinds of information can be obtained from the analysis of stellar
spectra?
5.4 Stellar Masses
Observations, Physical Quantities & Their Relationships
38
To understand how stellar masses are determined we need to learn about Kepler’s laws.
Kepler’s laws describe the motion of planets around the Sun (so they really aren’t laws).
Kepler tried to find patterns in the data related to the motion of Mars and other planets.
Analysis of Mars’ data led him to the conclusion that planets move around the sun
following elliptical orbits rather than circular ones, as Copernicus thought. He also
found out that Mars moved with different speeds at different times. He concluded that
the closer Mars was to the Sun, the faster it moved on its orbit. Kepler never explained
his findings, it was done later by Newton. What is important to us is his 3rd law, which
allows us to estimate masses of stars.
1.
2.
3.
4.
Using the data in the table below, develop a relationship between the
parameters. (Hint : Kepler discovered the relationship after he made a plot and
used logarithms.)
Planet
Semimajor Axis
(AU)
Period (Years)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
0.3871
0.7233
1.0000
1.5237
5.2028
9.538
0.2409
0.6152
1.0000
1.8810
11.86
29.46
Does the relationship you discovered make sense? Why?
Derive an equivalent relationship using Newton’s 2nd law and the law of
universal gravitation. After you discover the relationship, express it for distances
and times as measured in SI units.
To calculate the constant in Kepler’s law one must know the mass of the Sun.
What observational data can help us to determine the mass of Sun?
Predicting &Testing The Model
5.
The center to center distance from the earth to the moon is 3.84 x 108 m. The
center of mass of the Earth-moon system has been found to be 1.7 x 103 m below
the surface of the earth. Use this information and Newton’s “correction” to
Kepler’s 3rd Law (found by visiting the Web sites) to determine the mass of the
earth and the moon.
Test your prediction if possible.
Applying the Models
6.
Using the spectral observations on the following page, determine the mass(es) of
the star(s) in the system HR 80715.
39
To fully complete this exercise, you’ll need a little more information about a concept
known as the center of mass .
Reference : Center of Mass
If you have two objects like the Earth and the Moon, then the system of these two
objects has a point called the center of mass, which does not change its location when
these objects orbit each other (you might think that the Earth does not move at all, but
in fact it moves around this point too, as well as the Moon does. But because this point
is much closer to the center of the Earth than the Moon, we see that the Moon moves
around the Earth). If the distance of the mass M from this point is R and the distance of
m is r, then we can wrote the expression for the gravitational force acting on each object
as
GMm/d2 = M(R/T2) = m(2r/T2)
(where d = R + r)
Use this equation to prove that :
MR =mr
(1)
MV = mv
(2)
and
(where V = 2πR/T and v = 2πr/T)
40
Spectral Observations of HR 80715
Notes:
(1) The laboratory  for iron is 654.315 nm
(2) The number listed to the right of the observations represents time in days.
41
Chapter 6 :
The Hertzsprung-Russell Diagram
Now that we’ve we established the system of stellar parameters, it would be interesting
to find out if there is any relationship between different parameters of the stars. To do
this, one must examine data from different stars.
The H-R Diagram
Observations & Models
1.
In the table below you will find data on 65 stars. Note that the data for
Luminosity, Radius, and Mass are relative to the luminosity, radius, and mass
of the Sun! To find the patterns in this data, you need to graph it. Because the
luminosity of stars varies so much (about 9 orders of magnitude), it is convenient
to use semi-log paper to graph L, the ratio of the star’s luminosity to the Sun’s
luminosity (y axis) versus T, temperature (x axis). As an alternative, you may use
the Excel program on the computers across the hall.
(Note : traditionally astronomers graph temperatures backwards : the highest are
closest to the beginning of the axis and the lowest are at the right end.)
Use the appropriate web sites to compare your graph with the graph
conventionally known as Hertzsprung-Russell (H-R) diagram.
Star
Arietis
Bootis A
 Bootis B
Cas A
 Cas B
Centauri
 Centi
 Eridan y C
VZ Hydrae
Hydri
Lac 9352
Leonis
70 Ophiuch iA
70 Ophiuch iB
 Orionis
 Persei
Scorpii
Ursae Majoris
 Virginis
Ophuich i
Proper Name Luminosity Mass Radius Temperature Color
Sheratan
20
2.5
1.8
8800
White
0.5
0.8
0.9
5300
Yellow
0.19
0.6
0.8
4400
Orange
1.2
1.1
1.1
6000
Yellow
0.05
0.4
0.6
3700
Red
2900
11
6
19000
Blue
0.7
0.9
0.9
5600
Yellow
0.02
0.3
0.4
3300
Red
3.5
1.5
1.2
7200
Green
7
1.8
1.4
7400
White
0.04
0.4
0.6
3400
Red
Denebola
27
2.7
1.8
9000
White
0.4
0.8
0.9
5200
Orange
0.12
0.66 0.7
4200
Orange
3.0
1.4
1.2
6300
Green
Algol
200
5.2
3
12000
Blue
Dschubb
1200
17
7.6
28000
Blue
Alkaid
1100
8
4
15000
Blue
4
1.5
1.3
7100
Green
1600
19
8
30000
Blue
42
Spectral Class
A5 V
G5 V
K4 V
G0 V
M0 V
B2 V
G5 V
M4 V
F5 V
F0 V
M2 V
A3 V
K0 V
K6 V
F6 V
B8 V
B0 V
B5 V
F2 V
O9.5 V




Ursae Majoris
Austrini
Spica
Archernar
Regulus
Vega
Merak
Formalhut
1800
720
164
55
70
20
Altair
12
3.0
1.0
0.44
0.51
0.04
0.01
6000
10400
60000
60000
1500
6000
150
34
9200
115
160
300
9500
15000
700
1600
1
1.0
0.2
0.01
0.0004
WW Aurigae A
Aquilae
VZ Hydrae
UV Leonis
Ceti
Herculis
YY Geminorum
Krueger 60 A
Crusis
 Centauri
Orionis
Cygni A
Carenae
Castor C
Beta Crusis
Hadar
Rigel
Deneb
Canopus
Ursae Minoris Polaris
Capella

 Geminorum Pollux
 Cephei
Arcturus
Bootis
Aldebaran
 Tauri
Ophiuchi
Antares
Scorpii
Betelgeuse
Orionis
Sheat
Pegasi
Ras-Algethi
Herculis
Sun
Rigil Kent
Centauri A
Centauri B
Centauri C
Baranar ds Star
Wolf 359
14
10
5.5
3.8
3.6
3.0
181
2.1
1.12
0.95
0.8
0.78
0.58
21
15
43
42
15
14
4
4
16
4
5
6
24
27
7
9
1
1.0
0.6
0.2
0.2
7
5
3.2
2.6
2.4
2.0
1.9
1.5
1.05
1.05
0.8
0.89
0.6
0.3
10
10
22
44
60
90
13
15
220
18
45
50
600
750
100
100
1
1
0.8
0.3
0.3
0.001
0.1
0.2
2400
Red
M8 V
Lalande 21185
3400
43
23000
17500
13000
10000
9900
9000
8800
8100
6200
5800
5300
5100
3600
3300
28000
23000
11200
9300
7400
5300
5300
4600
5000
4400
3900
3300
3200
3200
3000
2700
5800
5800
4400
3000
2500
Blue
Blue
Blue
White
White
White
White
Green
Yellow
Yellow
Orange
Red
Red
Blue
Blue
Blue
White
White
Yellow
Yellow
Orange
Orange
Orange
Orange
Red
Red
Red
Red
Red
Yellow
Yellow
Orange
Red
Red
B1 V
B3 V
B7 V
A0 V
A1 V
A3 V
A5 V
A7 V
F7 V
G2 V
G8 V
K0 V
M1 V
M3 V
B0.5 V
B1 III
B8 I
A2 I
F0 I-II
G8 III
G8 III
K0 III
K1 I
K2 III
K5 III
M1 III
M1.5 I
M2 I
M2 II-III
M5 II
G2 V
G2 V
K4 V
M5 V
M5 V
Red
M2 V
Canis Majoris A
Sirius
28
2.7
1.8
9900
White
A1 V
Canis Majoris B
0.0027
1.1
0.02
9300
White
White Dwarf
UV Ceti
 Eridani
61 Cygni A
61 Cygni B
Canis
Minoris A
Canis
Minoris B
0.005
0.3
0.18
0.08
7
0.15
0.7
0.6
0.5
1.8
0.25
0.85
0.7
0.7
2.2
2800
4800
4400
4000
6500
Red
Orange
Orange
Orange
Green
M6 V
K2 V
K5 V
K7 V
F5 IV-V
0.0006
0.7
0.01
9100
White
White Dwarf
44
Physical Quantities & Their Relationships
2.
The stars on the H-R Diagram are all in the neighborhood of the Sun.
How were the values for L (luminosity relative to the Sun) and/or M (absolute
magnitude) determined?
3.
How were the temperature values (T) determined?
4.
What is the theoretical model (in this case, a quantitative model) that will allow
you to determine the radius (R) of a star when its temperature and luminosity
are known?
5.
Using the H-R data (only the Spectral Class V stars, which are those on
the Main Sequence) and log-log paper, develop a relationship between
luminosity and mass.
Testing the Model
6.
Test the model of #4 quantitatively using the data from the table for several stars.
Are your results consistent with the model?
Applying the Model
7.
The “Giants” are so named because they have a large radius.
How do we “know” this?
8.
The “white dwarfs” are so named because they have a small radius.
How do we “know” this?
9.
For the stars in the table:
Which star has the hottest surface?
Which star has the coolest surface?
Which are the largest stars?
Which are the smallest stars?
Which star is bluest?
Which are the reddest?
Which star is most like the Sun?
Which stars do not follow the intuitive temperature-brightness relationship for
stars?
10.
What does it mean if most of the stars are located on the region called “The Main
Sequence”?
11.
Suggest a possible reason(s) for why a star would not be on the Main Sequence.
12.
Summarize the stellar parameters that are used in constructing and using the
H-R diagram.
45
Chapter 7 : The Model of the Sun
7.1 For How Long Will the Sun Shine?
The H-R diagram shows that stars spend most of their life on the Main Sequence. Our
Sun is on the Main Sequence too, which means that it is
(a) going through the longest phase of its evolution, and
(b) a convenient nearby model that we can use to study Main Sequence stars.
What do we know about the Sun that might help us to learn about its interior? We
know its color (temperature), its size (mass & radius), the energy that is sends in all
directions every second (luminosity) and its approximate age : this can be deduced from
the age of the Earth that is estimated to be at least 4.5 billion years. Let’s try to figure
out how the Sun maintains its tremendous luminosity for billions of years and what is
the source of its energy. By doing so, we’ll have a good idea about how Main Sequence
stars “work”.
The Life Span of the Sun on the Main Sequence
Currently Accepted Knowledge
1.
How old is our Sun? Will it shine forever? Why not?
Different Models of Physical Quantities & Their Relationships
2.
What affects the length of time the Sun can shine? How would you express its
life time in terms of the energy that it possesses and the energy it loses every
second?
3.
How long can it shine using its heat energy?
(a) To calculate the heat energy of the Sun, we must first show that it is an ideal
gas.
(b) Calculate the heat energy of the Sun, assuming that it is made of hydrogen,
and that its average temperature is 50,000 K. In what state (gas or plasma) is
hydrogen at this temperature? How do you know? (The mass of the Sun is listed
in the data table).
4.
How long can it shine using its gravitational potential energy? Assume that half
of the total GPE can be radiated as electromagnetic radiation if the Sun shrinks to
a point. (Note : the gravitational potential energy of a single body of mass M is
given by GM2/R ; this allows you to estimate your own gravitational potential
energy!)
What other sources of energy might the Sun have? Can it use chemical energy?
46
Why not?
47
A Better Model for the Source of the Sun’s Energy
5.
What is fusion?
6.
What is a p-p cycle? What is a CNO cycle? (Check the Web Sites!)
7.
What conditions are necessary so that fusion might take place in the core of the
Sun?
Physical Quantities & their Relationships (for the Fusion Model)
8.
How much energy is radiated when a nucleus of helium is produced in a fusion
reaction?
9.
Assume that ~ 0.1 Msun is able to fuse in the core of the Sun. How long will fusion
power the Sun under these conditions?
10.
Calculate the temperature at which two protons can come close enough for
nuclear attraction forces to start acting on them. Why can’t they come close at
lower temperatures? (Check Web Sites!)
Applying the Fusion Model
11.
The actual temperature of the core is ~ 107 K. How can we resolve the difference
between this value and that of #10? Does it mean that fusion is impossible?
12.
What might be an indicator of the presence of the fusion in the core of the Sun?
Estimates show that, because of absorption, reemission, and scattering, it takes 10
million years for a photon radiated in the fusion reaction in the core of the Sun to
reach the surface. This means that the photons that we are seeing on the surface
of the Sun now were born very long time ago. What particle that is emitted in
fusion reactions, does not get absorbed on its way out?
13.
Assuming that the sun will be on the main sequence for 5 x 109 years, for how
long will a star with 10 x’s the mass of the Sun “live” on the Main Sequence?
Testing the Fusion Model
14.
What are the results of Solar Neutrino experiments?
7.2 What Holds The Sun Together?
Building a Model
We’ve decided that the Sun is powered by fusion, which releases tremendous amounts
of energy. If this is so, why doesn’t the Sun (or any Main Sequence star) simply explode
in an uncontrolled fusion chain reaction? Can you develop a model that can account for
why stars “hold together”?
(Check the Web Site on Hydrostatic Equilibrium!)
48
Reference : The Proton-Proton Chain of Fusion
The fundamental components of matter are called elementary particles. The most
familiar elementary particles are the proton, electron, and neutron. The existence of
another type of particle, the neutrino, was first postulated in 1933 by the physicist
Wolfgang Pauli to account for small amounts of energy that appeared to be missing in
certain nuclear reactions. Neutrinos were presumed to be massless particles that moved
at the speed of light. They interact very weakly with matter and most of them pass
completely through a star or planet without being absorbed. However, recent
experiments have shown that neutrinos do have some mass. This finding leads to
certain implications for cosmology and for the model of how our Sun produces energy.
The model for energy production in the Sun is that it taps the energy in the nuclei of
atoms through nuclear fusion, the joining together of atomic nuclei. Just as gases give
up gravitational potential energy when they come together to form a star, particles
release binding energy when the come together to form a nucleus. Since mass and energy
are equivalent, the binding energy released during fusion must correspond to a
decrease in the mass of the atomic nuclei that is produced. Or, as you may have heard,
E = mc2.
The primary model for fusion in the Sun is called the Proton-Proton Chain. This process
(illustrated on page 48) involves 3 basic steps:
+ e- + 
Step 1 :
1H
+ 1H
2H
Step 2 :
2H
+ 1H
3He
+ 
Step3 :
3He
4He
+ 2 1H
+ 3He
Since protons are positively charged, these reactions can only take place in regions of
very high temperature, where the velocities are high enough to overcome the electrical
forces that keep protons apart. As a result, the protons are able to move within 10-15 m
apart, at which point the attractive strong nuclear force takes over, enabling the binding
energy to be released. In the Sun, these temperatures must be on the order of 107 K.
These temperatures are reached only in the regions surrounding the center of the Sun.
But even at these high temperatures, it is difficult to force two protons together. On
average, it takes a very long time to do this (note the time intervals on the illustration).
This is a good thing!!! Because it takes so long for the protons to interact, the Sun has
lasted (and will continue to last) for a long time…a long enough time to enable the
slooooooowwww biological processes on Earth to produce the many complex forms of life
that now exist.
49
Illustration : The Proton – Proton Chain
50
Chapter 8 : Stellar Evolution
The next step in our understanding of stars will be learning about their evolution. This
deals with the question such as: Where do stars come from? Do they change over the
years? What will their life path depend on? Do they ever die? What happens to stars
after death? How do we know this?
Birth
Using Models to Determine Physical Quantities & Their Relationships
1.
Determine the size (radius) of a nebula, assuming that it has a temperature of 10
K and a density of 100 particles/cm3 so that this nebula can start contracting due
to its own gravitation.
2.
How many masses of the Sun are contained in this nebula?
“Childhood”
Applying the Model
1.
What is a proto-star? How do they form? Why doesn’t every gas/dust
interstellar cloud becomes a proto-star?
2.
Where in the Galaxy are likely places to search for proto-stars? What is the
observational evidence for the model of a proto-star?
3.
What kind of radiation would a proto-star be expected to emit?
4.
Explain the shape of the evolutionary tracks (shown on page 44) leading up to
the a star becoming situated on the Main Sequence.
5.
Explain how these tracks are related to the masses of the stars.
51
Stellar Evolution Tracks to the Main Sequence
52
”Adulthood” (The Main Sequence)
Using Models to Determine Physical Quantities & Their Relationships
1.
A main sequence star of 20 solar masses is about 10,000 times more luminous
than the Sun. Assuming the star to be pure hydrogen, how long would it take for
the star to convert all its hydrogen into helium?
Applying the Model
2.
Once the proton-proton reaction (fusion) begins in the core of the star, why isn’t
all hydrogen converted to helium instantly?
3.
When hydrogen is converted to helium plus energy inside stars, where does the
energy come from?
4.
What property of a star determines its location on the main sequence?
5.
Describe the basic differences between:
a) a main sequence A0 star and a white dwarf A0 star;
b) a main sequence M2 star and a red giant M2 star.
6.
Why is it believed that stars spend the major portion of their lives on the main
sequence?
7.
Why do upper main sequence stars (O & B) spend much less time on the main
sequence than do lower main sequence stars?
8.
Explain how Cepheid variables can be used to measure the distance to clusters.
9.
The H-R Diagrams for an open and a globular cluster are shown below. Explain
which is which. How would you test your explanation?
10.
Why is it useful to use clusters to study stellar evolution?
HR diagrams for open vs globular cluster goes here
53
“Old Age”
Using Models to Determine Physical Quantities & Their Relationships
1.
Suppose a star had the same mass, radius, and period as the Sun. If that star
evolved into a white dwarf of radius 104 km (~ Rearth), determine its density and
rotational period.
2.
The density of a neutron star is similar to the density of the nucleus of the
hydrogen atom. Use this information to determine the radius and rotational
period of a Sun-sized star which collapses into a neutron star.
3.
Determine the radius of a black hole with the mass of the Sun.
(Hint : the escape velocity for a black hole is the speed of light.)
Applying the Model
4.
What are possible end stages for stars?
5.
What property of a star determines how it will end?
6.
What will be the fate of the Sun?
7.
How does a main sequence star evolve into a red giant?
8.
When a main sequence star exhausts its hydrogen fuel in its core,
(a) what happens to the core of the star?
(b) what happens to the star’s outer layers?
9.
What two reactions are believed to occur in red giants, and where in the star do
they occur?
10.
What becomes of a red giant whose mass is ~ 2Msun?
11.
How is it known that dwarfs are dying stars, and not stars at some active stage of
evolution?
12.
What is believed to be inside a white dwarf? How do we know that this is true?
13.
What are neutron stars and how are they detected?
14.
What is degenerate matter? How is degenerate matter different that an ideal gas?
15.
How does degenerate matter create conditions under which hydrostatic
equilibrium cannot occur? What is the result of these conditions?
16.
Why is it difficult to detect black holes in space, if they exist?
17.
If black holes do exist but emit no electromagnetic radiation, then how might
they be detected?
18.
Why doesn’t a proto-star just continue to collapse until it becomes a planet or a
black hole?
54
“Death”
Applying the Model
1.
Under what circumstances will a Red Giant undergo a Type II Supernova?
2.
What are the possible outcomes of a Type II Supernova?
Give the criteria for each.
3.
Compare & Contrast the systems & mechanisms involved in
(i) Novae
(ii) Type Ia Supernovae
(iii) X-Ray Bursters
Life After Death
Applying the Model
1.
What are Supernova Remnants?
2.
What is nucleosynthesis?
3.
How do we know that our solar system is the result of a supernova?
4.
How do the main sequence stars eventually contribute to the content of heavy
elements in the interstellar medium?
Stellar Evolution – Putting It All Together
Applying the Models
1.
What is the predominant energy source for each of the following kinds of stars:
(a) pre-, (b) main sequence, (c) red giant, (d) white dwarf?
2.
Why is it that we can know so much about the formation of stars, which are so
far away, whereas we know so little about the formation of planets, one of which
is just outside the door?
3.
Why does fusion occurs peacefully in the stars on the main sequence but leads to
explosions in Novae stars? Supernovae Stars? Or on Earth? (hint: the reasons are
the same in principle but different in details).
55
Summary
In summary, the evolution of a star involves 6 stages (all of which are well-discussed at
the appropriate Web sites!). Listed below are the phenomena associated with those
stages:
I) Birth
Nebula
II) Childhood
Protostars
Evolutionary Tracks
III) Adulthood (the Main Sequence)
Clusters (open vs globular)
Cepheid Variables
IV) Old Age
Red Giants
White Dwarfs
Degenerate Matter
Neutron Stars (pulsars)
Black Holes
V) Death
Novae
Supernovae (Type Ia & Type II)
VI) Life After Death
Supernovae Remnants
56
57
Reference : The Magnetic Force Acting on a “Free” Charge Moving in a
Magnetic Field
When the charge q moves at a velocity v that is perpendicular to a uniform magnetic
field B, the charge will experience a force F that is perpendicular to both v and B. This
force is found by :
F = qvB sin
where is the angle between the tails of the magnetic field vector B and the velocity
vector v.
(note : vsinis the component of v that is perpendicular to B.)
The three vectors (vsin, B, and F) are related by the Right Hand Rule (if q is positive) :
If you position your right hand in the “open” position, with your thumb pointing in the
direction of vsin and your fingers pointing in the direction of B, then the force F is
directed out of your palm. (If the charge is negative, use the Left Hand Rule.)
The Right Hand Rule is illustrated below :
B
vsin
F
58
If F is perpendicular to vsin, how will the charge move? These criteria virtually define
a circular motion. Think of the earth going around the sun. The force of gravity is
directed toward the sun, while the earth is moving on a tangent to its orbit...F and v are
perpendicular!!! Think of a ball on a string being whirled in a circle. The string exerts a
tension force on the ball that is directed toward the center of the circle while the ball
moves on a tangent..again, F and v are perpendicular!!! So, guess what... the charge will
move in a circle! Physically, this allows us to say that :
qv’B = mv’2/R
where v’ = vsin& R = radius of the circle
Therefore,
R = mv’/qB
Problem : A positive charge q moves with velocity v in an extremely strong uniform
external magnetic field Bext. The velocity v is in the x-z plane and makes an angle with
the x-axis. The external field Bext is parallel to the x-axis.
Develop a model for how the charge will move in this field. (By the way, this is a
simplified version of a very common situation on guess what…stars!)
y
Bext
x

v
z
59