Physics, Astronomy, &Astrophysics

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Chapter 1 : Models of LightINTRODUCTION

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 Observations1. 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).

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• Exponential change (amplitude for damped motion, radioactive decay).

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).

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• Phenomenon of a white dwarf.• Phenomenon of a neutron star.• Phenomenon of a black hole.

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 21. 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 3The 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 41. 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 51. 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 celestialobjects? What should we study next?

1.2 The Mechanical Wave ModelIn 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

Observations1. 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.

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5. Observe what happens when two pulses oriented in opposite directions meet.

Modeling1. 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.

Observations1. What do you need to have if you want to create wave motion?

Observations & Models2. 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 & TestingBy 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.

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Vary the frequency of the source (your hand) and observe different waves. Describe your observations using the word “energy”.

Modeling1. 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.

TestingDesign an experiment to test your explanation.

(c) Interference and Diffraction of Mechanical Waves

InterferenceObservation

Observe the laser disc Ripple Tank scenes of vibrating water produced by 2 sources. Describe in words what you see.

ModelPropose a model for the pattern that you’ve observed.

Predicting & TestingUsing 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 Observations1. 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

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Propose a model that explains both the 2nd and 3rd observations.

Predicting & TestingUse the model to predict what will happen to the waves as the opening that they pass through gets wider. Test your prediction.

(d) Polarization of Waves

Observations1. 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.

ModelingExplain the difference between the effects of the box on different pulses. What waves can be polarized and what waves can’t?

TestingUse 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. 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.

2. 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?

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(b) Interference & Diffraction of Light

Design an ExperimentIf light is a wave, we should be able to use it to exhibit the characteristics of interference and diffraction. Therefore, given (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 Model1. 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 & Testing4. 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 Model7. 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

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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 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.12. Why should a diffraction grating be called an interference grating?

13. 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

S1

S2

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 nlonger 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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L

y

d

screen

are so small (<<10) that they can be considered to be equal. Therefore, we can saythat 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.

ObservationsUse 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.

TestDesign and perform an experiment to show that reflected light is

polarized.

Models and Testing1. 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

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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 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 ActivityObservations & Models1. Use a spectrometer to observe the spectrum of the light bulb. Develop

a model that explains this observation.

Predictions & Testing2. 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 ActivityObservations & Models4. 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?

Observations5. 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.

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7. Describe Plank’s solution of the problem. What is a “black body” and how does it work?

Observations8. Describe photoelectric effect experiments observed in class.

Models & Testing9. 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 Relationships19. 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 AtomsAs 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.

HistoryBy 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.

ModelsThe 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

Observations1. 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 & Models5. 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.

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8. What experimental facts cannot be explained with Rutherford’s planetary model? What does this model predict about the emission of light?

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 Quantities1. 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 & Testing5. 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.

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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.

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Sodium & solar spectrum go here

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Chart of solar spectrum lines (Appendix E) goes here

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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!!!

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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 + 2r)2](r) + … = kq1q2/r

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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

W = Uel = Uf – U0 = 0 – U0

kq1q2/r = -U0

U0 = -kq1q2/r

5) Total Mechanical Energy

Etotal = K + UChapter 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.

Observations1. 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)

Models2. What additional characteristic does an ideal gas have? What evidence do we have for this?

Physical Quantities & Their Relationships3. 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

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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?

(Hint: consider, among other things, what the escape velocity (see next page) would be for hydrogen in the earth’s atmosphere.)

11. 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.

12. 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).

13. Calculate the energy of a photon that can ionize the hydrogen atom.14. Calculate the temperature at which the hydrogen atom will be ionized.15. 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?

16. How often do molecules collide? (Calculate the number of collisions per second).17. 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?

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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/Re

2, it’s easy to show that

Ug = Uf – U0 = mgh

2) Escape VelocityLet’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

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Using this idea we can calculate the escape velocity.

Chapter 4 : The Doppler Effect

Observations, Models & Testing1. 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 Relationships4. 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?

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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?

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Predicting & Testing8. 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 Model10. 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.

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stationary source :

rotating source :

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Chapter 5 : Stellar ParametersIt 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?

Observations1. 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.

Models2. 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 & Testing3. 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 Relationships4. 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?

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8. Recall our expression for luminosity from Chapter 1 (section 1.5). What part of that expression represents brightness?

Applying the Models11. 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 MagnitudesA 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 Relationships1. 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)

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0.0 1 : 10.5 1.6 : 10.75 2 : 11.0 2.5 : 11.5 4 : 12.0 6.3 : 12.5 10 : 13.0 16 : 1 4.0 40 : 15.0 100 : 16.0 251 : 110.0 10,000 : 115.0 1,000,000 : 120.0 100,000,000 : 125.0 10,000,000,000 : 1

2. 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.

3. Determine the absolute magnitude of the Sun if its apparent magnitude is -26.5.4. 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 Model5. 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 SpectraThe 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!).

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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.

Observations1. Using Laboratory Exercises in Astronomy - Spectral

Classification, try to classify as many stellar spectra (numbered 1 – 30) as you can.

2. Which characteristics of a star can be determined by one night’s observing with a naked eye?

3. Which characteristics of a star can be determined by one night of telescopic observations, including auxiliary instruments?

Applying the Model4. 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:SiriusA1 ; Centauri G2 ; Arcturus K2 ; Rigel B8 ; Betelgeuse M2 ;

Crucis B0

10. 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

11. 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.

12. 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?

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13. What different kinds of information can be obtained from the analysis of stellar spectra?

5.4 Stellar Masses

Observations, Physical Quantities & Their RelationshipsTo 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. 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 0.3871 0.2409Venus 0.7233 0.6152Earth 1.0000 1.0000Mars 1.5237 1.8810

Jupiter 5.2028 11.86Saturn 9.538 29.46

2. Does the relationship you discovered make sense? Why?3. 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.

4. 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 Model5. 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.

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Test your prediction if possible.

Applying the Models6. Using the spectral observations on the following page, determine the

mass(es) of the star(s) in the system HR 80715.

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(2R/T2) = m(2r/T2)

(where d = R + r)

Use this equation to prove that :

MR =mr (1)and

MV = mv (2)

(where V = 2πR/T and v = 2πr/T)

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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.

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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 DiagramObservations & Models1. 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 Proper NameLuminosityMass Radius TemperatureColor Spectral ClassArietis Sheratan 20 2.5 1.8 8800 White A5 VBootis A 0.5 0.8 0.9 5300 Yellow G5 V Bootis B 0.19 0.6 0.8 4400 Orange K4 VCas A 1.2 1.1 1.1 6000 Yellow G0 V Cas B 0.05 0.4 0.6 3700 Red M0 VCentauri 2900 11 6 19000 Blue B2 V Centi 0.7 0.9 0.9 5600 Yellow G5 V Eridan y C 0.02 0.3 0.4 3300 Red M4 VVZ Hydrae 3.5 1.5 1.2 7200 Green F5 VHydri 7 1.8 1.4 7400 White F0 VLac 9352 0.04 0.4 0.6 3400 Red M2 VLeonis Denebola 27 2.7 1.8 9000 White A3 V70 Ophiuch iA 0.4 0.8 0.9 5200 Orange K0 V70 Ophiuch iB 0.12 0.66 0.7 4200 Orange K6 V Orionis 3.0 1.4 1.2 6300 Green F6 V Persei Algol 200 5.2 3 12000 Blue B8 VScorpii Dschubb 1200 17 7.6 28000 Blue B0 VUrsae MajorisAlkaid 1100 8 4 15000 Blue B5 V

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Virginis 4 1.5 1.3 7100 Green F2 VOphuich i 1600 19 8 30000 Blue O9.5 V Spica 1800 14 7 23000 Blue B1 V Archernar 720 10 5 17500 Blue B3 V Regulus 164 5.5 3.2 13000 Blue B7 V Vega 55 3.8 2.6 10000 White A0 VUrsae MajorisMerak 70 3.6 2.4 9900 White A1 VAustrini Formalhut 20 3.0 2.0 9000 White A3 VWW Aurigae A 181 1.9 8800 A5 VAquilae Altair 12 2.1 1.5 8100 White A7 VVZ Hydrae 3.0 1.12 1.05 6200 Green F7 VUV Leonis 1.0 0.95 1.05 5800 Yellow G2 VCeti 0.44 0.8 0.8 5300 Yellow G8 VHerculis 0.51 0.78 0.89 5100 Orange K0 VYY GeminorumCastor C 0.04 0.58 0.6 3600 Red M1 VKrueger 60 A 0.01 0.3 3300 Red M3 VCrusis Beta Crusis 6000 21 10 28000 Blue B0.5 V Centauri Hadar 10400 15 10 23000 Blue B1 IIIOrionis Rigel 60000 43 22 11200 Blue B8 ICygni A Deneb 60000 42 44 9300 White A2 ICarenae Canopus 1500 15 60 7400 White F0 I-IIUrsae MinorisPolaris 6000 14 90 5300 Yellow G8 III Capella 150 4 13 5300 Yellow G8 III Geminorum Pollux 34 4 15 4600 Orange K0 III Cephei 9200 16 220 5000 Orange K1 IBootis Arcturus 115 4 18 4400 Orange K2 III Tauri Aldebaran 160 5 45 3900 Orange K5 IIIOphiuchi 300 6 50 3300 Red M1 IIIScorpii Antares 9500 24 600 3200 Red M1.5 IOrionis Betelgeuse 15000 27 750 3200 Red M2 IPegasi Sheat 700 7 100 3000 Red M2 II-IIIHerculis Ras-Algethi 1600 9 100 2700 Red M5 IISun 1 1 1 5800 Yellow G2 VCentauri A Rigil Kent 1.0 1.0 1 5800 Yellow G2 VCentauri B 0.2 0.6 0.8 4400 Orange K4 VCentauri C 0.01 0.2 0.3 3000 Red M5 VBaranar ds Star 0.0004 0.2 0.3 2500 Red M5 VWolf 359

0.001 0.1 0.2 2400Red M8 V

Lalande 21185

3400Red

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M2 V

Canis Majoris ASirius 28 2.7 1.8 9900White A1 V

Canis Majoris B

0.0027 1.1 0.02 9300White White DwarfUV Ceti 0.005 0.15 0.25 2800 Red M6 V Eridani 0.3 0.7 0.85 4800 Orange K2 V61 Cygni A 0.18 0.6 0.7 4400 Orange K5 V61 Cygni B 0.08 0.5 0.7 4000 Orange K7 VCanis Minoris A

7 1.8 2.2 6500 Green F5 IV-V

Canis Minoris B

0.0006 0.7 0.01 9100 White White Dwarf

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Physical Quantities & Their Relationships2. 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 Model6. Test the model of #4 quantitatively using the data from the table for

several stars. Are your results consistent with the model?

Applying the Model7. 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.

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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 Knowledge1. How old is our Sun? Will it shine forever? Why not?

Different Models of Physical Quantities & Their Relationships2. 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!)

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What other sources of energy might the Sun have? Can it use chemical energy? Why not?

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A Better Model for the Source of the Sun’s Energy5. 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 Model11. 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 Model14. What are the results of Solar Neutrino experiments?

7.2 What Holds The Sun Together?

Building a ModelWe’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!)

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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:

Step 1 : 1H + 1H 2H + e- + Step 2 : 2H + 1H 3He + Step3 : 3He + 3He 4He + 2 1H

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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.

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Illustration : The Proton – Proton Chain

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Chapter 8 : Stellar EvolutionThe 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 Relationships1. 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 Model1. 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.

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Stellar Evolution Tracks to the Main Sequence

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”Adulthood” (The Main Sequence)

Using Models to Determine Physical Quantities & Their Relationships1. 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 Model2. 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

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“Old Age”

Using Models to Determine Physical Quantities & Their Relationships1. 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?

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“Death”

Applying the Model1. 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 Model1. 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 Models1. 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).

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SummaryIn 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 DwarfsDegenerate MatterNeutron Stars (pulsars)Black Holes

V) DeathNovae

Supernovae (Type Ia & Type II)

VI) Life After Death Supernovae Remnants

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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 : vsinis 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

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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!)

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x

vz

y

Bext