2. Energy conservation and some of its implications. · 2. Energy conservation and some of its...

What is it that we need to understand?

1. How we can use Newton’s theory of gravitation to find the masses of planets, stars, and galaxies.

2. Energy conservation and some of its implications.

3. How gravitational potential energy is liberated when a massive object gets smaller, and where this energy goes.

4. How mass can be converted into energy in other forms.

5. How angular momentum conservation affects the rate of spin as the radius from the rotation axis changes.

6. Quantized energy levels of atoms and molecules, and the implications for spectra.

7. Doppler effect: spectral line shift and/or broadening.

8. Effect of temperature on spectrum.

Conservation of Energy:

1. We can formulate the laws of nature as we now know them in terms of conservation laws.

2. The most important of these is the conservation of energy.

3. It says that energy can be transformed from one type to another by physical processes, but it can neither be created nor destroyed.

4. A car at rest at the top of a hill, given a tiny push, can appear to gain energy as it barrels down the hill.

5. But we say instead that it merely converts its gravitational potential energy into kinetic energy of motion in this process.

6. If the car runs back up another hill, it should stop at the same height where it began. This would convert the kinetic energy of its motion back into gravitational potential energy.

Because the total energy is conserved, in the absence of friction and wind resistance, it does not matter at all what path a roller coaster car takes or how many loop-de-loops it executes on the way, it will always come to rest again at its original height.In astronomy, we are exploiting this characteristic of conservation laws all the time to draw conclusions about the ultimate outcome of some process regardless of its details or the time it takes to happen.This is a very powerful technique.

Conversion of Gravitational Potential Energy into Heat:

1. The example of the car may not seem to have anything to do with astronomy, but it is actually not that far off base.

2. Imagine a star that is held up against gravity by the immense pressure of its hot gases in the central region where heat is being generated through nuclear reactions (we will come back to this presently).

3. Now suppose that the nuclear reactions run out of fuel and therefore cease.

4. Without the pressure they generate, the star will collapse under its gravitational force.

5. Just like the car, all the little chunks of the star will fall toward the star’s center, converting gravitational potential energy into kinetic energy of motion.

Nuclear reactions in the stellar core generate heat energy,

which produces the pressure that supports the star against

gravity.

In a steady state, like the state of our sun at its present stage

of evolution, this energy generation at the center of the

star balances the loss of energy by radiation at the

star’s surface.

When the nuclear fuel gives out, the pressure

support is reduced because the loss of energy by radiation

from the surface continues unabated. As a result, the star collapses inward.

The gases rushing inward toward each other collide, and convert the energy of ordered, inward motion into heat, which creates the

additional pressure necessary to support the star at a smaller

radius.

The process we have been describing that occurs toward the end of a star’s life also operates in a similar fashion when the star is

formed from an extended cloud of gas. In this case, energy is again lost through the cloud’s surface, reducing the heat energy content and pressure of the cloud, which

in turn allows gravitational potential energy to be converted

into kinetic energy of inward motion. Once again, the cloud is supported once more at a smaller radius, but the continued radiation

of energy through its surface makes the gravitational collapse

continue.

Conversion of mass into energy within a star:

1. Before Einstein, people believed in the conservation of mass.

2. But Einstein suggested that the conservation of energy was the most fundamental law, and that mass was just one particular form of energy.

3. Einstein’s famous equation E = mc2

tells us how much energy is stored in a mass m.

4. In a star like the sun, through a sequence of reactions, hydrogen atoms are converted into helium atoms, and in this process a small fraction (0.7%) of the mass of the original hydrogen atoms is converted into energy in the form of heat and radiation (light).

5. The sun converts 600 million tons of hydrogen into 596 million tons of helium, and a lot of energy, every second.

Even the non-mathematically inclined can see that four hydrogen atoms weigh more than one helium atom.

We should all be

familiar with the

conversion of mass into

energy.

Here the same process

that takes place in the center of the sun is used to liberate

energy in an uncontrolled

fashion.

These images and diagrams represent a 4 billion dollar

facility in California

that generates energy from

mass,as in the sun, using lasers and lots and lots of very high-tech

gear.

Momentum conservation:

1. Although mass is not conserved, in the absence of applied forces, momentum is.

2. Linear momentum, the momentum associated with linear motion, is just the mass, m, of the object multiplied by its velocity, v.Thus momentum is mass times velocity, or mv

3. The conservation of linear momentum can be easily observed on a pool table.

Fig. 6.6: Momentum conservation demonstrated on a pool table

No external force acts on the combined system consisting of the two pool balls, and hence the combined momentum of the pair does not

change. (An “elastic” collision is shown.)

Angular Momentum conservation:

1. Angular momentum is the momentum associated with spinning motion.

2. Angular momentum is conserved in the absence of applied forces.

3. Forces that act to alter spinning motions, and to change angular momentum, are called torques (twisting forces).

4. The angular momentum of a body of mass m executing a circular motion, with speed v, about an axis at a radius r is equal to the product

m×v×r

5. In the absence of torques, a reduction of the radius of this spinning motion by a factor of 2 must therefore cause the speed v to double, and both these changes make the number of rotations per second quadruple.

This behavior, a result of the conservation of angular momentum, is related to Kepler’s second law (equal areas are swept out in equal times)

This behavior, a result of the conservation of angular momentum, is related to Kepler’s second law

(equal areas are swept out in equal times)

The area of each triangle is r vΔt / 2 .

Because angular momentum is conserved,

the areas of the 2 triangles must be equal.

An Astronomical Example of Angular Momentum Conservation:

1. If the sun formed out of a spinning cloud of gas, then as this gas cloud contracted under gravity, it must have spun faster and faster (unless acted upon by an external torque).

2. The faster and faster spinning of the gas would have created centrifugal forces that would act in the opposite sense from the gravitational forces, reducing the tendency of the gas cloud to collapse further.

3. For the protosun to collapse to form the sun, it may be that a torque must be provided to reduce its spinning.

4. When we come to discuss the formation of the solar system, we will see how this might have happened.

Question:• What is a torque and why does the sun not apply a torque to a planet orbiting it?• Torque is an acceleration of azimuthal motion about a spin axis.• The sun can exert only a radial acceleration via gravity.• This cannot change the angular momentum of the planet, so it must remain

constant, and therefore we must have Kepler’s 2nd (equal area in equal time) law.• Torques change angular momentum.

Just as forces change linear momentum.Aside:

A planet simply passing by a massless object and not experiencing any force, gravitational or otherwise, conserves its angular momentum about the massless object. (Work it out if you don’t believe me.)

Another way to look at this:To apply a torque to an object, we must apply a force that does not act along a line that passes through the center of mass of the object.Think about putting a spin on the queue ball in a game of pool.The gravitational force between the sun and a planet acts along the line joining them, and this line must pass through the center of mass of the system. Consequently, no torque results and the angular moment must then be conserved.

Planet’svelocity

ϴ

ϴ

RV

The product giving the angular momentum, VR is the same in either case, because the cosine factors cancel

We see that angular momentum is conserved, even though in this case it might not appear to be a useful concept.

This diagram shows how we can derive the conservation of angular momentum from the conservation of linear momentum (which asserts that the velocity remains V and remains in an unchanged direction.

Review Questions (“Problems”) from your Textbook:At the end of each chapter in your textbook there are lists of

problems. None of these are assigned to you to write out. However, they are useful for you to review your reading and to be sure that you understand the chapter. Below are listed the problems from each chapter that concern the topics we will be discussing, however briefly, in class. Use them to review the material, if you wish.

Chapter 4, 3rd Edition: 9-12, 15, 16, 18, 19

Chapter 6, 3rd Edition: 9-1

Chapter 4, 4th Edition, Review Questions: 2, 3, 5 – 12, 15

Chapter 5, 4th Edition, Review Questions: 2, 4, 7, 8, 12, 14-18

Chapter 4, 5th Essential Ed., Review Q’s: 2, 3, 6-11, 13, 15

Chapter 5, 5th Essential Ed., Review Q’s: 1, 3, 4, 8-12, 15, 16

Review Questions (“Problems”) from your Textbook:

Chapter 4, 6th Essential Edition, “Review Q’s”: 5-11, 13, 15, 16.

Chapter 5, 6th Essential Edition, “Review Q’s”: 1-4, 8-11, 12, 15, 16.

Chapter 1, 7th Essential Edition, “Review Q’s”: 2-5, 9.

Chapter 2, 7th Essential Edition, “Review Q’s”: 1-9, 12-16.

Chapter 3, 7th Essential Edition, “Review Q’s”: 4-10.


Chapter 5, 7th Essential Edition, “Review Q’s”: 1-4, 8-11, 12-14.

Review Questions (“Problems”) from your Textbook:



Chapter 3, 8th Essential Edition, “Review Q’s”: 4-10.



Blackbody Spectra:

1. Common experience, especially for those who like to play with fire, shows that as an object is heated, it glows more and more brightly, first with a reddish color, then yellow, then ultimately white.

2. Stefan and Boltzmann quantified this knowledge through the Stefan-Boltzmann law, which states that the energy emitted per second per unit area from the surface of a perfectly light-absorbing object (a “blackbody”) is proportional to the temperature of the object raised to the fourth power:

E = σ T 4

3. The dominant wavelength at which this light energy is emitted decreases as (1/T), or equivalently, the frequency of the dominant emitted light is proportional to the temperature of the object. This is called Wien’s law.

What is a Black Body?A “black body” need not be black.In physics and astronomy we call an object a black body when:1) It is opaque and a good absorber of light, and2) The light it emits consists only to a very minor extent of reflected light that

originated in some other luminous object.The sun meets these criteria, and thus it is a “black body” to a very good approximation,

even though it certainly is not black.Every black body has the same characteristic shape of its spectrum of emitted light, as

shown on the previous slide.This happens because the object is opaque, and light must bounce around inside it many,

many times before some of it emerges.This means that the light has bounced so many times, giving up and receiving energy

from the atoms in the object, that its spectrum represents an equilibrium, a balance, with the energy of the atoms’ motions (the heat or temperature of the object).

Why do we call such objects black bodies?A black body is a good absorber of light. A lump of coal appears black because it

absorbs most of the visible light incident on it. This is the connection in our experience of “black” with “good absorber of light.” But not all good absorbers are black.

If you see an object that is a good absorber, you see mostly light that it emits and not light that it simply reflects from its surface.

Thus light that is emitted by a good absorber of light will have the special “black body spectrum” that is characteristic of the good absorber’s temperature and not characteristic instead of any properties of light from other sources that it reflects.

So, is the Moon a black body? When it is full, it certainly is not black. When we look at the Moon, we see mainly sunlight that it reflects. Nevertheless, the Moon reflects only 12% of the sunlight that strikes it, so it is actually a fairly good absorber of sunlight. It is also opaque, of course, so it could perhaps be considered as a black body.

Can the Moon really be a black body?The emitted light from the Moon is concentrated in the infrared, because the surface

of the Moon is cold.Our eyes do not respond to infrared light, and therefore we do not see this portion of

the Moon’s emission spectrum.The portion that we do see, the reflected sunlight, is not representative of all the light

the Moon gives off.Toward the end of this lecture, we will discuss the spectrum of the planet Mars,

which is shown on the next slide.Mars reflects only 15% of the sunlight that strikes it, so it is about as good a black

body as the Moon.You can see that a portion of the spectrum of Mars, the part at the right in the plot,

follows the shape of the classic black body spectrum quite closely.For Mars, like the Moon, there is a component of the spectrum, in the infrared, that

has a black body shape, but there is also a significant component of reflected sunlight.

An image of Mars taken with theHubble Space

Telescope8/24/03.

This is thesharpest

colorpicture

ever takenof Mars

from Earth.

An image of Mars taken with theHubble Space

Telescope8/23/03.

This wastaken just11 hours

beforeMars was

the closestto Earth

that it hadbeen in

60,000 years.

Even if an object is not a perfect black body, we can use the knowledge of the black body spectrum to measure its surface temperature.

The spectrum of Mars, on the previous slide, is a good example.There is a clear component of the spectrum of Mars that we can

identify as the “black body” emission from its surface.By locating the wavelength of the peak emission in this portion

of the spectrum, we can deduce the temperature of the Martian surface.

We can do this successfully in this case, because the reflected sunlight falls primarily in a completely separate range of wavelengths.

Spectra of Chemical Elements:1. Chemists knew for a long time that different chemical elements emit

different colored light when thrown into a fire.(Try throwing salt into a flame, and you should see the prominent yellow emission lines of the sodium in the salt.)

2. Bunsen developed a special colorless flame in order to use this technique to identify elements.

3. Kirchhof collaborated with Bunsen to develop a spectroscope, in which the colored light from the chemical in Bunsen’s flame was separated into its components in a prism.

4. They used their spectroscope to discover new elements through spectral analysis.

5. Helium was discovered in the spectrum of the sun’s corona during an eclipse 27 years before it was found on earth, in 1895.

Each chemical element produces its own pattern, a sort of chemical fingerprint, of spectral lines when it is heated in a

colorless (Bunsen) flame.

Elements extract from the continuous spectrum only certain wavelengths of light, and reradiate them in all directions.

Pulse of light from

red supergiant

April, 2002

Pulse of light from

red supergiant

May, 2002

Pulse of light from

red supergiantOctober,

2002

Pulse of light from

red supergiantDecember,

2002

Relation of the previous diagram to astronomy:In the diagram of Bunsen and Kirchhoff’s experiments, we can substitute a

distant star for the light bulb and an intervening cloud of cooler gas for the cloud that is shown.

The key point is that the star (or the light bulb) is a good black body, and hence its spectrum is close to that of the perfect black body (shown on a previous slide).

The gas cloud is not opaque, and hence is not a black body. It emits light in the set of spectral lines characteristic of the atoms and molecules from which it is made up. This light does not bounce around enough in the diffuse cloud to be brought into a black body spectrum by such scattering. The cloud is not opaque, and therefore any such scattering does not change its spectrum. We see the special set of “spectral lines” that the cloud’s atoms emit.

Absorption and Emission Spectra:Continuing to refer to the Bunsen-Kirchhoff diagram:As light from the black-body source (star or light bulb) passes through the

diffuse cloud, just those wavelengths of light that the cloud’s atoms characteristically emit will be absorbed from the star’s black body spectrum. This special set of wavelengths of light will be re-emitted by the cloud’s atoms in all directions.

If we look at the distant star through the intervening cloud, we will see the spectral lines of the cloud’s atoms in absorption. They will appear in the black body spectrum as dark lines (the absence of light that otherwise would have been there).

If we look at the cloud from an angle where we do not see the star behind it, we will see just the re-radiated light at these special wavelengths characteristic of the cloud’s atoms.

Absorption and Emission Spectra, Continued:By observing the patterns of emitted or absorbed wavelengths, we can identify

the types of atoms (the elements) that make up the cloud. This is a very powerful and much used technique.

In our diagram’s example, the intervening cloud had a lower temperature than the star behind it.

If, however, the cloud had been hotter than the star, we would see an emission spectrum when we look at the star through this hotter cloud. The atoms of the cloud, emitting their special wavelengths of light, would enhance the spectrum of the background star at each such frequency, not reduce it.

When we look at the sun, between us and its surface are tenuous regions of its corona which are not opaque and which are hotter than the sun’s surface. These regions therefore add emission lines, not absorption lines, to the sun’s black body spectrum.

The Hydrogen Atom – the Key to Understanding Spectra:1. Rutherford discovered in 1910 that the overwhelming bulk of the mass

of an atom is concentrated at its center in a very small nucleus.2. This discovery led Niels Bohr, who joined Rutherford’s group at

Manchester in 1911, to build a simple conceptual model of the simplest of all atoms, the hydrogen atom.

3. Bohr conceived of the hydrogen atom as a single electron orbiting a nucleus consisting of a single proton.

4. He proposed that the electron could only have a series of specific orbits, corresponding to specific orbital energies.

5. He proposed that when the electron jumps from one allowed orbit to another, it either absorbs or emits a photon whose energy equals the difference of the energies of the 2 orbits.

Bohr’s Model:Bohr’s model of the hydrogen model is extremely simple, and gives us a

means of understanding how the spectrum of only very special light wavelengths is generated.

In our modern understanding of the hydrogen atom, we think of the electron as not having a specific orbit, like a tiny earth going around a tiny sun, but as having a likelihood (probability) of being in any one particular place near the atom’s nucleus, its proton.

We can visualize this distribution of the likelihood of finding the electron as an “electron cloud.”

In the lowest energy, or “ground,” state, the electron cloud has a spherical distribution about the proton, as shown on the next slide.

When the electron is in the state of lowest energy, where it is most tightly bound to the nucleus, the probability of finding it at a given location depends only on that location’s distance from the nucleus, as shown here.

The ground state of the hydrogen atom:In Bohr’s model, with the electron orbiting the proton like the earth about the

sun, it is hard to understand how there can be a “ground state.” It should, it seems, be possible for the electron to get closer in toward the proton and become even more closely bound.

But in the modern view, where we think of the electron as an “electron cloud,” we can see that it is not possible for the electron to get closer to the proton than in the ground state.

In the ground state, we can think of the electron as essentially sitting right on top of the proton – it cannot get any closer.

In this view, the size of the electron cloud in the ground state of hydrogen can be thought of as the size of the electron itself. If we tried to push a whole lot of hydrogen atoms together in a box, we could then not make them take up less space than this. This consideration will turn out to determine the size of a white dwarf star.

Electron “degeneracy” and the size of a white dwarf star:

We will see as we discuss stellar evolution that the end state of our sun, billions of years from now when it no longer generates heat inside itself, will consist of material pulled so strongly together by gravity that it is held up by the fact that its electrons cannot possibly get closer together. We can think of the electrons in this “degenerate” state as essentially “touching.” It is the electrons that “touch,” because they take up much more space than the nuclei of the material, just as we can see in the picture of the hydrogen atom on the previous slide.

In a transition from a higher energy state to the ground state of the atom, as is shown here at one instant, the electron cloud is alternately concentrated on one side or the other of the central nucleus. As the electron cloud moves up and down, a wave of light – a photon – is emitted which carries away the extra energy of the excited state.

For the transition shown here, each oscillation of the electron cloud takes about a millionth of a billionth of a second (10-15 sec), and the transition is over after about 10 million such oscillations. The wave train of light that is emitted ( the photon) thus has about 10 million wavelengths.

http://www.physikdidaktik.uni-karlsruhe.de/software/hydrogenlab/elektronium/HTML/einleitung

_darstellung_3_uk.html

Here we make a comparison of transverse waves on the left, using waves propagating along a string (like a violin string), with longitudinal waves

on the right, using compressional waves traveling along a spring(sound waves in air work this same way).

We can think of what is oscillating along the direction of a light wave’s path as the electric and magnetic fields.

The direction and amplitude of the electric field tells us the size and direction of the electrical force that would be exerted upon any charged

particle located there. Thus charged particles along the light wave’s path would be pushed upward or downward as shown by the vertical arrows.

We will not discuss the magnetic field.

From these diagrams we can see what we mean by the wavelength and amplitude of a wave. The wavelength is the distance between wave crests, and

the amplitude tells us the height of the crests.

From these diagrams we can see what we mean by the wavelength and amplitude of a wave. The wave propagation speed is also shown. It tells us the speed with which the wave crests advance. The two waves at the right travel with the same speed, but because the lower one has half the wavelength, it has double the “frequency,” which is the number of wave

crests passing an observer per unit time.

The wave numbered 2 at the right has twice the frequency of the one numbered 1 above it. A light wave of twice the frequency (half the

wavelength) but the same amplitude causes a charge it passes along its path to move up and down twice as rapidly. This imparts more energy to

the charge, and we therefore see that the wave of higher frequency (shorter wavelength) has more energy.

All light signals propagate at the same speed in a vacuum, and we call this the speed of light. Light waves with different wavelengths (and

hence different frequencies) are given different names, as set out in this diagram. Astronomers get information about objects in the sky from

light of all these different frequencies. We design special telescopes to observe the different ranges of light wavelengths.

Going back to Bohr’s simple model, we see that transitions of the electron from the larger orbits, with higher energy, into the ground

state (the smallest orbit) cause the emission of light waves with greater energy and hence higher frequency.

Going back to Bohr’s simple model, we see that transitions of the electron from the larger orbits, with higher energy, into the ground state (the smallest orbit) cause the emission of light waves with greater energy and hence higher frequency.

For the hydrogen atom, the sequence of ever higher such energies and frequencies of light is very simple mathematically.

This allowed Bohr to come up with his simple model.Without the simple example of the hydrogen atom, we might never have

figured out how atoms really work.A series of “spectral lines,” or specific wavelengths (or frequencies) of light

that can be emitted or absorbed by a hydrogen atom, are shown at the bottom right on the next slide.

In one of your laboratory experiments, you will observe such spectra for hydrogen and other elements such as sodium or neon.

On the following few slides, we see a number of spectra.You will have a lab on this subject.Here you can get an idea of the variety of spectra and also their complexity.The spectrum of an element, like hydrogen or sodium, is like its fingerprint

or its bar code. It is unique and can be used by a trained person to positively identify the element.

Astronomers use these characteristic patterns of spectral lines in the light of distant stars to determine the chemical composition of those objects – a feat that would otherwise be impossible without going to the distant star and analyzing its material.

The wealth of information in the spectrum of light from an object is therefore a powerful tool for understanding the object.

Top: Hg lamp at low pressure

Middle: Fluorescent lamp and Hg lamp

Bottom: Hg lamp at high pressure

Solar spectrum

Spectrum of candle light

Electric lamp spectrum

Solar spectrum

Aluminum oxide spectrum

Iron spectrum

Spectra from discharge tubes

Hydrogen

Neon

High pressure Xenon

Top: Sodium spectrum. Middle: Sodium absorption in spectrum of incandescent light.Bottom: No sodium absorption with low temperature incandescent light

A diagram like this also applies to the case of light signals, except that the emitting object cannot travel faster than light.When we discuss relativity later in this course, we will come back to this point and explain it better.

Fig. 7.16

SpectralLines

How do spectra change with temperature?

1. From the earlier discussion of blackbody spectra, it is clear that as the temperature is raised, light of shorter wavelengths will be emitted.

2. For a gas of hydrogen, for example, that is not opaque (not a good absorber of light, and hence not a blackbody), an increase in its temperature will result in more powerful emission of the shorter wavelength lines in its spectrum.

3. This happens because the more violent collisions of the hydrogen atoms caused by the high temperature of the gas continually place electrons in the higher energy states from which high energy (short wavelength) photons may be emitted upon transitions to the lower energy states.

4. Thus the temperature of a gas through which a strong continuous spectrum is shining may be estimated by the relative strengths of its spectral lines.

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