Download Lect06-2-7-11

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.
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
nit area from the surface
s rface of a perfectly
perfectl lightlight
absorbing object (a “blackbody”) is proportional to the
temperature of the object raised to the fourth power:
E = σT4
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 call 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, and
2) 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
black body spectrum”
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.
1
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,
Mars which
hich is shown
sho n on the ne
nextt slide
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.
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,
3.
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.
2
Relation of the previous diagram to astronomy:
Absorption and Emission Spectra:
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.
Continuing to refer to the Bunsen-Kirchhoff diagram:
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, 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.
Bohr’s
model of the
hydrogen
atom.
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.
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 atom 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
(probabilit ) of being in an
(probability)
any one partic
particular
lar 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.
3
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
l
to get closer
l
to the
h proton than
h in
i the
h groundd 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 is 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
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.
This set of diagrams is intended to illustrate how the motion of
a charge q1 causes the charge q2 to move due to the
electromagnetic force between them.
There is a slight delay required for the signal that the charge q1
has moved to reach the location of charge q2.
This signal travels at the speed of light, and in fact is light.
4
We can make an analogy with the wave in the surface of a lake that is
caused by a floating object bobbing up and down.
This motion will cause waves to radiate outward, and these will cause
objects floating elsewhere on the lake surface to bob up and down as the
waves pass by.
These water waves, like light waves, are “transverse” waves.
This means that objects are caused to move perpendicular to the direction
of the wave propagation as the wave passes by.
5