Transcript Lect08-2-6-09

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.
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.
When the
nuclear fuel
gives out, the
pressure support
is reduced, and
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 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.
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.
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.
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 3 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.
The process is controlled, well sort of.
These images and diagrams represent a 3 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.
The process is controlled, well sort of.
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.
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)
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.