I said that everything in some sense, that was done for centuries after Newton
was filling in the details to F = MA. I don't want to give you the wrong
impression. There are a lot of fascinating details.
Newton's universe is rich and wonderful, and centuries of work by many, many, many
brilliant people went into producing insights and understandings.
And I wish I had the time to tell you about at least that part of it that I
know. But, we're an astronomy class.
What I'm going to do in the next two clips is give you sort of a highlight
reel of those of the aspects of what has been learned in the 250 years since
Newton and a little bit of the century since.
that will impact what we do and that we will need.
And, we will go through and try to develop, if not as deep an understanding
as we did for gravity which will be central, some intuition and some
understanding for some other important concepts in physics.
And as I said, we'll organize it by the M, the structure of matter, and the F,
the forces that act. [COUGH] And we will start.
Well, let's start with the M side. We'll start with what we know the end of
the 19th century about the structure of matter. By the end of the 19th century,
there we have a pretty comprehensive understanding of the
structure of matter. and its center is what is known as the
atomic theory of matter. All known matter is made up of a 100 or
so types of atoms, immutable units.
they're indexed by an number called Z that runs from one to
whatever a hundred, and roughly high elements with higher ZF, heavier atoms,
more massive atoms. And these characterize the chemical
elements. We understand some rules by which they bind together to form
compounds and molecules and various kinds of objects.
And bulk matter can appear in one of three common states.
We have solids, we have liquids and gases.
And in each, the bulk properties are understood as a consequence of the
microscopic dynamics of the atoms and the molecules that form it.
And, as a good and very helpful to us, example,
one of the forms of energy we discussed when we talked about non-conservation of
mechanical energy was heat. Friction, for example, converts kinetic
energy to heat. And
heat causes a object's temperature to rise, so we know what we mean when
something is hot. We mean, it has a high temperature.
in the context of an atomic theory, we have a good understanding of temperature.
Temperature is simply a measure of the average random motions of atoms and
molecules inside an object which explains why heat is sort of lowest common
denominator of energy. Heat is the energy that something has,
when it's doing its own thing. It's not moving in bulk, but its own
internal degrees of freedom are randomly fluctuating.
And for example, in an ideal gas, that's a gas that's made up of atoms or
molecules, that have no internal degrees of freedom
and are just non-interacting and bumping off each other in the walls. Then, you
can show that temperature gives you the average kinetic energy of a molecule or
an atom or whatever it is, is proportional to the temperature.
And temperature, in this expression, is measured in Kelvin.
Kelvin degrees are the same as centigrade degrees.
That just determines the units in which we measure this con, constant here, K,
which is called Boltzmann's constant. What is not arbitrary of course, is where
you put the zero of T because doubling the temperature relative to that zero
involves doubling the energy, and there is a temperature at which the average
kinetic energy of molecules dissents to complete zero.
This is absolute zero, or -273 or so degrees centigrade.
And this is the origin of that. And using these equations, you can see
that at higher temperatures the molecules bump around more.
A gas exerts a pressure on any container in which you put it.
The pressure is a force per unit area applied in, equally in all directions,
and the pressure times the volume for an ideal guess is given by that same
Boltzmann constant times the temperature times the number of atoms.
In this guess, most guesses are not ideal.
there are all kinds of subtleties here. But, the essence of what we see here is
that temperature is a measure of the random motion of the atoms and that the
pressure and the volume increase with he, rising temperature.
if you take a gas and you cool it at sufficiently low temperatures and if the
pressure if sufficiently high, we'll see that later.
you form a liquid phase which is similar to a gas in that it doesn't have a fixed
shape, the items are moving around, but they are weakly bound. And as a result,
you form a phase called the liquid which is almost incompressible, which means
under reasonable pressures it maintains a constant density.
The volume of a chunk of a given amount of liquid is pretty much fixed.
in both cases, in both phases, the density decreases with temperature.
What that is especially useful to us for? Is it tells us that if you have a
collection of fluid under gravity, then the worm fluid, which is expanded and is
therefore less dense, will rise because it floats above the
cooler liquid. And this brings us to the point that an
equilibrium with gravity the pressure in a collection of fluid
will not be constant and the same at all points on the container.
This is the case in the absence of external forces.
In the presence of an external force like gravity, pressure will, in fact, increase
with depth at a rate proportional to the density of the fluid.
And let's see a demonstration of this, and maybe it'll help us understand what's
going on. In this image,
what we see is very simply, me holding a slinky,
a spring. And, what you see when you look at it is
the following, not very surprising fact, which is that near the top of the spring,
the spring is more stretched than it is near the bottom.
And the way you understand that is quite simple.
A slinky stretches more the more you pull it, and the top half of the slinky is
holding up its own weight, as well as the entire half weight of the bottom half of
the slinky. Whereas, the coils of the slinky very
near the bottom are not holding up anything, and so they're relatively
relaxed and unstretched. The higher up the slinky you go, the more
of the weight the slinky is supporting. And therefore, if you look at it
carefully, you'll see that the degree to which the slinky is stretched decreases
uniformly from top to bottom. What wee see here is me holding a cup of
water. this is to demonstrate the decreasing
pressure with height or increase in pressure with depth.
Now, this is true for the air. We know that at high altitude, air pressure
decreases. But remember, the decrease in pressure is due to the extra weight of
the air, and since air is not very dense. We're going to assume that air pressure
is the same everywhere in this room. Not so for the cup because the density of
water is much larger. The water at the bottom of the cup, just
like the spring at the top of the slinky, is holding up all of the water above it,
whereas the water at the top of the cup is not holding up anything.
And so, I expect the pressure to increase from the top of the cup towards the
bottom respectful of the density of the water in the cup and there's a way to see
this. The way you see this is, we drilled holes
at the edge of the cup. And when I open those holes, if the pressure of the fluid
is identical to the pressure at the top, which is the pressure of the air around
it, then the fluid will happily stay in the cup held in by air pressure.
But, of course, it won't. When I remove my fingers, the fluid will
come splashing out impelled by the excess pressure at the bottom of the cup,
relative to the air pressure which is similar to the pressure at the top of the
cup. And in case you're not certain that this
is due to the weight of the water and to gravity, I will drop the cup. At which
point, fluid becomes weightless, the pressure at the top of the cup and at the
bottom of the cup are now the same, and indeed, the fountain ceases.
So, I hope I've convinced you that, in a fluid and equilibrium under the influence
of gravity, pressure increases with depth depending on the density of the fluid.
This is going to be important in astronomical context in understanding the
structure of things like planets and stars, which contain fluid and are
certainly held together by gravity. we talked about gasses and liquids.
matter comes in a solid state. In a solid, the positions of atoms are
roughly fixed, they can oscillate about those positions.
This allows a solid to maintain its shape under external force.
You can apply pressures and stresses and a solid will react, but only a little
bit. It will not, in fact, change its shape to
suit a container. the slinky we had was a very good example
of a solid object. It deformed in response to gravity, but
it didn't completely stretch out. It gave some small proportionate response
to the stresses that were placed upon it. And the larger those stresses the larger
the response, but the response was limited.
The slinky retained fundamentally its shape even under duress.
In all of these phases, fundamentally the interaction is between each atom or
molecule and those near to it. There is no long distance interaction
or there's a weak interaction between atoms in one side, on one side of the
slinky. And on the other, each side of the bit of
a slinky knows about the bits around it. And the result is that if you, for
example, compress the slinky at what one point or move it at one point, then the
perturbation travels through the material as each piece communicates the
information, if you wish, to the next bit over. And this mechanical deformations of
a solid, of liquid or a gas are called sound wave, and they travel with a speed
characteristic of the material. Now, the most familiar to us, of course,
are the longitudinal sound waves in the air, that is what we hear with our ears.
And let's use that slinky. There's a very nice demonstration of wave
propagation. To do that, let's see what happens when I
let go of the top of the slinky. Remember, the slinky was extended in such
a way that it was in equilibrium. The net force on every bit of it vanished
and this had to do with the fact that it was more extended at the top and less at
the bottom so that this compensated for the pull of gravity on every individual
piece. Now, when I release the top, the top of
the slinky starts falling down. But near the bottom, everything is still
in equilibrium. The bottom of the slinky is not falling.
What we see is a density wave propagating at the appropriate speed, determined by
the slinky, through the slinky. And it is only when this wave reaches the
bottom of the slinky that the bottom begins actually to fall.
Until that time, it is in equilibrium, and the information that something has
happened at the top has not reached it. What we're seeing is a sound wave. You
bang on something at one point and it takes time for that deformation to
propagate through the material and reach the other end.
In this simulation, we are going to be able to make some
waves. So, if I grab the end of this rope, the
dif, disturbance travels at the characteristic speed of sound down the
rope. And frequently we deal not with
random pertivations but with periodic oscillations,, usually in a sign wave
pattern as is done here. Each point on the rope is oscillating
periodically and they're all oscillating with the same frequency, which is the
frequency that this motor is oscillating with.
In this case, it's oscillating with a frequency of 50 something.
And so, frequency F, is the number of oscillations per second.
And so it's measured, since this is a number, in units of inverse second and
once per second is called a hertz. Which is why the frequency of your radio
signal is measured in megahertz if you're listening to FM radio.
And so, I can restart this oscillation, and what we see is that because it takes
longer amount of time for the wave to get farther down the string, this,
the, the position of the string here reflects what happened here a longer time
ago than the position of the string here. And so, the periodic oscillation of the
driving force causes each point on the string to go up and down.
Notice, the wave is moving. The string is not going anywhere.
But, also what it causes is because distance translates to time.
we get a periodic sinusoidal structure in space at any given instant, and that is
characterized typically, by the distance between subsequent peaks.
The distance along which the wave repeats, that's it's wavelength lambda.
And if you think about it, sitting at some point for a second F peaks will come
by. And so,
F peaks will come by, that means the wave will have transferred
a distance of F times the distance between peaks.
In other words, F times lambda is how far the disturbance travels in a second.
So, F times lambda is the speed of the wave which we characteristically call C.
Now, the reason we're dealing with periodic oscillations is a complicated
mathematical theory called free analysis that says, any deformation can be written
as a sum of periodic ones. But, Fourier analysis is
not some abstract mathematics, it's exactly what our ear does.
When your ear hears sounds, it analyzes them into a sum of harmonics and you hear
the pitch of a sound, you hear is given precisely by its frequency, and you can
hear three different notes and distinguish that there are three notes
there. this is biology and not physics, but it
is definitely true. And I can increase the frequency.
And what we see is that when I increase the frequency, the wavelength became
correspondingly smaller, the frequency times the wavelength is a constant the
speed of the wave. a wave is characterized therefore by it's
frequency or it's wavelength, given the speed, those are related, but also by an
amplitude. An amplitude is simply the magnitude of
the largest deformation. So, an amplitude of seventeen is a very small deformation
and an amplitude three times as large, is a very large deformation.
So, periodic disturbances, as we saw, are characterized by their frequency, F, in
hertz the number of cycles per second. if a wave is traveling at a speed C, a
periodic disturbance will produce a periodic wave in space at any given
instant with a wavelength lambda that is related to the speed by this important
relation. Lambda wavelength times frequency is
equal to the speed of the wave. The wave was characterized also by the
amplitude, which is the amount, the maximum amount of the pertubation.
And one of the important things about waves is, the string didn't move.
But clearly, waves carry energy from one place to the other.
You could imagine hooking something at the other end of the string.
I will vibrate this end, the wave will carry it and cause that object to vibrate
giving it kinetic energy. That kinetic energy came from my hand,
and therefore, a wave carries energy. in fact, it carries an energy if it's
periodic per unit time that is constant. Over time, energy is flowing down the
string. and so, you measure an energy flux in
joule per second. A joule per second is known as a watt.
This is the same unit of energy per unit time that you measure the intensity of
your light bulbs in. How much energy does it take to operate
that light bulb for a second. Now, this is a wave on a string.
of course, a wave, like a sound wave that travels in three
dimensions will carry an energy flux that is a density.
Because if the wave is propagating over a wide front each bit of the wave carries
an energy, so we measure the flux in watts per
meter-squared of wavefront. And that energy flux, in watts per
meter-squared, is proportional to the square of the amplitude.
Clearly, the larger the amplitude, the more energy the wave is carrying.
And the flux is going to be measured in watts per meter-squared.
Now, what happens when two waves meet is a very important property of waves.
Let's look at it in a demo and then we'll talk about it.
In this demo, we have a loudspeaker. The loudspeaker will be creating a
periodic wave. And we see the wave fronts are shown in
this picture. They spread away from the loudspeaker.
And what we see here is a measure of the pressure of the density that the
perturbation that the wave represents at one given point.
And we see the characteristic, periodic isolation that we're talking about.
And, one thing you can see is that, unlike waves on a string, we'll talk
about this. But, because these waves are spreading out, the amplitude of
oscillation is larger near the source, then it is far, And anybody who stood
very near a honking car horn is painfully aware of this fact.
we'll talk about that aspect of things in a little bit.
I just thought I'd just point it out while we're seeing it.
But, the interesting thing to note is what happens when, instead of one loud
speaker, we put two speakers. So now, the two speakers are both
creating wavefronts. And we see here what happens as the
wavefronts add, and you notice that I put my detect in a
very interesting point. the two speakers together,
each of which would've produced the noise with the amplitude we saw before,
produces essentially zero noise at the position of this detector.
But if I move the detector a little bit, I will see that it's is not because the
two speakers are not producing noise at all,
I see that indeed at this position, they're producing a far louder noise
together than either one separately. And in fact,
more than twice the sound that each would have produced separately, somehow the
combination of the two speakers is focusing sound into these regions where
the dark bands are and leaving these regions where the grey bands are, the
nodes, where essentially no sound gets to.
Now, the reason that this is possible is because waves can subtract from each
other. When these two waves propagate, it turns they essentially don't interact
with each other at all, that's the approximation that we're working in and
it's valid one for sound waves. And what that means is that the total
dis-perturbation at any point in space and time, is the sum of what it would be
if there was one speaker, and what it would be if there was the other speaker.
And, how does that, adding what one speaker
would produce to what the other speaker would produce, lead to zero?
Well, remember they're both oscillating at exactly the same frequency. So, there
can be a point where because of the time delay from this speaker being different
from the time delay from that speaker, due to the difference in distance.
when a peak from this speaker reaches, it is always timed to arrive at the same
time as a trough from this speaker, so that the two waves that these two create
are always counteracting each other. Waves add, but that addition can amount
to subtraction. And that phenomenon is known as
interference and its characteristic of waves.
So, let's summarize what we've learned here.
A hallmark of wave behavior is that waves are deformations or perturbations to some
equilibrium, and when two waves meet at the same point in space and time, the two
disturbances add. In other words, each wave behaves as
though the other were not there. We just add the disturbances.
What this allows, if the waves are of the same frequency, is that if the time delay
between one source and the other source relative
to your position, is suitably adjusted. In other words the distance difference,
which translates to the time delay, is suitably adjusted.
You can adjust it so peak from one wave precisely meets a trough from the other.
The two waves add in the sense that they subtract, and you can get nodes, you can
get regions of quiet in the vicinity of two speakers producing the same
frequency. this will be important to us.
It's a hallmark of wave behavior. Now, another property of waves that is
going to be useful to us is extremely useful in fact, is something called the
Doper effect. It's the well known phenomenon that the
sound from a moving source sounds different then the sound from a
stationary source. If a source is the noise is approaching
us, it sounds higher in pitch than where it's
stationary. If it's receiving, if it's moving away from us the pitch sounds
lower. the archetypal example of this is the
sound of a race car driving past you. I'll play it in the speakers, perhaps you
can hear it. [SOUND] That [SOUND] whine is the
transition from an elevated pitch as the car approaches you to a decreased pitch,
decreased frequency as the car recedes from you.
And the mathematical formulation of this is due to Doppler in the 19th Century,
and we can understand this rather straight forwardly.
Here is a source producing regular, regularly spaced periodic waves, and an
observer placed over here receives the waves with a time delay.
It takes time for each wave to reach there.
The time delay is the same for each wave so the time between subsequent floods is
the same as the time between subsequent emissions.
And this is a stationary source. The frequency with which his ear is
oscillating is the same as the frequency with which this source is producing the
sound. That is true.
Now, let's move on to the case where the source is moving and let's again place an
observer over here. And we see that what's going on is that
the wavefronts appear compressed on this side, they're compressed, of course,
because between the emission of one wave and the next the source moves.
And so, the distance between the wavefronts, as seen by the observer here,
is smaller than it would be were the source stationary.
And we can easily compute the relation between the two because the distance
between subsequent wavefronts as they arrive at this observer,
lambda, which is the wavelength that this observer sees, is equal to the distance
between the two wave fronts when they were omitted minus the distance which the
source traveled. Because each subsequent wave front has a
shorter distance to travel because in the interim between emitting one wave front
and the following, the source traveled distance given by v times the time
between wave fronts. What's the time between wave fronts?
The time between emissions is one divided by the frequency because this is the
number of seconds. And so, remembering that lambda times f
is c, I can write f as c over lambda. And I get lambda is lambda zero minus,
let me write this correctly. lambda is lambda zero minus v lambda zero
over c, or Doppler's formula, lambda is lambda zero times one minus v over c.
notice that this v was taken to be positive when the source was approaching
the observer and observer on this side, of course, sees the wave dilated.
The wavelength looks longer, the pitch is lower as the car is receding from you.
Each subsequent wave front has a longer distance to go.
That is compensated in this formula by considering v to be negative if the
source is receding, and positive if the source is approaching you.
We talked about heat energy and we need a few properties of heat because heat will
be a dominant form of energy in some environments that we talk about.
And so, it's a property of heat that if you have a hot object in a cooler
environment, then the hot object will lose energy the
environment and this process will continue indefinitely or until such time
as the temperatures equilibrate. Because as the source looses heat, the
environment gains heat and warms up. And eventually, if the temperatures are
equal, then we have equilibrium, and heat flow stops.
But so long as you have warm sun in cold space, the sun will continue to lose heat
to space. And until such time as either space heats
up or sun cools down, this process will continue.
There will be a constant flux of energy out of the sun into space.
And, there are several ways by which this transfer of heat from a hotter to a
cooler body can occur. One is called conduction, this is where
heat is transferred through continuous contact.
And this is not an in, great importance.
In astronomy, heat conduction on astronomical scales is rare.
A second phenomenon is convection. This is basically like trucking.
This is what happens when the physical motion of the fluid carries energy.
this works very well if you have volume of fluid that is heated from
below. That happens when you boil water on your
stove, it also happens in stars, say, where a
hot core is heating the atmosphere from the inside.
What happens is that at the bottom of the pot or the bottom of the atmosphere the
fluid is heated. The heated fluid expands.
The expanded heated fluid is lighter, less dense then the surrounding fluid.
And therefore, floats to the top and rises.
Cooler fluid descends to take it's place. And what you have is a energy transfer
from the bottom to the top by heated fluid physically moving from one part to
the other. And this does occur in astronomical
phenomenon. None of this, of course, is helpful to
the sun in losing its energy to space because the sun is not in contact with
anything nor is there any fluid around it to convect.
The way the sun loses energy to space is radiation.
It turns out that every object, unless its at zero temperature, glows and that
phenom, that mechanism of energy loss is radiation heat transfer, and essentially
the radiation question is sunlight. Sunlight is the way the sun attempts to
heat up space, losing energy because it's warmer.
And we'll come back to this calculation in a moment.
But, let's understand something about this
heat flux, energy flux problem.
And so, the sun is hot and so it radiates.
And we'll learn how to compute how hot it is and how much it radiates.
But for now, let's assume that the sun is radiating energy at a constant rate.
Why is the rate constant? Because the properties of the sun are
unchanging, the properties of space are unchanging. So, the rate at which the sun
loses energy to space is therefore unchanging.
We call that rate the luminosity, and it's measured in joules per second, in
watts. The sun like a light bulb produces a
certain amount of energy or loses a certain amount of energy to space every
second and we'll figure out this number, obviously.
It's a great many light bulbs.
Now, what interests us often is how bright the sunlight is at some given
location in space. And what that implies is put some uniform
detector, say, an eye with a fixed surface area.
And as we saw or as this diagram shows, if this is something the size of your eye
or of your detector, the farther it goes, the less sunlight it traps.
Because essentially, the sun makes produces radiation in all directions.
And so, at any distance, the sunlight is spread uniformly across
the surface of a ball [SOUND] or a sphere of radius R.
And, if you are at some distance and you place your detector, your detector takes
up some fraction of that sphere. If you are twice as far, the sphere is
larger and your detector now takes a smaller fraction of this larger sphere.
And the result will be that your detector, since all of the suns rays pass
through this sphere and all of the sun rays a little later pass through this
sphere as well, the net total amount of energy passing
through each sphere is the same. But the total amount passing through a
fixed detector decreases because the spheres grow larger whereas the detector
does not. Mathematically, the statement is that at
a distance R, the radiation is distributed uniformly on the surface of a
sphere. So, energy per second received by a
square-meter of detector will be the luminosity divided by the surface of a
sphere across which the radiation is spread uniformly.
And so, the sun energy is diluted by a factor of R
squared as it travels out into space. Light, of course, is going to be of
incredible importance in our understanding of the universe.
It's the main way we perceive our environment on Earth.
It is certainly the main way that we receive information from distant places
in the universe, so the properties of light will be extremely important to us.
So, light light, of course, is the main way we as
humans perceive the world around us. It's also the main way that we as
astronomers learn about distant objects. The nature and the properties of light
are going to be extremely important to us.
And the first of these is the speed. So, light carries energy away from the
sun to Earth to warm us. and the speed at which light propagates
is huge. It's about three times ten to the eighth
meters per second. this may, this large number makes
accurate measurement difficult. First, really good measurement was in
1850 by Foucault and Fizeau, though there were earlier less precise measurements.
Now, in a classic experiment in 1670, our old
friend Newton made the discovery that the sunlight, white colorless light from the
sun, in fact contains all colors of light.
he found a way to split a beam of white light into beams of various colors.
He was using a prism, today we use a defraction grading.
And he conjectured that the nature of light was that light was a stream of
particles, of course, he had all the mechanics know how to understand the
stream of particles, light moved in straight lines, light particles which
upon no forces act. Moved at a constant velocity, it seemed a
lot like mechanical particle model of light would be likely.
This is one of the rare cases where Newton was wrong because in 1799, only
120 years later, Young observes light waves interfering.
So, he makes the observation that you can take
light and you can construct these interference effects where light waves
add and subtract, and you get nodes. This interference was a hallmark of wave
phenomena, so Newton, for once, was actually wrong.
Light is a wave. But never count Newton out.
We'll come back to this point in the next clip.
And that same Young, three years later, shows that
our eyes while they are extremely accurate in giving directional
perspective. So, they tell us very precisely which
direction light is coming from, that's how we construct our image of the world,
are in contrast relatively poor as fully analyzers.
Unlike our ears, which can distinguish a great variety of sounds and hear several
simultaneous tones and distinguish them, our eyes are very poor for an analysis
machine. They only are sensitive to the relative intensity of three colors, red,
green and blue, and we'll see in a minute what that leads to.
What we're going to do is an experiment somewhat similar to what Newton did.
So, we have here a beam of white light, and we're going to put a diffraction
grating in front of it, and the diffraction grating splits the white beam
so that the white stripe in the center is somewhat dimmed.
And on both sides, we find the familiar rainbow structure. And so, we see a
stripe of red flaring to orange and yellow and green, and into blue and
violet. And,
we can see what happens when we put a filter in front of the light,
we get red light. And when we feed red light into the
diffraction grating, what we see in the spectrum is that the green and the blue
are missing. Similarly, when we feed green light into
the defraction grating, the blue and the red are missing, and only green light
comes in. So, the defraction grating is not
coloring the light. It is deflecting whatever green light it
gets to that region. Again, putting a blue filter in front of
the beam, we see that only the blue remains in the spectrum. The blue filter
absorbs the green and the red, the green filter absorbs blue and red, and the red
filter absorbs green and blue. these are, it turns out, the primary
colors it has to do with our eyes. here's what happens when you put a cyan
filter in front. A cyan filter absorbs the red light, and
so lets through the blue and the green. Similarly, if we use a magenta filter.
A magenta filter absorbs the green, letting through the blue and the red.
So, what we see as magenta is blue light and red light.
There is also, we can put a yellow filter in, and this
is important. We see that what we perceive as yellow
light is, in fact, light that is missing its blue.
The green and the red are present, the blue is absent,
and we see yellow. And so, if we superpose the yellow and
the magenta filters, the yellow filter absorbs the blue.
The magenta absorbs the green, and what is left is red.
And indeed, superimposing those two filters, we get
red. If we superimpose yellow and cyan, the
yellow absorbs the blue, the cyan the red, what's left is the green.
And we see that very nicely in this picture, and we can keep playing.
We can superimpose the cyan and the magenta, the cyan absorbs the red, the
magenta absorbs the green. Putting them on top of each other, we're
going to be left with blue. So,
this has to do with the way our eyes see colors.
the important thing for us to take away here is that a filter that absorbs the
blue looks yellow. We cannot distinguish this yellow from
the yellow in the spectrum there in between the red and the green that is
pure yellow light. We saw that as Young showed us, light is
a wave, and also as Young showed us, our eye, in fact, has three color detectors.
One sensitive to red light, one sensitive to green light, and one sensitive to
blue. And what we perceive as color is merely
the relative intensity at which those three detectors are excited.
So, yellow light that excites the red and the green approximately equally.
Because in the spectrum, the other light is between red and green cannot be
distinguished from a combination of actual red and green light.
And so, our eye, unlike our ear, is not a very good free analyzer.
so much for biology, but this will be important in understanding some phenomena
later. And then, once we know that light is a
wave, it's not too shocking that color is a property of the wave.
Color turns out to be the frequency of light
visible light has a frequency of order ten to the 12th hertz
even at the rather high speed of light, we can compute the wavelength is pretty
short. It's between 400 and 700 nanometers.
A nanometer is a billionth of a meter, ten to the minus nine meter, and these
are the units in which, typically we measure the wavelengths of visible light,
at least in astronomy. And so, 400 nanometers, the shortest
wavelengths visible light is the blue light as this indicates.
And the red end of the spectrum is the longer wavelengths.
And the order of the colors in the rainbow is according to their
wavelengths. So, that violet light in fact, is at the
short end of the spectrum, green is somewhere in the middle around 550
nanometers is green light and so on. Now, for something completely different.
we talked about gravitation as the force that dominates astronomy.
But, in most of our lives, gravitation of the Earth is important.
But, when we interact which each other or with objects around us, gravitation is
completely unimportant. The gravitational force of a human being
is negligible compared to the other forces we interact with.
The force the tunes out turns out to dominate most of physics is
electromagnetism. And so, we need to understand some things about this.
And, the simplest version of electromagnetism is the force between two
static charged objects. So, there is in the universe a thing
called electric charge. some objects carry electric charge.
You can charge your hair by combing it on a dry day, and the knowing force that
makes your hair stand on end on a dry day is this electrostatic or Coulomb force
which has a formula that will appear familiar.
The force between two objects of charges, q1 and q2, is given by a constant times
the product of the charges, divided by the square of their distance.
This might appear familiar, it's that square of the distance again.
It's not exactly a coincidence. The big difference between electric
charge and mass, or between a Coulomb's law and Newton's law is that Newton's
law, remember had FGMM = GMm over R squared.
And these numbers were positive, and the force was attractive.
Always. Gravitation is a universally attractive
force. Coulomb interactions can be attractive or
repulsive. It turns out that the force is attractive
when the two charges are of opposite sign, so positive and negative charges
attract, and charges of like sign repel with an equal magnitude.
But the sign, the direction of the force, depends on whether the charges are equal
or opposite. This in turn explains why this force does
not dominate the universe, and this force does.
It's not because gravitation is stronger. In fact, we have an unvarying measure of
strength in which gravitation will be a far weaker force than electromagnetism.
But, because opposite charges attract, most
objects are neutral. In the presence of charge, if you have
something that's charged positively, it will attract negative charge until such
time as it reaches equilibrium. In other words, as neutral.
This is not possible for mass. There's no concept of negative mass.
In fact, it's worse than that. We'll discuss the instability of gravity.
This will have profound consequences so I'm not belaboring an obvious point.
And charge, like mass, momentum, energy, etc., is conserved.
You can neither create nor destroy charge.
Though, of course, charge can be transferred from one object to the other.
So, when you comb your hair, the comb is charged with one charge, your hair with
the opposite charge. when atoms form an ionic bond, they
exchange charges and so on. But, the total charge in the universe or
in any enclosed region in the universe is conserved.
And then, you see,
in gravitation, I kept wanting to talk not about the gravitational force,
but the gravitational acceleration. So, you talk about the Earth for example,
generating around it a gravitational field.
The gravitational field is the statement that were a mass to be placed there, it
would experience a force. And, you can compute the force per unit
mass without knowing what the size is of the mass that you would introduce.
And in a similar way, you talk about an electric charge producing an electric
field which is the force per unit charge that would act were a charge to be placed
there. So, a charge create an electric field
around it because another charge were it to be brought there, would be effected.
And a charge is effected by electric field.
A similar phenomenon of a field region around an object in which other
objects might be impacted is encountered with magnets.
A magnet creates around it, something that we can call a magnetic field.
In the presence of such a field, any other magnet, like this little compass,
will align itself in a particular direction at any given point in space.
Of course, it will align itself so that its north pole faces the south pole and
so on. And so, [SOUND]
We can imagine that this region of space has some property called a magnetic field
which is the statement that, were you, remember, there was a field here even
before I brought the compass, that told you that were you to bring the compass,
it would point in such and such a direction.
Now, magnets and electricity initially appear completely unrelated.
And it is known, however, that electric currents in fact produce magnetic fields
in the vicinity of an electric current. Electric current flowing around the coil
behaves very much like a magnet. We call this an electromagnet.
And, we see that we have a magnetic field in the vicinity of an electric current
moving charges create magnetic fields. So this is the first relation between
electricity and magnetism, but there's another relation which is that you have a
changing magnetic field, it will cause charges to move so that this magnet has
absolutely no impact on the charges in this coil.
But if I move the magnet so that the magnetic field in the coil changes, then
while the change is on going, current will flow and current flows means
there's a force on these charge carriers. This implies an electric field.
A changing magnetic field causes an electric field and I can, of course,
cause the change by moving either the magnet or the coil.
And so, changing magnetic fields essentially can be thought of as
generating electric fields. This has profound consequences.
Let's see what they are. The fact that moving charges can create a
magnetic field was discovered in 1820 by Orsted.
And, it turns out that moving charges are also affected by magnetic fields.
We saw that when we moved the coil in the presence of a magnetic field,
Faraday realizes the phenomenon that we were discussing, which is that a changing
magnetic field is tantamount to an electric field.
30 years later, Maxwell writes down what are the collected set of equations
describing electric and magnetic fields. And, in particular, the phenomenon he
needs to add is that a changing electric field creates a magnetic field.
In the presence of a changing electric field, there's a magnetic field.
And the combination of these two phenomena leads in Maxwell's equations as
part of the solutions to propagating waves.
Propagating waves in which essentially an elec, you set up a changing electric
field which creates a magnetic field which creates an electric field which
creates a magnetic field. You solve the differential equation that
this leads to, and you find that it describes a propagating wave.
And Maxwell computes from the properties of magnets and currents and Coulomb's
law, properties that have been measured in the lab, he can compute the velocity
of these waves that he's discovered. And their velocity surprise, surprise, is
precisely c. It co, he didn't get it this precisely,
but it coincides with the speed of light that Fizeau and Foucault had measured
eleven years previously. So, Maxwell has very good reasons to
imagine that he has discovered that light is a wave, as Young said.
In fact, we now know what it is a wave of,
it's an electro-magnetic wave. The disturbance that propagates in space
when a light beam travels through it is an electro-magnetic wave, is an elec,
disturbance in the electric and magnetic fields which have come to take on sort of
a life of their own. Remember, there were ways to calculate
what would happen if you put a charge. And suddenly, they create each other an
they propagate through space far away from the charges that might have created
them in the dim distant past, they have essentially their own existence.
And so, we do talk about these fields as important objects, the understanding that
light is an electromagnetic wave also tells you that you can in principle
imagine oscillating a charge or creating an oscillating electric field at any
weave, wavelength or any frequency. So, we know that we see light from between
400 and 700 nanometers in wavelength. What, what about the solutions to
Maxwell's equation with wavelengths outside this region?
And it turns out that there is an entire huge electromagnetic spectrum out there
ranging from at very high frequency and very short wavelength gamma rays with
wavelengths of down to ten to the fourteen, ten to the fifteen meters and
below. Except though, below that are very hard
to produce, through the x-ray spectrum.
The drugs from wavelengths of about ten to the minus nine to ten to the minus
twelve meters, and then the ultraviolet light which is the
chunk of the spectrum, immediately beyond the deep violet light at 400 nanometers.
that is the shortest wavelength our eyes can see.
And then, on the other side, to the left of the red light we have infrared light
and below that, microwaves and radio waves whose wavelengths can be hundreds
of meters. and so there's this entire spectrum.
And in fact, we're familiar with it, we use it.
our eyes are sensitive only to this tiny little bit of the electromagnetic
spectrum. Our eyes have all kinds of
inefficiencies. But, in this case, our eyes are well
adapted to where we live. if you look down here at the bottom of
this plot, this is the transparency of the atmosphere.
And you see that, for example, gamma rays and x-rays are not very useful despite
Superman claim to the contrary, x-ray vision is not very useful because x-rays
are very quickly absorbed in the atmosphere, they do not penetrate.
Visible light penetrates the atmosphere. A little window in the infra red
penetrates the atmosphere. And then, radio waves penetrate the
atmosphere, well we know that, because that's how we watch TV.
But the wavelengths of radio waves are too long you would need your eyes to be
antenna. They would need to be on the order of
meters in size. We do not have eyes meters in size.
There would also be other limitations. It would be hard to see features of the
universe smaller than a few meters across.
So, given our need for fine resolution, our eyes are well adapted to the
conditions under which we evolve. But the universe, on the other hand,
produces light in, in the entire, across the entire spectrum
from high energy gamma rays to radio waves.
And it's important in order to collect information about the universe to be able
to observe it in all possible frequency bands.
And indeed, with technology, people have developed ways to observe the universe in
all of these frequencies. And at every time that we've developed a
new technology and a new way to look at the universe, there have been new and
often surprising discoveries made. note that observations in many of these
bands, in the ultraviolet and x-rays and in gamma rays for example, need to be
made outside the atmosphere. Putting a very good gamma ray detector on
the ground will not buy you anything, because gamma rays from space will not
penetrate. So, these observatories including,
actually, infrared observatories, tend to be space-born or at least high altitude
balloon flights. radio telescopes can be placed on, are
placed on the ground. Those are these very large dishes that
receive radio waves. And over the course of the class, we will
have occasion to use information collected by all of these bands.
So, it's the richness of the spectrum corresponds to the richness of phenomenon
out there. We're almost done with our discussion.
One last statement now that we have our understanding of the spectrum.
A hot object radiates, that's what we disussed.
Everything radiates as long as it's warmer than the
surroundings for a dense, dark object. So, for an object that essentially
absorbs light that hits it, such as you, me, the Earth, and it turns out, the sun,
the radiation is almost completely characterized, and it's precisely
completely characterized for some idealized black body, an object that
absorbs any light that falls upon it, by the temperature.
It turns out that the nature of the radiation produced by a warm object,
if it's dense, it's essentially a property of light and thermodynamics and
has nothing to do with the object itself. And this is called black body radiation
and it has two important properties. One is, that hotter objects are blue.
This despite the fact that blue is a cold color and red a warm color.
To our intuition it turns out that hotter objects radiate shorter wavelength.
In other words, a blue glow is hotter then a red glow because a red glow has a
longer wavelength. This is summarized in Wien's displacement
law which says that the wavelength that which an object emits more radiation than
any other wavelength, the wavelength of maximum emission, multiplied by the
object's temperature, is actually a constant. So that if you double the
temperature of an object, the wavelength at which it emits is halved.
And Wien's constant is written over here. It's got units of meters times Kelvin, so
wavelength times degree. The other and more familiar statement is
that hotter objects produce more light. This is encoded in the Stefan-Boltzmann
relation which tells us that the rate at which H meters squared of an object
radiates. Of course, a big object radiates more
than a little object if they're the same temperature.
Two light bulbs make more light than one light bulb. So, we need to normalize to
the rate at which an object produces radiation per meter squared,
that's F in watts per meter squared. And the rate at which each meter squared
of an object radiates is proportional to the fourth power of the temperature in
degrees Kelvin. And so doubling the temperature leads to
a sixteen fold increase in radiation and the constant, the Stefanu-Boltzmann
constant, sigma has this numerical value in our units.
And its units are watts per meter squared per degree Kelvin to the fourth.
Let's take a look at what this black body radiation looks like.
Here, we have a plot of the relative intensity in various wavelengths as a
function of the wavelength from near zero to a few times ten to the minus six
micrometers. And so, the visible band is located over
here. And the
black body curve has this characteristic shape.
As I said, it's independent of the material. And the curve I have plotted
here is the curve for an object that glows that has a temperature of about
3,000 degrees Kelvin. 3,000 degrees Kelvin is about the
temperature at, of the filament in an incandescent light bulb.
And what you will notice is that most of the light produced by an incandescent
light bulb's black body radiation, is in fact not light.
It is invisible to us. It is infrared radiation, which heats us,
if you underst, if you talk about people not liking incandescent lamps as being
very inefficient, this is because due to the limitations of the filament.
Filaments melt if you heat them too hot. most of the light produced by an
incandescent lamp is in fact not light, it is just heat.
And so, on the other hand, if we heat the object a little bit.
This is an object with a temperature of say, 6,000 degrees.
This is closer to the sun. And you see that much of the light
produced by the sun is, in fact, in the visible spectrum.
It's very nice of the sun to be adjusted to our atmosphere so that much of the sun
light can, in fact, penetrate our atmosphere, and we can use it to see.
And you also see that the factor of two in temperature leads to this huge
increase. The previous light bulbs graph has been
scaled down here, it looks very small. remember, doubling the temperature
increases the total radiation by a factor of sixteen.
And, it has also moved the maximum from, according to Wien's law, from the
infrared in this case, down into the visible.
And we can add another object say, with an even higher temperature.
let's say, 12,000 degrees. So, another doubling in this case,
the spectrum again increases to the point where the sun's spectrum is negligible
and the peak is now in the ultraviolet. So, we're seeing Wien's displacement to
the left, hotter temperatures mean bluer light.
We're also seeing the Stefan-Boltzmann law in action,
hotter temperatures mean way, way, way more light.
These are all important observations. Let's see what we can do with them.
[SOUND] That was a long, video clip. There was a lot going on there.
We got to go through two and a one-half centuries of physics.
Remember, what we're doing is we're packing our toolkit so that we know what
it is we need to address understanding astronomy.
So, don't despair. rather than trying to summarize all the
disparate things we talked about, let me do an example of an application to actual
astronomy of what we learned about that might clarify some things.
So, lets talk about the sun. We said that the sun radiates its heat.
Let's see what all our various results tell us about solar radiation.
So, we can measure the brightness of sunlight on Earth in watts per meter
squared. How do you do that?
Not put our a meters squared of detector and measure how much sunlight hits it.
Put out a meter squared of something very dark that absorbs all the sunlight,
measure how much energy that thing receives.
And you'll find, the number is called the solar constant, and it is 1,361 watts per
meter squared. So, a meter squared of sunlight will run
a very small hair dryer, or light ten incandescent light bulbs.
So, we know the intensity of solar radiation, in the vicinity of Earth.
Now, we know how far the sun is. The sun is one astronomical unit or 150
million kilometers away. And so, if we know how bright it looks
and we know how far it is, we know that we can compute its
luminosity. Remember, we had that B, the brightness,
is the luminosity divided by the area of the big ball whose radius in this case,
is the distance to the sun or one astronomical unit.
And so, we can com, solve this we know the brightness, we measured it, we know
the distance. We can write an expression for the
luminosity and it's four pi times the distance to the sun squared times the
brightness of the sun. And so, we put in the numbers.
This is 4 pi times 1.5. I'm rounding.
Times ten to the eleven meter squared, times 1361.
And we evaluate the calculation and you find that the sun is a very bright light
bulb, indeed. It has a luminosity of 3.8 times ten to
the 26 watts. That's a lot of energy lost by the sun.
Now, we can measure the radius of the sun
since we know the distance. we can make a small angle calculation.
We measure the angular size of the sun in the sky, and we find that the radius of
the sun is 6.96 times ten to the eighth meter,
that's about 700,000 kilometers. And from this,
we can figure out something else. Because what we know now is that, think
of the sun as a ball. Every square meter of the ball is
radiating light to the outside, you count how many square meters there are on the
surface of a ball. That would be 4 pi times the radius of
the sun squared. If I multiply that by the flux, the
amount of power in watts emitted by each square meter, this should equal L. Since
I know the radius of the sun and the luminosity, the total amount of energy
that's being emitted, I can figure out the flux that each
square meter emits. Now, the lumina, the luminosity, as I
said, is the area of the sun times the flux.
And so, I could compute the flux. the nicest way to do this is to use a
scaling relation, as usual. So, the scaling relation is going to be
that I have that F, which is L over 4 pi times
the radius of the sun squared. and I have that b is L over four pi times the
distance to the sun squared. I can cancel everything around.
Find that L is b times the distance to the sun divided by the radius of the sun
squared. And this relates again, a flux to a flux
through a dimensionless quantity. I recommend this method of computing and
this is 150 million, this is 700,000. That's a big factor.
Every square meter of the sun produces 6.3 times ten to the seven watts, a
square meter of sun produces 63 megawatts of light emitted out towards the
universe. That's a large number, but we can go
further. We have the flux and we know the
Stefan-Boltzmann law, we know that the flux is also equal to sigma, T to the
fourth. Since the sun emits approximately as a
black body, it's dense enough. And so, since we know the flux, we can
compute the temperature, and we can actually find the temperature
of the part of the sun that we see. It's called the photosphere.
It's somewhere outside on the surface of the sun.
And, we find that the temperature is the fourth root of F divided by sigma, the
temperature of the radiating surface of the sun of the photosphere is about 5800
Kelvin. I can use Wien's Law because I know the
temperature to figure out the wavelength of maximum emission for the sun.
The wavelength at which the sun emits more light than anything else.
I take Wien's constant, 0.0029 meters is the value of Wien's constant.
I divide by the temperature of the sun, and I find that the sun produces more
light than any other frequency at a wavelength of 503 nanometers.
And if you look up the table, this is somewhere between the red and the blue,
and indeed it is green. But Ronin, you tell me every
kindergartner knows the sun is yellow. Yes, but not really.
Every kindergartner knows the sun appears yellow.
Why that is consistent with the maximum emission being green, we'll see in the
next clip. In the meantime, go and catch your
breath. We have gone through many, many different
topics here. We have a whole grocery list of new ideas
and new concepts to digest and to learn how to use.
And we'll start applying those very soon.