All of you have looked at rainbows, but very few of you
have ever seen one. Looking at something is very
different from seeing it. And today I will make you see
the rainbow in a way that goes way beyond the beauty that we
can all experience, a way that you will always
remember. And I would like to start
asking you fifteen perhaps simple questions about the
rainbow. The first question then is
would any one of you remember if you see a bow whether the red
color is outside or whether the -- the red color is inside?
And then I wonder about the radius of the bow.
If this is a bow in the sky, something like this,
here is the horizon, it's clearly a perfect circle,
and so the perfect circle has somewhere a center.
And so that means there must be a radius R.
You can measure that radius in terms of how many
degrees and so what is roughly that radius.
You've never measured it but is it ten degrees,
is it twenty, thirty, fifty,
sixty? The length of the bow.
Is there a difference, do you sometimes see a very
long bow, sometimes a very short one?
What is the width of the bow? You see colors here.
How wide is that strip of colors in degrees?
Perhaps some of you have noticed that there is a
difference in light intensity between inside the bow and
outside the bow. Maybe you've never seen it,
and if there is a difference where is it brighter,
inside the bow or outside the bow?
What time of the day would you see bows?
Would you see rainbows in the north, east, south or west?
Is there perhaps a second bow in the sky?
And if there is a second one, where should you look for the
second bow? And if there is a second one
what is the color sequence of the second bow?
Is the red on the outside or is the red on the inside?
And then you can ask the same question, what would be the
radius of the second bow?
And what would be the width of the second bow?
All these first twelve questions in principle you
should have been able to answer if you really have seen a
rainbow. The last three is more
difficult. The question is are the bows
polarized? In what direction are they
polarized? And are they weakly polarized
or are they strongly polarized? Who knows the answer to twelve
questions, to the first twelve questions?
Who knows the answer to more than
ten? Who knows the answer to nine?
Eight? Seven?
Six? Five?
Four? Do I see a hand at four?
Good for you. Five, four, three?
Three, good, that's already good.
Two? One?
And who knows the answer to zero?
Most of you, right?
I haven't seen a lot of hands though.
All right. So I've made my point.
You've looked at rainbows but you've really never seen them.
And I'm going to make you see them today.
What you see here on the blackboard is one drop of water.
I put the sun for simplicity at the horizon.
Later I will put it a little bit higher in the sky.
Light from the sun hits this raindrop.
I've only drawn one narrow beam which hits the raindrop right
there. And you see here the angle of
incidence, which with Snell's law we call theta one.
I call it I here because it's nicer for me,
more descriptive, it means incidence angle.
Right at that point A some of the light will be reflected and
some of the light will go into the water.
We call that refraction. And Snell's law will tell me
this angle R. Whatever goes in there reaches
point B where there is a transition back to air and so
some of that light will come out here and some of that light will
be reflected inside. And then when it reaches point
C again there is a transition from water to air.
Some of that light will be reflected inside the water.
And some of it will come out. And as far as the geometry is
concerned, if this angle is R, then this angle is also R,
this is also R, and this is also R.
And the angle here is I. That follows from Snell's law,
and I'll leave you with that. Notice that the light came in
like this but it comes back like
this. So the direction has changed
over the angle delta. And the angle delta is very
easy to calculate in terms of I and R.
Delta is a hundred eighty degrees plus two I minus four R.
I want you to check that at home.
The four Rs come in here. One, two, three,
four, and the two I's come in here and
there. If now I think of all possible
narrow beams of light that can strike this raindrop,
one that would strike it here would have an I of zero degrees.
And then here would be ten degrees and twenty degrees and
thirty and forty. And the largest value for I is
when the light strikes here, would be ninety degrees.
And so I can calculate for all these values of I,
which obviously all of them occur, sunlight strikes this
raindrop, and all these angles for I are present.
So I can calculate now for all these angles of I what the value
is for R and then I can calculate what delta is.
R follows from Snell's law and delta follows from this
geometric relationship. And what you will find now very
much to your surprise, that there is a minimum value
for delta which is about hundred and thirty-eight degrees.
That means this angle phi here has a maximum value which is
very roughly about forty-two degrees.
And I will show you some numbers.
You can download this, by the way, this is on the Web,
under lecture supplements. Here all I have done I've taken
I to be from zero to ninety degrees, all these angles are
possible, with Snell's law, using an index of refraction of
one point three three six,
that you see at the bottom, I calculate R and then in the
last column using that relationship I calculate delta.
And indeed you see that delta starts at a hundred eighty
degrees when I is zero. And then goes to a minimum of
roughly a hundred thirty-eight, after which it increases again.
And this now is crucial, is key to
an understanding of the rainbow.
Imagine now that I have one drop of water here.
And sunlight comes in at all angles of I, not just at one,
but all angles of I. Whatever you see here has of
course axial symmetry. It is a spherical drop.
And the light comes in like this.
So light can go this way but it can also go this way.
And it can also go this way and this way, so there's complete
axial symmetry, so this whole drawing you can
rotate about this line here. And everything holds then in
axial symmetry. So therefore if phi maximum,
if this angle phi maximum is forty-two degrees,
then the light that will go back in the direction of the
sun, the light that goes through the
journey A B C and then comes out of the raindrop,
that's all I'm talking about, now, I'm not talking about this
light that sneaks out here, it is this journey,
A, refraction at A, reflection at B,
and then coming out at C. That light comes out in the
form of a cone. And the half -- top angle of
the cone must be roughly forty-two degrees.
And so I will go -- I'm going to draw that cone for
you. Like so.
And like so. And you have to think of this
as a cone. It's completely symmetric,
axial symmetric, about this line.
And this angle here then is roughly forty-two degrees.
No light can go here. Because that would mean that
phi would be larger than forty-two
degrees and that's not allowed. Now comes something very
important. The index of refraction of red
light for water is about one point three three one.
And that translates into an angle for phi max which I can
calculate now -- it translates into an angle of phi max which
is about forty-two point four degrees.
But blue light has a slightly different index of
refraction, therefore has a slightly different angle for phi
max, and the blue light has an index of refraction of something
like one point three four three. Notice I have blue light and I
don't use violet light. Violet is much harder to see
with our eyes. So I always refer to it as blue
light. And that has a value for phi
max which is approximately forty point seven degrees.
A different index of refraction means of course that if you know
I that R is slightly different, using Snell's law,
so you get slightly different values for R,
and so you get slightly different values for delta,
so you get a slightly different value for delta minimum,
you get a slightly different value for phi max.
What does this mean now? That means if you look at this
cone of light that goes back into the direction of the sun,
that the outer edge, the outer
surface of that cone, which has the largest possible
angle, this angle is now forty-two point four degrees,
must be red light, because blue light cannot come
out in that direction. Because the maximum angle for
blue is forty point seven, is not forty-two point four.
So it's only red light that can come out when the angle -- the
half angle of the cone is forty-two point four degrees.
And the blue light is not going to make it until this
angle here -- I'll put it in here -- is forty point seven
degrees. If you look inside the cone
with an half top angle of forty point seven degrees all colors
can come back. All this is saying is that for
red light phi maximum is forty-two point four.
It can also therefore be forty point seven.
It can be thirty. It can be twenty.
It can be ten. It can be zero.
Light that comes in here at I equals zero
reaches point P, comes straight back.
That's allowed. Phi would be zero then.
If all the colors can make it back inside this cone that I
have given here a blue color, that would mean that your
brains will tell you that it is white light.
Because you see red light, you see blue light,
you see green light, you see yellow light,
and so your brains will tell you that that is white light.
If I had a screen here with a small opening to let the
sunlight in and I asked you what would you see on this screen
having only one water drop, then you would see the
intersection of this cone of light with this screen.
And it would look as follows. The outer edge,
the outer circle, which is the intersection of
the cone with your screen, would be red.
And then there would be an inner portion whereby all colors
can come back. So that would be white light.
And then as you go further in from the red,
until you reach that white portion, then the last color
that will be added is the blue. And you can already of course
sense that all the action about the rainbow occurs here.
And here there is no light. It's
going to be dark because there is no way that phi can be larger
than forty-two point four degrees.
And if light would appear on the screen outside the red it
means that phi would be larger than forty-two point four
degrees and that's not allowed. So it's dark here.
It's white light here. It's red light here.
Then as you go further in you will finally see the other
colors. And this really is now the key
to the geometry of the -- of the
rainbow. I'm going to put you here,
so here you're standing, and let the sun again be near
the horizon. It's always nice when you make
a picture. The easy reference that you
have. You're standing here.
Sunlight is coming in in this direction.
And there is rain here. If it's also raining here you
won't see a rainbow.
Because the sun will not be able to hit these raindrops.
So it's essential that it's raining in the direction away
from the sun but that you can still see the sun.
So here are these raindrops. All right.
You're looking in this direction in the sky.
You're looking up in the sky like so.
And I pick here one raindrop and only one but what I'm
telling you holds for all the raindrops in that direction.
What will that raindrop do? That raindrop will produce a
cone of light which goes back in the direction of the sun whereby
the edge of the cone is red light and this angle is
forty-two degrees, forty-two point four,
whatever. What do you see?
Nothing. Because there is no light that
can come from this raindrop in your
direction. There is no light because that
would mean that phi is larger than forty-two degrees and
that's not allowed. So you look high up in the sky,
you will not see any light coming back from that raindrop.
And think of the whole thing as axial symmetric,
right? Not only there,
but there and there you will not see any light.
Now I am looking say at some raindrops which are here.
I pick one. It holds for any one in that
direction. And so now I draw a line to
this point. This is what I'm looking.
I look in this direction. I take this raindrop.
I could have picked this raindrop.
I could have picked this one. Would have made no difference.
What is that raindrop doing? Well, that raindrop is throwing
back at the sun light in the form of a cone and the cone has
this angle forty-two degrees, forty-two degrees.
What do you see now? Look.
You're looking straight into that cone.
You're nowhere near the edge. So you see white light.
Because green light comes back at you, red light comes back at
you, everything comes back at you.
So you will say ha, I see white light.
Not only there, low on the horizon,
but the whole thing is axial symmetric, also there and there.
So you haven't seen a rainbow yet.
But now suppose I ask you to look
up in the sky at a very specific angle and we'll make
the angle forty-two point four degrees.
So you're looking at the sky in the direction somewhere here,
I'm not quite sure that I have the angle just right.
So I'm looking in the sky here. Pick one raindrop.
But you could pick any one, any other other one.
There's nothing special about that one.
So the sunlight comes in like so.
What is that raindrop doing? Well, it's throwing a cone of
light back to the sun. And it just so happens that you
are only looking at the outer surface of the cone where only
red light can go because this is the famous angle of forty-two
point four degrees. And so you see red light.
And since the whole problem is axial symmetric,
you would see red light if you look forty-two point four
degrees in this direction but also forty-two degrees in this
direction, away from this direction to the sun.
And so now you see how the bow is being formed by zillions and
zillions of small water drops which each of them in their own
way contribute to your rainbow. And we'll now put the sun a
little higher in the sky. So I'll put you here now.
And let's say the sun is now like
so. So, you look at your shadow
here on the ground. The sun is there.
There's your shadow. Here's your shadow.
Here's your head -- your shadow.
And this is the reference line that we're talking about.
That is this line. That was this line.
And so where would you see a rainbow?
You have to look now forty-two degrees away from that line.
Forty-two degrees away from that line.
That would be red then and a little lower would be your your
blue. And so if I sketch in there the
bow, it would be sort of like this and then the blue would be
on the inside and here you would have white light and this would
then be the red. It's always relative to this
reference line. This would be roughly -- I call
it forty-two degrees, but according to my
calculation, there it would be be forty-two point four.
So whenever you know where the sun is, you look at the
shadow of your own head and you have to go forty-two point four
degrees away from that direction from your eye to the shadow.
And so what you're seeing now is that if the sun is low in the
horizon then the bow will be high above the horizon and when
the sun is rising then the bow goes down and goes down and goes
down and by the time that the sun is forty-two degrees above
the horizon, you're not going to see a rainbow anymore unless the
water is right where you are. If the water is at a distance,
you won't see a rainbow. So the higher the sun in the
sky, the less likely it is that you will ever see a rainbow.
Before I show you some -- some slides and before we're going to
answer some of the questions, uh, I want to mention to you
that what I went through here is refraction at A,
reflection at B -- reflection at B and coming out at C.
You can go through the same geometry and allow for one more
reflection at C and then let the light come out,
so it would come out at point D and if you did that,
you can convince yourself that there is indeed a second rainbow
and we call that the secondary. We call this the primary.
And the secondary has a radius -- this one of course has a
radius of forty-two point four degrees for the red and the blue
would have a smaller radius.
The secondary has a radius for the red of fifty point four
degrees. So in the red,
it's fifty point four degrees, and in the blue,
it's larger. So the blue is outside
fifty-two point five degrees. Is that what it is?
Fifty-three point five. The secondary bow is fainter
and it is also wider because you see this separation in terms of
angle is larger than the separation there.
That's only one point seven degrees
and this is, uh, more than three degrees.
And so there is a secondary bow and the secondary bow is only
about roughly ten degrees above the primary.
So if you see your primary at forty-two degree angles away
from your shadow there and there and there, go another ten
degrees and you should see a second bow.
And you should see the colors reversed.
Red is there on the outside and blue is on the inside.
Red is on the outside and blue is on the inside.
Red is on the outside and blue is on the inside,
but here it is reversed. The secondary bow,
the red is on the inside and the blue is on the outside.
So let's answer some questions now.
We'll put the questions back up again and you will see that you
can now without any difficulty already answer twelve questions.
The first twelve. Red is outside.
That is non-negotiable, right?
That follows immediately from what we'd talked about,
but that's not an issue anymore.
The radius is about forty-two degrees and the length of the
bow, well that depends on if the sun is high in the sky then the
length will be very small because this whole s- arc will
then go down and there will only be a little bit left.
It could also be that it's only raining here and it's not
raining there. So depending upon where there
is rain and how high the sun is in the sky, that will determine
the length of the bow. The width of the bow,
you would think, perhaps naively,
that if you subtract the forty-two
from the forty -- forty point seven from the forty two point
four that that would give you the width of the bow which would
then be one point seven degrees in angle.
However, you would overlook then that the sun is not a
point, but that the sun in the sky has a dimension of half a
degree and each point of the sun, of course,
makes its own little rainbow, so you really have to add
roughly half a degree. So, the width of the bow is
more like, something like two degrees rather than the one
point seven, maybe two -- two point two degrees.
It's a little wider than what you would think.
It has to do with the finite size of the sun.
A comparison of the light inside and outside,
clearly inside the bow, there must be a lot of white
light which you don't see outside the bow -- big
difference. You will see a slide shortly.
The time of the day -- well the sun has to be a little bit low,
so you don't expect it at midday and you want rain as
well, so late afternoon, early morning would be ideal
and you can figure out uh in what direction to look.
If it's early morning, the sun rises in the east,
you would see the bows in the west and late in
the afternoon when the sun is in the west, you would see the
bows in the east. Yes there is a second bow.
It is about ten degrees higher in the sky than the primary.
And the colors are reversed. The blue is outside and the red
is inside. And the radius is about
fifty-two degrees. And the width of the bow is
again the difference between these two numbers which you have
to add about half a degree. So you would think on the basis
of this that it is more like three degrees and you have to
add roughly the half a degree, so it's about three --
three-and-a-half degrees. So let's now look at some
slides. And the first slide that you
will see is a -- is a drawing that I made.
Which uh is meant to repeat some of what I told you.
So you see a person standing there and here you see the
direction to the sun. Your shadow would be this
long. This would be your head of the
shadow, here would be your feet. And if there are water drops
here and if there's no interference between the
sunlight and the water drops then in this direction forty-two
degrees you would see this water drop red.
And then the water drops which were here you would see white
light from those water drops and maybe from this one you would
see red as well as blue. And if you have water near your
feet there's no reason why you can't see the rainbow there
either but of course you would have to lead some
garden hose to produce water there.
The next slide then. This is a drawing made by the
maestro himself, Newton, who was the first to
fully understand the workings of the rainbow.
You see here the primary bow and for those of you who are
sitting close you can see the direction of the sunlight going
in at A, reflects at B, coming out at C.
Which he calls E, but that's a detail.
And here you see then the secondary whereby the light
comes in, one reflection, two reflections,
and now it comes back at you and that gives you the
secondary. And so this is red and this is
blue whereas here this is blue and this is red.
The next slide. So if the sun is high in the
sky, I've done this many times when I was uh watering my
garden, you get this for free, you might as well do this,
it's great fun, you're standing there,
gives you an immense feeling of power, and is completely clear,
there's not a -- not a cloud in the sky, there's no rain
anywhere, but you spray water around you and you see a
beautiful rainbow encircling your feet.
And always what matters is this forty-two degree angle relative
to the direction from the sun to you and this is where the shadow
of my head would be. Very easy to do and I would
recommend that you try that when you get a chance.
The next slide is a painting from the eighth century,
Turkey, I see a hand here -- probably makes reference to the
Bible, but I think it reads in the Bible I do set my bow in the
clouds, it's very nice and very dandy, the fact that the colors
are wrong, the sequence of the colors,
that's a detail, it's a nice picture,
but red has to be on the outside and blue has to be on
the inside. Small detail.
Next slide. Ah yeah.
A -- a few years ago, actually it's more than a few,
when I was first lecturing eight oh three I knew I was
going to talk about the rainbow, ideal for eight oh three.
And so I wanted to make some rainbows myself in my backyard
in Winchester. And so there is water coming
out of a um water hose and the sun was behind me
and I took this picture and you see indeed all the ingredients
that we just discussed. Notice the red is on the
outside and the blue is on the inside.
You see all this white light. That is that light that comes
back from the water drop but if you go here where the angle of
phi would be larger than forty-two degrees,
which is not allowed, you don't see any light coming
back at you, so you look straight through and see the
forest without any white light. So you really see already here
sky so to speak is bright here and darker there.
I needed help from my daughter. And the next slide will show
you that. The poor darling was suffering
badly. It was January,
it was freezing cold, she was crying.
She was really crying and I felt guilty but I said look,
you know I really need these slides for my eight oh three
class, it'll only take you half-an-hour or so,
and she still remembers it, I had email exchange with her
yesterday and she said dad I was crying, it was an awful thing
what you did to me. But look, but look.
You know, you got to do something, and when you're
daughter of a scientist occasionally you have
to suffer a little bit. And so she's holding here the
-- the water hose and you see the same thing red outside blue
inside, and clearly the white light, you see this is that
white light that I discussed with you earlier.
The next slide will show you then a real rainbow -- oh no,
this is the um this is the artificial one I made uh over my
driveway, if you want to see the secondary, you need a real dark
background, because the secondary is quite faint.
And so that's why I did it over my driveway and you see here the
primary, red on the outside, blue on the inside,
and here you see the colors reversed.
You can also see perhaps that it's a little wider.
But it's always much s- it's much fainter,
so it's -- it's hard to tell. And then the next slide is one
that's a wonderful slide made by Doug Johnson in New Mexico,
Socorro, it is where the radio telescopes are,
the very large array, now look, red on the outside,
blue on the inside. The sky you got to admit is a
hell of a lot brighter here than it
is there. And you may never have noticed
that, you've -- you've looked at it but you've never seen it and
then here you see the secondary red on the inside blue on the
outside. There is a phenomenon that we
have never discussed uh in eight oh two yet, it's coming up I
think next week. And that phenomenon we call
diffraction. Snell's law cannot deal with
diffraction. That occurs when water drops
are very very small, say, smaller than a tenth of a
millimeter. What you then get over and
above the bow you get areas in the bow whereby you get as we
call that destructive interference,
the waves begin to kill each other and you get dark bands and
you can actually see that here. With a little bit of
imagination you see here a dark band.
And when that is the case the water is always extremely small
in size and we give this a name, we call this supernumerary
bows. It's not so uncommon.
If the water drops become exceedingly small,
let's say, smaller even than fifty microns,
then this diffraction phenomenon becomes so important
that in addition to the dark bands all the colors are
beginning to wash out over each other, and that creates then a
white rainbow. Plus the dark bands.
And a student of mine who was in my eight oh three lecture
sent me years later the next slide when he was -- his name
was Carl Wales, this picture he took,
it's three hundred forty miles from the North Pole,
he was at Fletcher Island at the time, this picture was taken
at two AM at night in July when the sun is above the horizon.
And so this must be the result of very fine water drops which
somehow are there in the atmosphere.
Doesn't look like it's raining but there must have been small
water drops. Fifty microns or less.
And here you see the white-colored rainbow and you
also see beautifully the supernumerary bows,
you see the -- the dark bands in there and the next slide is a
close-up of that. So you see here the white
rainbow plus the dark bands which is the phenomenon the
result of uh of diffraction. Now before I will discuss the
polarization of the bows, because I still owe you answers
to the last three questions, now that I'm at it,
I want to show you some phenomenon which are quite
common which you may never have seen
even though they're quite common and they're rather
spectacular. The first slide that comes now
shows you what we call the twenty-two degree halo.
It's very common around the sun, it's very common around the
moon. The red is inside.
It has nothing to do with water.
It is the result of ice crystals way up in the
atmosphere. You can see it both in the
summer as well as in the winter because it's very cold way up
there also in the summer. This is very common.
The reason why you and I don't see
it that often, because who wants to look in
the direction of the sun. This angle is only twenty-two
degrees, it's not so far. But I would advise you to at
least keep an eye on the moon. Because the moon also has this
twenty-two degree halo. And of course it's much easier
to look in the direction of the moon.
That doesn't uh -- any no problems for your eyes.
This is very common that I see this at least three or four
times per month and I always look for it of
course. The next um slide shows you
both the twenty-two degree halo as well as the forty-six degree
halo which is way more -- way way less common,
it's very rare to see the forty-six degree -- I've seen it
only a few times. It's very rare.
In addition to the twenty-two degree halo and the forty-six
degree halo you sometimes see bright spots here.
You see them here and you see them here.
They're really not circles. They are arcs.
And they have names, they call them uh sun dogs,
mock suns, they have various names.
I've seen this often in in Boston, the forty-six degree
halo is rare. All of this as the result by
the way of um ice crystals. Ice crystals way up in the
earth atmosphere. And there is a phenomenon that
many of you may have seen from an airplane.
Uh if your airplane flies over clouds and you look at the
shadow of your airplane onto the cloud you may have noticed
colorful rings around the shadow of your
airplane. And the next slide is such an
example. I took this picture several
years ago. You are always right at the
center of the circle. So you can see that I was
sitting just behind the wings. This is the result of
diffraction. Got nothing to do with Snell's
law. It is the result though of very
very fine water drops but not in the sense of refraction and
reflection the way that we discussed it with the rainbow.
This is the result of diffraction.
It's extremely difficult, the explanation of the glory.
And even ten years ago, I read an enormous article
about it, I think it was in Scientific American.
Which made an utter attempt to be very quantitative about the
explanation, which is not easy. The radius of this bow,
if you want to call it a bow, depends on the size of the
water drops. If the water drops are very
small the radius is substantially larger.
So as you fly over clouds you may fly over different clouds
with different size of water drops and on a very short time
scale will you see the radius of this glory change.
It can be very rapidly. It could be on a time scale of
seconds to minutes. It's very dramatic.
It's very clear. And these rings can extend,
there's just not one ring, but you can have several rings.
And so this is the result of what we call diffraction.
Now I want to mention to you another phenomenon which is um
also the result of diffraction. And that you can see when there
is fog. If you have fog fog consists of
extremely small water drops, just like the ones in clouds.
I mean fog is a cloud, let's face it.
Then if you for instance had the headlights of a car you
could see your own shadow onto the fog away from the direction
of the car. And that means you could see
then around your head the same kind
of stuff that you saw here. A glory.
We call this a fogbow. That's just a name.
The Germans have a much better name for that,
they call it Heiligenschein. And heilige person is someone
who is a saint and so you see this around your head,
so it is sort of the -- the radiation that you would expect
from a saint. Quite a few years ago I was
invited to visit the Soviet Union.
And they took me to the Caucasus, to show me the
six-meter telescope which was at the time the largest telescope
on earth. And I went there for a few
days. And I noticed much to my
surprise that every night at five o'clock there would be fog
coming out of the valley overtaking the telescope so you
could make no observations at night but that was a detail and
what I realized is that the fog was coming up,
the sun was there, that was the west,
and the valley was at the east, and so I said to myself my
goodness if only at the right moment of time I am there then
the sun is my light beam and my shadow will fall onto this wall
of fog coming towards me, this is my chance to make a
wonderful picture of this Heiligenschein.
And I'll show you that I succeeded.
I will first show you a picture of this bizarre observatory.
This is the six-meter telescope in the Caucasus.
And this is indeed you could every night around five-thirty
really like a wall the fog will just coming
towards you overtake you. You can see the timing of my
picture was extremely crucial. If you wait a little bit too
long then the fog overtakes you and then the sun would obviously
not be effective anymore. I wouldn't see my shadow on the
fog. So I only had a small time
window, maybe thirty seconds, maybe one minute.
But I succeeded and you will see w- Saint Walter coming up.
There is Saint Walter. You can see here my arm and you
can see here my camera.
The camera must be exactly at the center of the glory.
And that's where it is. I have the camera right here
and here you see the rest of my body, isn't that terrific?
This can be used as proof sooner or later.
OK. Back to our last three
questions. The last three questions deal
with the polarization of the bow.
Are the bows polarized? And the answer is yes,
they are enormously polarized. Why are they polarized?
You can answer that question. But I will answer it for you.
The light that contributes to the rainbow is the light whereby
the minimum angle for delta is about a hundred thirty-eight
degrees and if you go back to that
first transparency you will see that that's the case when the
angle of incidence is about sixty degrees.
You can download that again. And that means that the angle
of refraction is about forty degrees.
Forty degrees is very close to the Brewster angle.
The Brewster angle if you go from water to air you -- medium
where you are is water and you bounce it off this medium,
so you go from water to air, the tangent of the Brewster
angle is N two divided by N one. N two is where you're going.
That's air. So this is one.
N one is where you are. That's the water.
Let's just call that the index of refraction one point three
three. If you calculate now what the
Brewster angle is you will find that the Brewster angle theta
Brewster is about thirty-seven degrees.
That is only three degrees different from this angle which
is the crucial one which contributes to the rainbow.
So you're only three degrees away from the Brewster angle.
So what happens as this light comes in i will make a a
separate drawing