Download Theme 2.2 The Seasons The Sun`s Special Location In the previous

Theme 2.2
The Seasons
The Sun’s Special Location
In the previous unit we found it useful to begin with a situation in
which we imagined the earth sitting alone in empty space, surrounded by
a vast number of very distant stars. In fact, of course, the earth is
very close to one star, the sun, which provides warmth, and gives us
life. We will now consider the implications of our presence near that
star.
To understand the apparent motion of the sun, we have to think about
two motions of the earth: the first is the spin of the earth, the fact
that it rotates on its own axis once every 24 hours, something we've
already discussed with respect to the remote stars. To remind yourself
of what this motion looks like, take a look at the computer simulation
shown here, which is a very much sped-up version -- of course the
actual rotation of the earth takes 24 hours.
In addition to this spinning motion, the entire earth is actually
moving through space, going around and around the sun in a big orbit.
That orbit of course takes one year. Here's a computer simulation
showing the combined effect of the earth spinning as it moves through
space. This will help you to visualize the situation.
The Zodiac
It may seem hard to believe that the earth itself is really moving
around the sun, you might think that we would feel the effect somehow.
This was why, in part, the ancient Greeks believed that the earth
itself stood still at the middle of the universe. We'll return to that
question later, but for the moment, note that it doesn't really matter
which is correct, whether the earth goes around the sun, or the sun
goes around the earth, we will still see the same annual behaviour of
the various zodiacal constellations appearing at different times of
year. In February, we will see Leo overhead at night as you can see
from the figure, whereas the stars of Aquarius are hidden from view in
the daytime sky. Six months later, in August, the situation is
reversed, and again it doesn't matter which of the objects, the sun or
the earth, is actually moving.
The Seasons
That behaviour of the zodiacal constellations is merely one yearly
cycle that we have to consider, the second, of course, is that there
are seasonal changes, we progress through spring, summer, fall and
winter, and very dramatically so here in Canada. Why does this happen?
Well there are two obvious possibilities. One is, that the earth is
sometimes closer to the sun and therefore it's warmer in summertime.
In other times of the year it would be farther away, which might give
rise to our winter. The second possibility is that the orientation of
the earth, in its orbit around the sun, somehow matters.
We'll now consider which of these is correct.
The Effect of Varying Distance
The distance of an object from the sun can certainly have an effect on
the heat and cold it experiences. Some comets for example, have very
elongated orbits, when they're far from the sun, they're very cold,
when they're close to the sun, a lot warmer. In fact, gases that are
frozen in the comet, boil off and produce the conspicuous tail. Mars
is similar in that its orbit is not a perfect circle around the sun.
In the diagram on the right, the smaller circle shows the orbit of the
earth, which is very close to a perfect circle, meaning we're nearly
always at the same distance from the sun. Mars, as you can see, is
less circular: at certain times it is closer to the sun, at other times
rather farther away. This means that it experiences changes of heat
and cold that are planet-wide. When Mars is far from the sun, it's
winter over the whole planet; when it's close, it's summer.
One important piece of evidence however, comes from the fact that in,
on the earth, the seasons are not all in phase. When it's the winter
in Australia in June, July and August, it's summer in the Northern
hemisphere. I know that myself through having lived in Australia for
several years. Here's a calendar that makes the point, showing the
snowy regions of Australia in June, inviting you to go skiing there.
So the distance from the sun can't be the thing that determines the
seasons across the planet. In fact, we're actually closest to the sun
in early January, which is the depth of our northern winter. At that
time we're about three percent closer to the sun than our average
distance, and in June and July, we're actually farther away.
The Effect of Changing Angles of Illumination
To understand what really causes the seasonal changes, let's consider
the effects of light landing at an angle on a surface. If you have a
flashlight aiming straight down onto the floor, the light is very
concentrated. If you tip it through some angle, the light is more
spread out, and that means the illumination, and any heating effect,
are much more dilute. The same is true for the sun landing on a
surface.
Even without considering the seasons, you can see why it is that this
leads to cool Arctic regions, compared to the hot tropics. The light
landing on the tropics, with the sun almost directly overhead, is much
more concentrated than the light landing at the edges of the globe, in
the Arctic regions and the Antarctic. The same reasoning tells us why
it becomes cool in the evening, when the sun is setting in the west and
the light is more spread out.
So the conclusion, which we'll explore a bit further in the next panel
or two, is that the seasonal changes are caused by changes in the
apparent height of the sun above the horizon. That's shown here in
these two figures, which are drawn from computer simulations. They
show where the sun would appear to be at about midday on December 21st,
and on June 21st, as seen from Kingston. You can see that in December
the sun is very low in the sky, in June it's very high in the sky. But
why, what makes this important difference?
The ‘Tip’ of the Earth’s Rotation Axis
In effect, the best way to think about this is to consider the sun as
just being yet another star, although an extraordinarily bright one of
course. During our winter, it acts like a southern star, rising low in
the southeast, moving through a short arc across the southern sky, and
then setting in the southwest. During our summer, it acts like a
northern star, rising in the northeast, climbing high into the sky and
travelling through a long arc until it sets in the northwest. But why
are there these differences, what makes a change in its position north
and south?
The reason is that the rotation axis of the earth, the axis around
which it rotates once a day, is tipped relative to the flat plane that
describes its orbital motion around the sun once a year. Look at the
top left-hand figure in this panel. The red arrow indicates the
direction of the North Pole, and you can imagine yourself standing
there with the earth spinning under your feet, turning you around once
a day. As the earth moves around the sun, I think it's straightforward
to realize that, if you were at the equator, you would see the sun pass
overhead every day, day after day, with no change. From the North
Pole, the sun would always look low on the horizon from your
perspective, and likewise, from the South Pole, there would be
perpetual twilight. From intermediate latitudes, the sun would pass
through the sky the same way every day, and there would be no seasonal
change.
The reality is shown in the figure on the bottom right. On June 22nd,
the tip of the earth's axis, which is about 23-1/2 degrees relative to
the plane of the orbit, means that Canada and North America are tipped
towards the sun. We receive the sun's rays more directly, have more
heating effect, and see the sun high in the sky like a northern star.
Six months later, we're on the far side of the sun, and we are tipped
away from the sun. That means we receive less heating and
illumination, and the sun appears as a southern star. Of course, from
Australia, that's their summertime, from their point of view the sun is
high in the sky, but it's our winter.
This figure summarizes those thoughts. In late June, when the sun is
at its highest point in the sky from our perspective, you can see that
it rises in the northeast, it follows a long arc across the sky giving
us many hours of daylight, and sets in the northwest. In December, by
contrast, it rises low in the southeast, follows a short arc across the
sky, giving us short daytime, and sets in the southwest. In between,
on March 21st and September 21st, it rises due east, passes across the
sky, and sets due west.
Solstices and Equinoxes
These days are given special names. From December to June, the sun
gradually transitions from being a southern star to being a northern
star, thanks to a combination of the tip of the earth's axis and our
orbital motion around the sun. By June 21st, the sun has reached the
farthest northern point on this progression, and will start to slowly
again drift back towards the south. This momentary pause in the northsouth motion, leads us to refer to this as a solstice (“the sun has
stopped”), and that's the day in which we have maximum daylight hours.
December 21st marks the end of its southern drift, after which it will
again start to drift slowly north, and the days will start to become
longer. That is referred to of course as the winter solstice, December
21st, the day of minimum daylight hours. On March 21st and September
21st, the sun is directly overhead the equator, everyone on earth sees
12 hours of daylight and 12 hours of nighttime, and these two dates are
referred to as equinoxes (“equal night”).
The Equator and the North Pole
To finish this part of the discussion, let's think about what we would
see from two very special locations, first of all the equator, shown on
the left-hand panel here. Because the sun, from our point of view,
seems sometimes to be a northern star and sometimes a southern star,
you can see it will not always follow the same path across the sky.
But as we see in the diagram, the effects on illumination and heating
are not very dramatically different, and this explains why there are
not big seasonal changes in the tropics.
If we look on the right-hand panel though, we see something rather
different. Here we're at the North Pole. At the North Pole, when the
sun is a northern star, it travels around parallel to the horizon, as
we discussed earlier, but well above the horizon, so we have perpetual
daylight. This is the “land of the midnight sun” in June, and for
several weeks or months surrounding that. By the time we reach the
equinoxes, March 21st and September 21st, the sun is down right on the
horizon, and we're in a state of twilight. By December, of course, the
sun is well below the horizon, and you're in perpetual darkness for
months at a time. Very dramatic indeed.
Mars and Uranus
Incidentally, we noted earlier that the seasons on Mars are mainly
determined by its changeable distance from the sun. When Mars is close
to the sun, it's summer over the whole planet; when it's farther away,
it's winter. That's not the case for the earth, as we've seen.
There's one planet however that has a very extreme tip, that is Uranus,
which is almost exactly on its side as it orbits the sun. That's shown
here. What this means is that the North Pole of Uranus sometimes has
the sun directly overhead, which of course never happens on the earth,
and for long periods of time it will be illuminated directly, with the
South Pole in complete darkness. As it orbits the sun, that will
change, and the South Pole will eventually have the sun directly
overhead, with the North Pole in darkness. So Uranus has the most
extreme imaginable seasonal changes.
Two More Conservation Laws
We've talked about the cause of the seasons, but there's actually a
little bit of physics that I should introduce at this stage in support
of what I've been telling you. I'm going to introduce at this stage
two more conservational laws. Perhaps you'll remember that we met the
conservation of energy earlier. I'm going to introduce very briefly
the conservation of linear momentum, which is something we'll come back
to at a different context later. But importantly, we'll consider the
conservation of angular momentum, and see what it has to do with the
seasons.
Linear Momentum
To develop a conceptual understanding of what we mean by linear
momentum, think of a football game in which the fullback is trying to
plunge through the defensive line to score a touchdown. In general,
that would be easier if the fullback is big, and moving rapidly, and he
can burst through the tackles. A moving object has a quantity known as
linear momentum which is given by the product of its mass (that is to
say its weight, how much material it contains) and its velocity. Don't
worry about the equation, but just try to understand conceptually what
this implies.
It may not be obvious but linear momentum is a conserved quantity.
Consider a billiard game for example. If the cue ball is moving
rapidly towards an object ball and runs into it, the cue ball may come
to rest, but the object ball moves off with the same linear momentum
that was being carried by the cue ball in the first place. A rocket in
interplanetary space is an example of the same thing: the rocket may be
sitting at rest with nothing moving in particular. If you start to
fling material out of the end of it, like hot exhaust gases for
example, you now have some material moving off in one direction, and
the rocket itself must move off in the opposite direction so that the
total momentum is zero, as it was before: some material moving in the
positive direction, some in the negative direction. That's the
conservation of linear momentum.
This doesn't seem to square with common sense! We all know that things
are sometimes moving and sometimes at rest. Look, for example, at the
picture to the lower left, where we see the young girl doing a standing
jump. To start with, she's at rest; a moment later, she's moving to
the right, with some amount of linear momentum. And then she's at rest
again when she lands on the floor. How does this square with the
conservation law?
Well, the answer is she sets herself in motion to the right, by pushing
off on the floor. This gives her rightward motion, but the floor
itself, and indeed if you think about it, the entire earth is pushed
backwards a little bit, by her muscular action. In total, the linear
momentum is conserved, and adds up to the same as it always was, so it
can be exchanged from one object to another. It can be very
complicated, you have to think about other forces too; for example, the
billiard balls on the table gradually slow down and come to a halt
because of friction.
Just for fun, here's a link to a video of a piece of physics apparatus
called Newton's Cradle. Watch this and notice the interesting way in
which the balls behave. The behaviour is completely explicable in
terms of the conservation of energy and linear momentum: the balls have
no choice but to behave in the rather neat way that they do.
Angular Momentum
That's enough for linear momentum for the moment, we'll return to that
later. Right now I'd like to discuss the conservation of angular
momentum. The word momentum should again make you think of the product
of mass and velocity. A large, heavy object moving with high speed has
a large amount of momentum, sort of ‘punch’ or ‘drive.’ But the
angular movementum means that we should also think in terms of an
element of turning, or rotation.
Start by thinking of something like a stone on the end of a string
moving around in a circle at high speed. The angular momentum for that
small particle, which has a mass, m, moving at a velocity, v, is given
by the equation you see. Don't worry about the equation, just
understand this conceptually. If you want to think about the angular
momentum for a large distributed object, like a figure skater spinning,
you have to think about all the independent atoms moving in those
independent orbits around the spin axis, and then that way you can add
up the total angular momentum for a distributed body.
Again, the remarkable fact is that angular momentum is conserved.
Suppose for example, some material were to move inward, towards the
axis of rotation. If you look back at the previous panel, this means
that r, the distance of the atoms from the centre, becomes smaller. If
you want to conserve the same amount of angular momentum, that means v,
the velocity, has to increase, so, as the object shrinks, it will spin
faster. An excellent example of this is given by a spinning figure
skater, who draws her arms in to speed up the rate of rotation.
This panel presents links to two videos showing me doing two simple
experiments that demonstrate the conservation of angular momentum. I'm
sitting on a rotating piano stool, and in the top figure, if you follow
the link, you'll see that I'm holding two lead balls in my outstretched
hands. As I pull them towards me, my rotation speeds up. As I move
them back out, my rotation slows down: the conservation of angular
momentum.
In the bottom panel, if you watch the video, you'll see that I set a
bicycle wheel in spinning motion. If I turn the bicycle wheel over,
the rotation of the atoms is now in a sense opposite to what it was
before. For the conservation of angular momentum to hold, something
else has to move in the opposite direction so the total angular
momentum remains the same. You can see what happens is that I start to
rotate on the freely-spinning piano stool. (By the way this experiment
can make one feel very nauseated.)
Stability
So you've met two new conservation laws, but what's the point? The
conservation of angular momentum has a particular relevance here. The
spin of an object means that it has a certain stability. It can't just
flop around, because it has a certain spin pointing in a certain
direction, and the conservation law means that that can't dramatically
change. This explains for example, the behaviour of a Diablo, shown on
the left panel here. This is otherwise known as a Chinese yo-yo. When
this is set spinning rapidly and thrown high in the air, the spin means
that it maintains its orientation, and it's easy to catch on the string
as it falls back down. If it were flopping around, that would be very
difficult to do. I've added a link to a homemade video of my son
throwing a diablo probably a hundred feet into the air and catching it.
To emphasize this point, here's three more links to short videos, two
of which show the behaviour of bullets leaving the barrel of a gun, and
arriving at a target. They're set spinning by the rifling, the grooves
inside the barrel of the gun, and that provides the bullet with a
stability and helps it maintain its direction in flight. We see the
same behaviour with a football, which is thrown with a spinning motion,
so it maintains its orientation. That means it will move through the
air in the most streamlined way possible and reach its target more
readily.
And back to astronomy at last. The conservation of angular momentum
explains why we have reliable seasons, because it reassures us that the
spinning earth maintains its orientation in space. Year after year we
can count on the fact that it will maintain that tipped orientation,
that the seasons will reproduce faithfully from one year to the next,
and indeed if you think about the orbital motion of the earth around
the sun, that movement too has a certain stability because of the
conservation of angular momentum.
Precession
That stability is not guaranteed forever, however; we have to think
about the effect of other forces. Just as the billiard balls on the
pool table slow down because of friction, and the conservation of
linear momentum doesn't keep them moving forever, so too forces that
act on the earth can change its orientation as it spins and orbits
around the sun. The axis of the earth right now points close to
Polaris: we have a Pole star, and will do so for many centuries. But
that slowly changes in an effect known as precession, because the
gravitational effects of Jupiter and other planets.
As you know, Polaris is currently the Pole star, almost directly above
the North Pole of the earth. If you come back about 12,000 years from
now, you'll discover that there's a different Pole star, Vega, quite a
bright star; that will be very dramatic. But there are periods in
history in which there is no Pole star. For example, about 2,000 years
ago, at the time of the birth of Christ, there was no Pole star, and
4,000 years ago, in the time of the Egyptians, the Pole star was one
called Thuban.
So the consequences of the conservation of angular momentum include the
constant duration of the days, the steady 24 hour period of the day,
the earth doesn't dramatically speed up or slow down in its spin. And
the orbit of the earth around the sun is likewise stable. The length
of the year is reliable, and predictable.