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Now you have the pieces you need to
construct

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our very first climate model
of a naked planet.

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We have sunlight that's coming from the
sun

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and coming all in one direction
toward the Earth.

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And then we have Earth that is shining
it's own light, based on it's own

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00:00:27,690 --> 00:00:30,050
temperature, with light that's going in
all

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00:00:30,050 --> 00:00:32,730
different directions around the surface of
the Earth.

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00:00:32,730 --> 00:00:34,710
And what we're doing, to

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00:00:34,710 --> 00:00:37,350
solve for the temperature of the Earth,

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00:00:37,350 --> 00:00:40,470
is we're solving for the condition where
the energy

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00:00:40,470 --> 00:00:44,490
coming into the planet balances the energy
that's leaving.

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00:00:44,490 --> 00:00:46,760
That's the eventual steady state.

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00:00:46,760 --> 00:00:49,940
An analogy to that would be if we had

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a kitchen sink, and you turn on the faucet
and the

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00:00:52,640 --> 00:00:55,950
water starts to build up in the sink and
it gets

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00:00:55,950 --> 00:00:58,800
deeper and deeper.  And as it gets deeper in
the sink,

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00:00:58,800 --> 00:00:59,740
there's more pressure

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of water pushing it down the drain, and so
the rate at which the water

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00:01:03,730 --> 00:01:08,520
leaves the sink is a function of how deep
the water is in the sink.

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And so, the water level in the sink, if it
starts out all the way down, will rise,

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until it comes close to the level which
the water budget balances and stays there.

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Or if you start out with too much water,
it'll sink

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until it goes to that steady state value
and stay there.

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If this is the rate of water

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flow in from the faucet, and you started
out with

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00:01:32,820 --> 00:01:35,120
no water, there would be no more
coming in

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than out initially, and so the water would
build up.

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00:01:38,380 --> 00:01:41,720
And it would tend to relax to that
condition of balance.

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00:01:41,720 --> 00:01:44,480
Or, if you walk up to it with a big bucket
and dump a big bucket all at

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00:01:44,480 --> 00:01:46,800
once in there, you'd start with too much
water,

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and it would relax downward, toward that
steady state value.

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00:01:50,150 --> 00:01:52,850
So, what we were doing in this
calculation,

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00:01:52,850 --> 00:01:55,830
is going right for that steady state
value.

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00:01:57,650 --> 00:02:00,240
So, we're looking for the condition
where

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the energy in is equal to the energy out.

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00:02:04,930 --> 00:02:10,300
We'll look at this in pieces.
The energy going out is

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00:02:10,300 --> 00:02:16,190
given by the Stefan-Boltzmann formula,
epsilon, sigma, temperature to the fourth.

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00:02:16,190 --> 00:02:20,010
Where the temperature is the temperature
of the Earth, and we're assuming for now

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00:02:20,010 --> 00:02:23,040
that it's all the same temperature because
this is a very, very simple model.

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00:02:24,410 --> 00:02:27,620
Epsilon for the Earth is pretty close to
one.

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00:02:27,620 --> 00:02:31,392
Most solids and liquids, most condensed
matter

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has a pretty good black body properties.

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00:02:34,650 --> 00:02:36,630
And so it has epsilon pretty close to one.

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00:02:36,630 --> 00:02:39,480
The exception to that we'll get to is
greenhouse gases.

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00:02:39,480 --> 00:02:41,020
A very important exception.

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00:02:41,020 --> 00:02:43,740
But for now we can kind of just call that one and
not worry about it

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00:02:43,740 --> 00:02:45,870
and this is the Boltzmann
constant which

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00:02:45,870 --> 00:02:46,890
you can just look up in a book.

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00:02:48,160 --> 00:02:54,190
This tells us the energy flux in watts
per square meter, but if we want to

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00:02:54,190 --> 00:03:00,790
do the planet overall, we have to multiply
by the area of the surface of that sphere.

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00:03:00,790 --> 00:03:08,678
So this area, to get rid of the watts per
square meter, in the area of the sphere,

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00:03:08,678 --> 00:03:13,130
is 4 pi R squared.
Now on the other side of the equation, we

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have the energy coming in from the sun,
and this is given by a solar constant.

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00:03:20,270 --> 00:03:23,920
Which is a number in watts per square
meter which is determined by how

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bright the sun is, but also how far away
we are from the sun.

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00:03:27,180 --> 00:03:32,750
It's how bright the sunshine is if you
look at it straight on.

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00:03:32,750 --> 00:03:33,750
at the distance

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from the sun to the earth it's about 1350
watts per square meter of area.

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00:03:39,940 --> 00:03:41,480
If you had a solar cell.

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00:03:41,480 --> 00:03:43,920
One square meter in size,

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00:03:43,920 --> 00:03:47,420
You could about run a normal sort of hair
dryer

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00:03:47,420 --> 00:03:51,050
on that, that's kind of the energy flux
coming through that.

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00:03:51,050 --> 00:03:55,260
But not all of that energy actually is
absorbed as heat by the planet.

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00:03:55,260 --> 00:03:59,810
Some of it gets reflected back out to
space and never gets absorbed.

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00:03:59,810 --> 00:04:02,710
And that fraction is reflected in the

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00:04:02,710 --> 00:04:06,250
albedo, which is given the Greek letter
alpha.

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00:04:06,250 --> 00:04:10,550
So, 1 minus albedo is the fraction that
gets absorbed.

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00:04:10,550 --> 00:04:13,340
The value of the albedo for the

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00:04:13,340 --> 00:04:16,900
Earth is about 30%, because of the clouds,
mostly.

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00:04:16,900 --> 00:04:19,410
Reflecting light back out into space.

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00:04:19,410 --> 00:04:21,790
This is in watts per square meter

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00:04:21,790 --> 00:04:26,702
but it's in per square meter of
this energy, that's coming from the sun,

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00:04:26,702 --> 00:04:30,110
looking straight on, and we
have to multiply it

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00:04:30,110 --> 00:04:34,830
by an area, to get the total
energy coming in.

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00:04:34,830 --> 00:04:42,020
But this is a bit tricker now, because the
area of the earth isn't all

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00:04:42,020 --> 00:04:46,340
facing directly at the sun, perpendicular
to the way the sun is coming.

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00:04:46,340 --> 00:04:50,390
So we could do a complicated integral,
where we add up all this,

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00:04:50,390 --> 00:04:53,540
the square meters of the earth and figure
out which ones

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00:04:53,540 --> 00:04:57,060
are sort of oblique and so they're not
getting as intense sunlight.

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00:04:57,060 --> 00:04:59,570
But there's a tricky, easier way to do it,

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00:04:59,570 --> 00:05:04,350
and that is to realize that the amount of
light that is intercepted by this

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00:05:04,350 --> 00:05:10,370
planet is given by the size of the shadow
of the earth.

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00:05:10,370 --> 00:05:13,280
That shadow is all directly
straight onto

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00:05:13,280 --> 00:05:15,790
the sun, and so it's oriented in the right
way.

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00:05:15,790 --> 00:05:22,620
This area is not the area
of the sphere that we had before.

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00:05:23,650 --> 00:05:27,930
but it's the area of a circle which is
only pi R squared.

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00:05:32,040 --> 00:05:35,760
So, if we equate those two sides we have

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00:05:35,760 --> 00:05:39,890
the solar constant, times a fraction
that's absorbed,

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00:05:39,890 --> 00:05:45,490
times the area of the shadow, and here's
the infrared emission, the black-body

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00:05:45,490 --> 00:05:52,450
radiation, and the area of the sphere, but
we can now divide by pi R squared.

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00:05:52,450 --> 00:05:56,480
In fact, we can even divide by 4 pi r
squared.

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00:05:56,480 --> 00:05:57,760
To get this

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00:05:57,760 --> 00:06:02,580
equation, which is in the most convenient
form, it's now in units of watts per

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00:06:02,580 --> 00:06:06,010
square meter of the surface of the sphere,

99
00:06:06,010 --> 00:06:08,890
so per square meter of the Earth's
surface.

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00:06:08,890 --> 00:06:13,290
So what we have now is an equation that
only has one unknown,

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00:06:13,720 --> 00:06:15,810
and that's the temperature of the planet.

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00:06:15,810 --> 00:06:18,190
We can rearrange that, and
calculate what the

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00:06:18,190 --> 00:06:22,900
temperature should be, for any given
combination of the solar constant,

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00:06:22,900 --> 00:06:23,780
and the albedo.

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00:06:26,260 --> 00:06:29,470
This is a table of how that
calculation goes

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00:06:29,470 --> 00:06:33,700
for the Earth and our sister planets,
Venus and Mars.

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00:06:33,700 --> 00:06:38,420
The solar constant is much higher for
Venus because Venus is closer

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00:06:38,420 --> 00:06:42,300
to the Earth than the Earth is and it's
lower for Mars.

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00:06:42,300 --> 00:06:46,900
That's because the intensity goes down as
you get further from the source.

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00:06:46,900 --> 00:06:50,610
The reflectivity of Venus is very, very
high.

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00:06:50,610 --> 00:06:51,480
That's because Venus

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00:06:51,480 --> 00:06:53,449
is covered with clouds all the time.

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00:06:54,630 --> 00:06:57,950
Venus is the second brightest thing in the
night sky.

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00:06:57,950 --> 00:07:01,280
Not because Venus is so hot that it's
shining its own

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00:07:01,280 --> 00:07:04,210
light, it's not a star.  

117
00:07:04,210 --> 00:07:06,830
It's just reflecting the light that's
coming

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00:07:06,830 --> 00:07:09,130
in from the sun because of these clouds.

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00:07:09,130 --> 00:07:13,690
And then Mars doesn't really have clouds

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00:07:13,690 --> 00:07:16,550
or much of any ice, it's mostly

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00:07:16,550 --> 00:07:19,680
darkish rock and so it has a fairly low
albedo.

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00:07:19,680 --> 00:07:23,330
Here is where we calculate the
temperature of

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00:07:23,330 --> 00:07:29,020
these planets given their solar constant
and albedo values.

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00:07:29,020 --> 00:07:32,440
And it's interesting, Venus, because it's
so

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00:07:32,440 --> 00:07:35,760
reflective, would be even colder than the
earth.

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00:07:35,760 --> 00:07:37,750
Even though it's closer to the Sun.

127
00:07:37,750 --> 00:07:41,960
It's just wasting all that energy
by reflecting it out to space.

128
00:07:41,960 --> 00:07:45,344
And then the Earth is warmer than
Venus and then

129
00:07:45,344 --> 00:07:49,110
Mars is cool again because it's so
far from the Sun.

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00:07:50,150 --> 00:07:52,360
But the real interesting thing comes when
we compare

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00:07:52,360 --> 00:07:56,190
it with the real temperatures that the
planets actually have.

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00:07:56,190 --> 00:07:59,950
Venus is much much hotter than we just
predicted

133
00:07:59,950 --> 00:08:03,070
Earth is hotter than we predicted, and so
is Mars.

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00:08:03,070 --> 00:08:07,210
They are all the same, the climate model as
we've done

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00:08:07,210 --> 00:08:09,415
it so far is always too cold.