We now have achieved our goal of 
understanding the motions of the sky, 
predicting which stars will be visible, 
which season, where on Earth, at what 
time of the night. 
there are some other things we'd like to 
understand. 
we talked about the orbital motion of the 
Eearth and how it affects which stars are 
visible in the sky when, 
of course, there's another phenomenon on 
Earth that repeats with the periodicity 
of once a year, namely the variation of 
the seasons. 
The climate on Earth changes once a year, 
and in fact, the original interest in 
astronomy, much of it, was derived from 
the fact that by understanding or 
watching which stars were visible in the 
sky, say early in the evening, you could 
predict the coming of harvest time, or 
the time to sew, or the inundation of the 
Nile, or whatever. 
All other phenomena that were periodic 
with annual periodicity, because they 
depended on the seasons. 
And the reason why the Earth's orbital 
motion is related to 
seasonal changes in temperature, of 
course, has to do with the fact that if 
you ever see one of these globes, it's 
always depicted in a tilted version. 
We all know that the globe is tilted by. 
Whoops, I had it upside down. 
Tilted by 23.5 degrees, 
but, this is interesting. 
We're out in space. 
23.5 degrees, with respect to what? 
Well, it turns out that the globe is 
tilted by 23.5 degrees relative to the 
plane of the ecliptic, the plane in which 
the earth orbits the sun, in other words 
if I want to imagine that the earth 
orbits the sun in a horizontal plane then 
I must hold the globe titled by 23.5 
degrees, rather than vertical with the 
north celestial pole, therefore, often 
that direction 23.5 degrees away from the 
vertical. 
Contrary-wise, if I want to present 
things with the north celestial's pole 
vertical, above me, then the plane in 
which the earth orbits the sun, and of 
course, the plain English the sun appears 
to orbit the Earth will be tilted by 23.5 
degrees away from the horizontal because 
it was horizontal in this frame of 
reference and I have tilted everything 
23.5 degrees this way so that. 
The sun's orbit around the Earth, or the 
sun's motion along the celestial sphere, 
is not along the celestial equator, but 
along a circle tilted 23.5 degrees, with 
respect to the celestial equator. 
This trajectory of the sun moving to the 
east along the celestial sphere is called 
the ecliptic and it's a circle tilted 
with respect to the celestial equator by 
23.5 degrees, and if you take two circles 
and tilt one with respect to the other, 
they're hinged, so there are two points 
at which they continue to meet. 
Those are the points where the hinge 
meets and so those two points are the 
points where the ecliptic meets the 
equator. 
Those are the positions along the Sun's 
motion when it is along the celestial 
equator and, the names of those points 
are the vernal and the autumnal equinox, 
for reasons we'll talking about in a 
minute. 
And these it turns out are the positions 
which by convention we chose to define 
the merid, prime celestial meridian of 
zero hours of right ascension. 
So when, 
the sun is at the vernal equinox, it is 
at 0 hours of our dissension, 
this means that the sun is overhead at 
Sidereal time 0. 
Sidereal time is off from Solar time by 
12 hours. 
And so, the sun is at the vernal equinox, 
if you look back at the last lesson on 
March 21st, 
and it is at the autumnal equinox, 180 
degrees away 6 months later, at 12 hours 
of our dissension on September 21st. So 
March 21st and September 21st are the 
days of the year at which the sun is 
along the celestial equator and has 
celestial declination 0. 
And between those two after the sun 
passes the vernal equinox it moves into 
the northern hemisphere of the celestial 
sphere and its declination rises until at 
the full apex of its tilt. 
Its declination is 23.5 degrees north and 
then this happens on or around June 21st 
then it goes on September 21st it meets 
the equator and on or around December 
21st the sun is farthest south that it 
goes its declination is 23.5 degrees 
south. 
And, let's see how all of this relates to 
what we understand about the seasons. 
This demonstration will explain to us 
what the tilt of the Earth's axis, or 
equivalently in the Sun's motion around 
the celestial sphere, has to do with our 
seasons. 
So, 
what we have here is here's our celestial 
sphere of view, 
the green line is the celestial equator. 
The tilted line is the ecliptic, the 
motion, the line in which the Sun moves 
around the celestial sphere and, what we 
are seeing is that the Sun is located now 
at the line of 0 hours of right 
ascension, in other words, it is the 
vernal equinox. 
This is the position of the sun on March 
21st. 
Notice that at, on March 21st, the sun is 
at 0 hours at sidereal midnight. 
And so, it's sidereal midnight when the 
sun is overhead. 
In other words, when it is noon, indeed, 
sidereal time is off from local time by 
about 12 hours at the vernal equinox. 
So this is the position on March 1st. 
The sun is on the equator. 
What this tells us, 
if we switch to an orbit view is that the 
earth's tilt at this point is such that 
the sun is neither north nor south. 
In other words, the tilt is perpendicular 
to the direction to the sun, and if we 
look from above, we see that as the Earth 
rotates about its axis the line 
separating day from night, the sun side 
from the dark side of the earth, goes 
right though the North Pole and 
equivalently through the South Pole, so 
over the course of a day. 
Every point on earth, spends a half of 
it's time on the sun side, in daylight 
And a half of it's rotation on the dark 
side, at night time. 
Hence the word equinox, night and day, 
are equal length, everywhere on Earth and 
this happens, whenever the sun hits the 
ecliptic. 
In another words, on March 21st, when the 
sun is at the vernal equinox and again 12 
months 6 months later on September 21st 
when the sun is at the autumnal equinox. 
Now what happens as we moving back to the 
celestial sphere, 
as time goes by, the sun moves around, 
not the equator, but the ecliptic. 
So a few months later, the sun is now 
north of the celestial equator. 
It's declination at this point its right 
ascension is 3 hours. 
So let's move it a little bit more. 
its right ascension is 4 hours, that 
means, that about 2 months have passed. 
It is now, if it was March 21st it is now 
May 21st and the Sun's declination is 20 
degrees north, which means the Sun is 
well north of the celestial equator. 
That means that the Sun is directly 
overhead at a point of it's a terrestrial 
latitude 20 degrees. 
You see the direction from which the 
sun's rays are reaching Earth. 
And you see that the lines separating 
dark from light on Earth no longer goes 
through the poles, but that in fact the 
whole region around the north pole is 
encircled 
such that it is always in the sunlight. 
If we look from the Sun, we will see that 
this whole region around the North Pole 
is always visible from the Sun, 
and therefore, as the Earth rotates, the 
Sun never sets at the pole, or points 
sufficiently close to the pole. 
And the Sun impinges overhead at latitude 
20 degrees. 
And, as we move the sun farther along the 
ecliptic, it reaches its northernmost 
point in June, on June 21st, when the 
sun's declination is 23.5 degrees. 
The sun is now at the point where it's 
maximally north on the celestial sphere. 
What that means from the point of view of 
the orbit is that the Earth's North Pole 
is tilted, the direction in which the 
Earth's North Pole is tilted which is 
always towards, in the same direction is 
now the direction of the sun. 
So now, points all the way to within a, a 
circle of 23.5 degrees latitude around 
the north pole are visible from the sun 
and therefore have a continuous sunlight 
throughout 24 hours, 
whereas, a full circle of radius 23.5 
degrees around the South Pole is 
invisible to the sun, and the sun never 
rises there, and the sun is overhead at 
the tropic of, of Capricorn, the north of 
Cancer, the northern tropic, where the 
sun is overhead at a latitude of 23.5 
degrees. 
And this means that this is the point at 
which sunlight impinges most directly. 
This will be the point at which solar 
heating is, is most intense on Earth. 
And as the Earth continues it's orbit 
around the sun or equivalently, as the 
sun continues its path around the 
celestial sphere, it again reaches an 
equinox on September 21st at which point 
the separate lines separating light from 
dark passes through both poles. 
The sun is overhead at the equator and 
night and day are of equal length 
everywhere on the planet. 
And as the sun continues its path to its 
December solstice. 
In December, the Sun is, 
as far south of the celestial equator as 
it gets. 
It's at a declination of negative 23.5. 
At this point, the sun is overhead at a 
latitude of negative 23.5. 
The entire Antarctic Circle latitudes 
within 23.5 of the south terrestrial pull 
have 24 hours of sunlight and latitudes 
to within 23.5 of the north terrestrial 
pole have 24 hours of dark. 
The sun never rises in the North and 
never sets in the South and maximal 
heating is at southern temperate 
latitudes. 
And I hope that this picture, if you play 
with it a little bit, will clarify the 
relation between the 
orbit of the Sun around the celestial 
sphere, the tilt of the Earth's axis 
relative to the Sun, note that the, where 
its axis points in the same direction 
towards the celestial North Pole as the 
Earth rotates around the sun and 
the changing, changes in length of day 
and night and in terrestrial heating. 
So, let's summarize this. 
Let's summarize what we've learned. 
between, [COUGH], March 21st and 
September 21st, the sun is north of the 
celestial equator, increasingly so until 
June, and then decreasingly so from June 
through September. 
And then from September through March, 
the sun is south of the equator, 
increasingly so until December, and then 
decreasing from December through March. 
When the sun is, either, say to the north 
of the equator, then days are longer in 
the northern hemisphere and the sun is 
higher in the sky in the northern 
hemisphere, because it's declination is 
closer to the declination of your Zenith, 
if you're in the northern hemisphere. 
We saw that your, Zenith is located in 
declination equal to your latitude. 
The sun will closer to that, 
Zenith, at its highest point if you're in 
the Northern hemisphere, when the Sun is 
north of the celestial equator. 
And, 
we saw that the Sun's rays impact the 
earth more directly in the northern 
hemisphere, so climate is warming in the 
North and cooling in the southern 
hemisphere. 
And of course, the reverse is true from 
September to June when the Sun is south 
of the equator. 
Inside the Arctic circle, at least for 
some part of that time, the Sun becomes 
circumpolar and the Sun comes so far 
north that it's close enough to the 
celestial north pole never to set four 
regions North of the Arctic Circle, for 
at least one day, of course, at the 
precise North Pole, the sun is 
circumpolar for 6 months. For precisely 6 
months, so long as the Sun is North of 
the celestial Equator, the Sun never sets 
at the North Pole and the Sun never rises 
at the South Pole and then the inverse is 
true as when the Sun is South of the 
celestial equator. 
And, twice a year, at the equinoxes, 
March and September 21st, day and night 
are of equal length everywhere. 
And then, there are these two tropics, 23 
latitude, plus or minus 23.5 degrees. 
If you live somewhere between those two 
tropics, there will be some day a year 
when the sun is directly overhead at 
noon, and passes through your Zenith. 
The sun, interestingly, is over only at 
your Zenith, directly overhead on the 
Equator, 
precisely on the equinox. Let's apply 
what we've learned to answer the 
question, 
how high the sun is at noon? And since 
we've started with Aristotle, let's 
continue with Aristotle. 
We'll be working at Athens latitude of 
37.7 approximately, degrees north. 
Remember, that noon is the time when the 
sun is highest in the sky, 
that's its meridian crossing, 
and so it's altitude or its Zenith angle, 
will be determined precisely by the sun's 
declination, and at the equinoxes. 
March or September 21st, the sun's 
declination is 0 degrees, Those are the 
days when the Sun crosses the celestial 
equator and for the course of a day we 
can imagine the sun fixed along the 
celestial sphere because it only moves by 
about four minutes along the celestial 
sphere. 
And if the sun is at declination 0, then 
at noon, it's Zenith angle. 
Is just the difference between its 
declination and the declination of our 
Zenith, which is given by our latitude. 
Its Zenith Zenith angle is 37.7 degrees, 
which means its altitude is 90 degrees 
minus 37.7, which is I believe, 52.3 
degrees. 
At the summer solstice and the summer 
solstice is June 21st, so this is the 
northern summer solstice, the sun has 
reached a northern declination of 23.5 
degrees north. 
The Zenith angle, therefore, at noon, is 
the difference between our latitude and 
the sun's declination, which at this 
point is only 14.2 degrees. 
So, at noon in midsummer, the sun in 
Athens will reach an altitude of 75 point 
eight degrees in the sky. 
On the other hand, on December 21st, the 
sun's declination is now negative 23.5 
degrees, 
so its Zenith angle. 
When it's as high in the sky as it gets 
at its meridian crossing is the 
difference between plus 37.7 and negative 
23.5, which is if I got it right. 
61.2 degrees, which means the sun's 
maximum altitude in the winter is only 
28.8 degrees. 
You know that, 
the sun solar heating in Athens is 
minimal in December 21st/. 
December twenty-first is not by far the 
coldest day of the year in the northern 
hemisphere. 
the coldest day is usually somewhere 
around February. 
There's something like a thermal inertia. 
There's a time, 
it takes time for the Earth to respond to 
the, the change in solar heating. 
there are many other complicated factors 
that govern climate heat exchange between 
the equator and the poles and so on. 
But to good approximation, solar heating 
is maximal at the summer solstice, 
and therefore, in June, and about 2 
months later in August, 
the temperature has reached maximum. 
By that time, solar heating is in decline 
and with the, the same 20 month lag, the 
Earth starts cooling off. 
Solar heating is minimal in December, 
always in the northern hemisphere, of 
course everything is reversed in the 
south. 
And 2 months later the Earth is as cold 
as it's going to be. 
And while we're discussing solar days and 
the Earth's orbital motion, there are two 
other small imprecisions in what I said 
that I need to correct. 
One is, I said that 24 hours are adjusted 
to be a mean a solar day and you can ask, 
what's so mean about a solar day? 
this was mean in the sense of average. 
It turns out solar days are not all th, 
same length. 
why is this? 
Well, the Earth's rotation about its axis 
is extremely uniform. 
We'll talk later about how it fails to be 
uniform, but it's almost exactly uniform 
certainly on the levels of precision that 
we're talking here. 
So the Earth's, well, the length of a 
sidereal day is very constant. 
But the length of a solar day is not, 
remember, the difference between a 
sidereal day and a solar day is 
associated to the sun's motion along the 
celestial sphere. 
The sun moves to the east by 4 minutes, 
every day, 
and therefore, there is a 4-minute 
difference between solar and sidereal 
days. 
But, the sun's motion to the east along 
the celestial sphere is not uniform. 
So, the first cause of this is that, the 
sun moves uniformly if the Earth orbits 
the sun uniformly. 
Even the the sun would move uniformly 
around the Earth, but it would move along 
this tilted path of the ecliptic. 
What this means is that, 
at, near the equinoxes, the sun's motion 
is not parallel to the celestial equator. 
It's not purely in right ascension, it's 
also changing declination, since, where 
as near the solstices the sun's motion is 
parallel at the maximum, and minimum. The 
sun's motion is now parallel to the 
celestial equator, and so, its motion and 
right ascension is slower near the 
equinoxes and faster near the solstices 
and the four minutes is an average. 
And that is one reason why the length of 
solar days is not uniform. 
There's another correction to this. 
In fact the rate at which the Earth 
orbits the sun or equivalently, the sun 
orbits the earth, is not precisely 
uniform even along the ecliptic. 
This is because the Earth's orbit is not 
precisely circular, and the Earth is, in 
fact, very slightly near the sun in 
January than it is at any other time of 
the year. 
At that time, the sun's apparent motion 
in the sky was fastest. 
And 12 months later it is slowest, and so 
even along the ecliptic, the Sun's motion 
is not precisely uniform. 
Moreover, 
I kept talking about the fact that the 
Earth's north pole maintains its 
orientation in space as the Earth orbits. 
So that, it always points it in this 
direction of the celestial north pole 
where sits some particular star, say the 
pole star. 
And this, it turns out is also 
inaccurate. 
we are building up more and more 
precision into our model, and in fact, 
the Earth is spinning about its axis. 
It's acted upon, as we will see, by the 
sun and the moon applying tidal forces. 
And like a spinning top spinning on a 
table and acted upon by gravity the 
Earth's axis wobbles in the same way that 
a spinning top wobbles except it's a 
little bit tricky. 
A spinning top will wobble, in the same 
sense in which it is spinning. 
The Earth's 
polar axis actually wobbles, in the 
opposite sense to the sense it rotates. 
In the other words, the celestial north 
pole moves to the west along the 
celestial sphere. 
So, the Earth's North Pole does not 
always point in the same direction as 
I've been saying so far, 
instead it's tilted by 23.5 degrees to 
the orbit, but as the Earth rotates to 
the east, the celestial pole very slowly 
wobbles in a big circle in the sky of 
radius 23.5 degrees, about the 
perpendicular to the orbit. 
And this wobble called the precession 
takes about 26,000 years. 
What that means is that the point that we 
define as the celestial north pole moves 
in with time. 
And what this means, is that, what we now 
call the Pole Star, will in a few 
thousand years no longer be the Pole 
Star. 
The Earth's North Pole will face in a 
different direction. 
More intriguingly, 
the celestial equator changes as the 
earth wobbles, 
the tilt of the celestial equator 
relative to the orbit or of the ecliptic 
relative to the celestial equator, is 
always 23.5 degrees but in different 
orientations. 
What this means is that the equinoxes, 
the hinges at which the point at which 
these two circles are hinged move around 
the celestial equator. 
Hence, this whole wobble leads to what is 
called the precession of the equinoxes. 
The vernal equinox shifts, then the 
coordinates of all stars shifts because 
the origin, 
the intersection of the equator, with the 
meridian, including the vernal equinox 
moves, and so, the coordinates of a given 
star change. 
This is most clear if you think about it. 
The star which now has the celestial 
coordinates declination 90 north, which 
is the north star, will not have those 
same coordinates very long as the Earth 
continues to wobble. 
And, so when you look at right ascension 
and declination for a given star it will 
be given in terms of some epoch, 
most typically epoch J2000. 
That means these coordinates are given 
relative to the position of the celestial 
equator and the vernal equinox, 
as of the orbital parameters on January 
1, 2000. 
Back in Athens, it's still November 27th. 
What I've done is I've, attempted to have 
the software keep us centered on the 
north celestial pull, so that we will be 
following the north celestial pull as 
time goes on and I will make time move by 
centuries. 
And what you will notice is that over 
time the celestial pull moves. 
the celestial pull started out very near 
this pull star of Polaris and the, the 
celestial pull moves and it moves in this 
great big circle in the sky. 
So that 
by the time you get to 10,000 or 11,000 
AD Vega is closer to being a pole star, 
and of course, in 26,000 years, the pole 
star will of the, the celestial pull will 
have completed its wobble and we will 
again have Polaris as a good North Star. 
Let's try to see that again, and in this 
at this point we should pay attention to 
the position of the vernal equinox. 
Of course, as the pole star moves, so too 
does the vernal equinox. 
Remember that it is currently in the 
constellation of Pisces, you can see that 
its moved quite a distance along the 
equator. 
remember, it completes a full circuit of 
the equator and over this time and I'm 
now dialing time back to the present, and 
the vernal equinox is moving to the east 
along the celestial equator. 
And as we approach the present, 
we find that around the present 1000 
years ago, the vernal equinox is indeed 
in the constellation Pisces, 
and as you move time forward, you will 
see that on or about 2600 A.D., the 
vernal equinox leaves the constellation 
of Pisces and enters the constellation of 
Aquarius. 
That means we are currently living in 
what is called the Age of Pisces and as 
of 2600 and some, we will be in the Age 
of Aquarius. 
If you look at 20, 2000 B.C. 
or so the time that pyramids in Egypt 
were being constructed, the North Star 
that we use today was nowhere near the 
north celestial pole. And in fact dating 
of pyramids, is based to some extent, on 
astronomical conjectures of what it was 
the Egyptians used as a north pole and 
there's real information to be gleaned 
from this wobble of the Earth's pole. 
We started out trying to understand which 
stars were going to be visible when. 
This was, in itself, enriching, along the 
way, we figured out the reasons for the 
seasons. and, 
we started investigating this sort of 
arcane phenomenon of the wobble of the 
Earth's axis and the precession of the 
equinoxes and came up with an 
understanding of ways to date the 
Egyptian pyramids based on which North 
Star they aligned to. 
So, we're making progress in 
understanding the moving parts of this 
model of the universe that we're 
constructing. The most conspicuous 
absence is the moon and it is to that 
that we turn next.