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.