. 
What did we see out there? 
Well, we saw that the universe around us 
is pretty fixed in it's shape as far as 
maintain their pattern. 
And then, 
we saw in addition that the entire 
pattern appears to rotate around us 
moving from east to west. 
Now, we have a modern understanding of 
all of this, of course. 
The modern understanding is that we live 
in a large three dimensional universe, 
the stars are very, very distant and 
that's why they appear not to move 
because they're so far away that their 
motion is irrelevant. 
And the reason the entire fixed universe 
appears to us to be rotating from east to 
west is simply that we, living on the 
earth, are rotating from west to east and 
so from our point of view things appear 
to be rotating. 
If you think everything is spinning, it's 
probably you that are. 
And so we live on this rotating spaceship 
earth and its daily rotation from west to 
east makes the universe appear to us to 
be rotating daily, from east to west. 
Very nice. 
We have an understanding what goes on. 
the picture that we see in the sky 
however, is a two dimensional picture. 
We do not have a good way of measuring or 
noticing the distance to stars. 
All stars appear to us to be essentially 
very far away. 
Though how far, 
will be a story that we'll have to 
develop later. 
Before we get there, at the level of just 
watching the planetarium show, that is 
the heavens, what we need to describe is 
direction. 
So we may as well assume that all the 
stars are at one very fixed large 
distance from us. 
In other words, they span a great big 
globe surrounding a terrestrial globe. 
We call it the celestial sphere. 
And the positions of the stars are fixed 
on the celestial sphere, because that is 
how they are. 
So here's Aristotle in 350 B.C. 
looking down his nose at the ancients who 
thought that it required gods in heaven 
to maintain the stars in their fixed 
positions and their regular periodic 
motion. 
Notice, Aristotle, there are laws of 
nature that compel the stars to do what 
do we see them doing. 
that is very much the spirit in which we 
will work, though we will try to take it 
a step further and actually understand 
those laws, [COUGH] comprehend them by 
making measurements down here on earth, 
because our laws are going to be 
universal. 
For Aristotle, there's a set of laws 
governing terrestrial phenomena and a 
completely different set of laws 
governing phenomena in the heavens. 
But, before we get to all of this, we 
need to describe the planetarium show. 
Where it is that we see which stars at 
what time? 
How the whole thing moves? 
And for this, we can follow the 
mathematical model, which is what 
Aristotle is describing, 
which is this picture of the stars fixed 
on this large celestial sphere. 
Inside this sphere, the earth rotates 
daily, from 
west to east or if you want, the entire 
celestial sphere rotates daily from east 
to west with the earth fixed inside it. 
From the point of view of lines of 
vision, 
well the, the celestial sphere rotates or 
the earth rotates inside the stationary 
celestial sphere is a matter of complete 
indifference that the two are completely 
equivalent. 
So we have the celestial sphere on which 
the stars are fixed. 
And what we need to get a more precise 
mathematical description of where stars 
are is we need to be able to label points 
on the celestial sphere, because 
somewhere on the celestial sphere, say 
over here is the constellation Orion, 
somewhere on the other side of the 
celestial sphere, say over here is the 
constellation Cygnus. 
You can say a star is in Orion, but if 
you want to be more precise than that, we 
need to say where exactly in Orion it is. 
We need the way of specifying positions 
on this sphere or directions in the sky. 
Luckily, we know very well how to 
describe positions on a sphere, because 
remember again, we live on a sphere. 
We use precisions on the sphere all the 
time. 
We specify points on earth by giving 
their latitude and longitude, 
those are the coordinates on the earth, 
reminds us what that means. 
Well, the earth, because it spins as an 
axis with poles where the axis meets the 
surface, a north and a south pole. 
Splitting the distance between them is 
this great circle of the equator and we 
measure latitude as the angular distance 
north or south of the equator. 
So that the north pole is at latitude +90 
and the south pole is latitude -90 and 
picking the latitude gives you a 
particular circle, a parallel, say the 
38th parallel. 
If you live at the latitude at 38 north, 
you live somewhere along this circle. 
To specify where along the circle, we 
split the earth into these orange slices 
with great circles passing through the 
poles, those are lines of longitude or 
meridians. 
And if you know what parallel and 
meridian you are on, that's a street 
address and then you know where you are. 
Now picking those 0 of latitude is very 
natural, it was the equator that split 
the distance to the poles. 
Picking the 0 of longitude is a more 
tricky business. 
We know that we measure longitude from 
this particular meridian, the prime 
meridian. 
What makes it prime is that it passes 
through the Royal Observatory in 
Greenwich, England. 
Of course, that's a political decision 
which tells us that there is no natural 
way to set a 0 meridian. 
You just pick one and then you measure 
longitude east or west of that in 
degrees. 
So that say Aristotle in Athens, Greece 
is 37 point something degrees north of 
the equator and about 24 degrees east of 
the prime meridian that specifies where 
Athens lies. 
Let's review in a minute what we exactly 
mean by this. 
So here's a model of earth and on this 
model, I can add the poles, because the 
earth rotates. 
So I will orient the earth as we usually 
do with the north pole on top and here 
are two diameters through the center of 
the earth, the axis, and another 
perpendicular diameter. 
Here is the earth's equator, as best I 
can draw it. 
And we are going to place some observer 
at some particular point on the surface 
of the earth. 
And, the position of that observer is 
determined by measuring this angle, which 
is his latitude. 
And that determines a line of latitude, a 
parallel, on which the observer lives, 
but we need to specify which of the 
points on this parallel is the point 
occupied by our particular observer. 
Notice that this parallel is, in fact, 
the trajectory of the observer's motion 
as the earth makes its daily rotation, 
the observer who is now here, will, 12 
hours later be on the other side. 
And in general, this is your trajectory 
as you move around the world once a day. 
And so, to specify that, of course we 
drew our lines of longitude, our 
meridians. 
We draw some meridians and pick which one 
of these you live on and that tells you 
where you, where along this line of 
longitude you live. 
Now, where you are on earth, of course, 
also determines among other things. What 
it is of the sky you can see? 
So, if you live at this particular point, 
then this roughly measures your horizon, 
you can see the part half of the sky that 
lies above your horizon and not the half 
that lies below it. 
Your horizon changes as the earth 
rotates. 
12 hours later, you will see this half of 
the sky and fail to see that half of the 
sky. 
We have coordinates on earth. We know how 
to specify coordinates on a sphere. 
Let's take this to the next level and 
apply the same idea to design coordinates 
on a celestial sphere. 
So, here we have the model that I was 
describing. 
We have the earth sitting in the middle 
of a large celestial sphere. 
You could imagine that the earth is very 
small, the celestial sphere very large. 
Celestial sphere 
inside the celestial sphere, the earth 
rotates from west to east or equivalently 
the celestial sphere rotates from east to 
west. 
Either way extending the earth's north, 
south axis we find an axis for the 
relative rotation. 
So this is the celestial sphere like the 
earth comes equipped with a north and a 
south pole, 
that the north celestial pole is the 
point that you would see in the sky. 
If you stood at the earth's north pole 
and looked up straight up, 
and similarly, the celestial south pole 
is what you would see if you stood at the 
earth's south pole and looked straight 
up. 
And so, we naturally have, splitting the 
distance between them, 
a celestial equator which is either the 
set of all points that are 90 degrees 
away from either of the poles. 
Or, if you wish, take the earth's equator 
imagine lighting a light bulb inside that 
earth there and projecting the shadow of 
the equator all the way to the celestial 
sphere, that gives you the celestial 
equator. 
And once we have a celestial equator, we 
can measure the latitude of a star. 
Again, just as we do on earth by 
measuring the angular distance from the 
celestial horizon to the star, 
so that this star here is at a latitude 
of north 44.7 degrees. 
We call celestial latitude declination. 
As I move the star around the sky, its 
declination will change. 
Declinations south of the celestial 
equator are negative, declinations north 
of the celestial equator are positive. 
And, now as the Earth rotates we observe 
if you want as the celestial sphere 
rotates from east to west, the star will 
move along this line of latitude. 
And just as on earth, we need to add to 
the specification of declamation some 
specification of longitude. 
And just as on earth, we pick randomly a 
line a meridian, 
a projection of some meridian on earth at 
some time a circle that crosses both 
poles. 
We call that the 0 longitude. 
Celestial longitude is called right 
ascension. 
So this is the 0 right ascension circle. 
it is chosen so that it intersects the 
celestial equator and then some 
particular star, in this case, this star 
is in the constellation Pisces. 
And so choosing the longitude of that 
particular star in Pisces, we call that 0 
longitude and we measure longitude east 
along the celestial sphere away from 
that. 
So that, for example, a star might be 90 
degrees away, 
you'd call that celestial longitude 90 
degrees. 
Or if it were 90 degrees to the west, 
you'd call that celestial longitude 270 
degrees. 
In a twist, celestial longitude is 
measured not in degrees but in hours. 
So instead of a full circle being 360 
degrees, the full circle describes 24 
hours of right ascension. 
That means a conversion rate of 15 
degrees per hour. 
So that a star that is 90 degrees away 
from the prime meridian, instead of being 
a celestial longitude 90, will be said to 
be at right ascension 6 hours. 
Remember, this is quite reasonable, 
because the celestial sphere rotates from 
east to west, which means if you're 
looking overhead and you see the seventh 
meridian, 
wait an hour. 
In an hour, the celestial sphere rotates 
15 degrees and the eighth meridian will 
be overhead. 
Wait 24 hours, the celestial sphere will 
have completed a full 360 degree 
rotation, you'll see the seventh meridian 
again. 
This is why measuring right ascension in 
hours increasing to the east totally 
makes sense. 
Now, before we proceed, there's a 
technical question we would need to ask, 
which is, I said the celestial sphere is 
large. What do you mean by large? 
Well, 
the statement I want to make, is that we 
need the celestial sphere to be large 
enough for this model to be 
mathematically accurate. 
That the lines of sight from any two 
points on earth to a given star can be 
taken to be parallel. 
In other words, we can assume that 
everywhere on earth is effectively in the 
center of the celestial sphere. 
And so in particular if in this picture, 
these two points, 
say A and B, are two different points on 
earth and this point O is the position of 
the star. 
I need to imagine the two within the 
accuracy of our measurements, the line 
from A or from B to O are in fact 
parallel or this angle, which I shall 
call alpha, is smaller then anything we 
can measure. 
The size of the celestial sphere here, 
representing the distance from points on 
earth to the stars, would be represented 
in this picture by this radius R. 
So what I need is a relation between 
distances and angles and the relation is 
going to be something we are going to use 
repeatedly in the class. 
So let's develop it and understand it. 
So, I have here this circle of radius R 
about the point O, 
and where I start with is the formula for 
the circumference S of the circle, 
and we all remember that the 
circumference of a circle is 2 pi R. 
Suppose I want to know the length of the 
segment of this circle, this arc between 
A and B mathematically, we denote this as 
AB with a little squiggle on top 
reminding me that this is the arc. 
So, this is not too hard. 
Imagine slicing a pie. 
To figure out how much crust you get, you 
take the amount of crust in the entire 
pie and then figure out how much your 
slice is. 
Your slice is a fraction of the entire 
pie that is a ratio of alpha, your angle, 
to 360 degrees. 
This tells you how much of the pie you 
have and I multiply that by 2 pi R and 
that gives me the arc length between A 
and B. 
And if I perform the full division here, 
this is alpha divided by 360. 
Excuse me. This is alpha / by 57.3 
degrees * R. 
Now the point that I want to make is that 
if the angle alpha is in fact small, then 
the difference between measuring the arc 
length from A to B and the actual 
distance between A and B, which is the 
straight line distance along the green 
line is very small. 
And so, for small alpha, the distance 
from A to B is similar or approximately 
given by the arc length. 
And this expression, called the small 
angle formula, is the one that we will 
use very often in the class. 
It relates distances to angles. 
I can rewrite it as alpha / 57.3 degrees 
is AB / r. 
So what this tells us is for example, if 
we can measure angles with some 
precision, we need the ratio of all 
terrestrial distance to the celestial 
sphere to be small enough that the angle 
you get from this expression is smaller 
than the precision of any of the 
measurements we do. 
Now you may see the same formula written 
with very different numbers. 
So compare what I wrote to this 
expression and it's, they look very 
different. 
this brings up another important point 
that I do not want to miss. 
neither of these expressions, of course, 
is wrong. 
they're just expressing the same angle in 
different units. 
In measuring small angles, we often 
measure units in fractions of a degree. 
And for the same Babylonian reasons that 
give us minutes and seconds, we measure 
fractions of a degree in arcminutes, 
sixty of which make up a degree and in 
arc seconds, sixty of which make up an 
arcminute. 
And if you perform the computation, you 
find that 57.3 degrees corresponds to 
this great number 206,265 arcseconds. 
So that these two expressions are in fact 
the same expression. 
the same angle is written as the 
different numbers of different units, 
which reminds us, that, when we're doing 
physical science numbers are in fact the 
ratios of one physical quantity to 
another physical quantity. 
If I say that my height is 178 
centimeters. 
That means its 178 times some fiducial 
unit, which is a centimeter. 
And so when we specify a relay, a 
physical property by a number it's very 
essential to remember the units in which 
it is expressed. 
And often, if you misunderstand the 
units, you get incorrect answers. 
And at this point you may be saying, wait 
Ronin, what about your consistency. 
You never specified in this expression, 
units for the length A, B, 
and for the length R. 
Those are two physical properties and 
what units are you measuring them, 
is that in light years? In millimeters? 
In angstrom? 
And the answer, of course, is that this 
expression will be correct no matter what 
units I use for A, B, and R as long as I 
use the same units. 
Because, remember, one way we wrote this 
formula was that alpha / 57.3 degrees is 
AB / R. 
Now we see that on the left-hand side, I 
have the ratio of two angles, 
an actual number. 
And on the right-hand side, the ratio of 
two lengths, 
and the ratio of the number representing 
AB, and the number representing R will be 
the actual ratio of the lengths so long 
as they're expressed in the same units, 
whatever those units are. 
But, this reminds us, A, 
that we have to be careful when doing 
physics. 
The correspondence between quantities and 
numbers involves units. 
And B, that the best way to avoid unit 
errors is to always write our expressions 
in terms of ratios of objects of the same 
unit, then those numbers are actually 
meaningful. 
Here again is our favorite view of the 
sky from Athens at nine pm on November 27 
and what I've had the software add here 
in green is the celestial coordinate 
grid. 
So here we have the celestial equator and 
if we raise our gaze a bit, then we can 
take a look at the celestial north pole 
here. 
And 
we see the lines of right fixed 
declination are concentric circles around 
the pole. 
And we see the lines of fixed right 
ascension are the lines of longitude 
spanning out of the pole. And, we see 
right here the prime celestial meridian, 
which as promised, meets the celestial 
equator in the constellation Pisces over 
here. 
And so we have the appearance of the 
celestial coordinate grid in the sky over 
Athens. 
And what we can see is that, as time 
progresses. 
notice, that here, we have, 
I will mark for us the position of the 
prime meridian. 
This is where 0 hours of right ascension 
is, 
and then, to east of it to by 30 degrees. 
We have the 2 hour right ascension line 
and 4 hour right ascension line and so 
on. 
And the reason that this makes sense is 
that, if I let 2 hours go by, by 
magically moving the clock. 
Notice that the 2 hour right ascension 
line, the sphere having rotated 30 
degrees is now exactly in the position 
where the 9 hour right ascension 
previously was. 
0 hours of right ascension has rotated 
over. 
And so hopefully this clarifies the way 
in which right ascension measures the 
rotation of the celestial sphere. 
To summarized what we've learned in this 
quick first video, we've learned that we 
can imagine the stars as fixed on a large 
celestial sphere, concentric with earth 
and surrounding it. 
the size of this sphere is governed by 
our favorite formula, the small angle 
formula. 
So the, angle, angle between the lines of 
sight to a star, for example, is AB / by 
R, 
where R is the radius of the celestial 
sphere, distance to the stars. 
AB is some distance between two points on 
earth and we pick R so big, that this 
angle is too small to be measured. 
the motion of the stars is carried by the 
fact that the sphere rotates daily from 
east to west. 
We measure positions on the celestial 
sphere by giving declination celestial 
latitude in degrees north and south of 
the celestial equator. 
And right ascension, measuring from some 
particular meridian in hours of right 
ascension moving to the east and this 
corresponds to the rate of rotation of 
the celestial sphere.