We now have a mathematical way to 
describe the position of a star in the 
celestial sphere. 
So we know where it is in the sky. What 
we need to do to finish our project is to 
figure out how to relate where a star is 
in the sky to where on earth it's visible 
at what time. 
And so let's understand exactly what this 
translation process is so we can make our 
next move. 
The challenge that we have to meet is 
explained in this simulation. 
Here we have the celestial sphere again 
surrounding the earth. 
The stars are fixed on the celestial 
sphere. 
We've equipped the celestial sphere with 
its equator and its prime meridian so 
that we can tell 
positions of a star relative to these 
coordinates. 
And then we have an observer here at some 
point on Earth. 
We want to understand what that is that 
the observer sees at any given instant as 
the whole thing rotates. 
And what the observer sees is the half of 
the sky that is above his horizon. 
So his horizon is this circle here. 
Which is an extension of the plane of the 
Earth near where their observer is to 
meet the celestial sphere. 
And this point over here. 
The point that he looks straight up above 
over his head will be his zenith. 
And so the observer's point of view is 
given by this diagram, extending his 
horizon. 
And then rotating to make the observer 
vertical. 
To describe where a star is, he will 
describe how high it is above his 
horizon. 
That's called the star's altitude. 
Here's a star at an altitude of 30 
degrees, and here's a star at an altitude 
of 43 degrees. 
And very often, we will express the same 
aspect of a star's position in the sky. 
By giving instead, its angular distance 
from the zenith. 
This is called the zenith angle, which is 
simply 90 degrees minus altitude. 
Once you've given either the altitude or 
the zenith angle, you've parameterized 
essentially a line of constant latitude. 
And we need an analog of longitude. 
we measure local longitude away from a 
particular meridian. 
The, meridian we use is the one line in 
the sky that. 
Starts at our northern horizon, goes 
through Earth's zenith, and intersects 
the horizon due south of us, so this is 
your locum meridian. 
It splits the sky into an eastern half 
and a western half and to des, designate 
the position of a star, we measure its 
angular distance from this meridian 
moving to the east, so that a star that 
is due east will be at azimuth 90 
degrees, and then various altitudes. 
This is how you describe to someone which 
way they need to look to find a star. 
Summarizing again our coordinates in our 
local view. 
We measure the position of a star by 
giving either its altitude above the 
horizon or its angular distance from the 
zenith. 
These two add up to 90 degrees and it's 
azimuth. 
The angle from north measuring east from 
zero to 360 degrees. 
And then we add our special points. 
The zenith the point directly overhead at 
altitude equal 90 degrees or zenith angle 
equals to zero degrees. 
And the zenith has no, azimuth in the 
same way that the North Pole has no well 
defined longitude. 
We have the horizon. 
The collection of all the points at 
altitude equals zero degrees, or zenith 
angle equal 90 degrees. 
And then we had the local meridian. 
The line that divides the sky into an 
eastern and a western half. 
This is the line that meets the horizon 
in both the north. 
And the southern points. 
And so this these are all the points with 
Azimuth equal zero degrees for the 
Northern half from the Northern horizon 
to the zenith or 180 degrees for the part 
from the Southern horizon to the zenith 
that is the local meridian. 
These are all the stars that are neither 
east nor west of us. 
Now what we need to understand is how to 
translate positions on the celestial 
sphere in the celestial sphere 
coordinates to positions in the sky in 
this altitude in this Azimuth Coordinate 
System. 
And so, here we have our little Earth, 
inside a large celestial sphere. 
Remember, the celestial sphere should be 
huge. 
We have the poles of the celestial sphere 
here. 
We have the celestial equator over here, 
and two diameters drawn for convenience. 
And so we will place our observer over 
here at some particular latitude. 
And, this observer will come with a 
horizon, a half of the sky that he is 
able to see and we've drawn this before. 
This is the line, notice that it should 
go through the center of the celestial 
sphere, because the celestial sphere is 
so large, and therefore, this is the 
visible half of the sky, this is 
2-dimensional version of what we did 
before in three dimensions and what this 
makes clear is if you look at the 
geometry for a moment, then this angle 
determining the latitude of the angle 
between this diameter and this line. 
Well this line is vertical so 
perpendicular to the horizon. 
These two diameters have a 90 degree 
angle between them, the net result is 
that this angle here is also the 
observer's latitude which means, 
remembering that this is the horizon, 
that your latitude is the angle from the 
horizon to whichever celestial pole is 
visible. 
In other words, if you can find the north 
star or the south celestial pole in your 
sky you now know which direction is north 
or south depending which hemisphere 
you're in and you also know your latitude 
because the higher the latitude higher 
the pole is in the sky clearly if you are 
at the pole then your latitude is 90 and 
the pole is directly over head, so you 
can use some understanding of the stars 
to figure out where you are. 
And in what direction you're going, stars 
are very useful for navigation. 
the other thing that this diagram shows 
us is that if this observer looks 
directly overhead, he finds his zenith, 
and his zenith is over here. 
And what this shows us is that his zenith 
is, at any given time, going to be at a 
declination, a distance from the 
celestial equator also equal to his 
latitude. 
And then, as the earth or the celestial 
sphere rotates, he will see different 
points on this line. 
A fixed declination directly overhead so 
the only points that are directly 
overhead ever for a given observer are 
the points whose declination is equal to 
your latitude. 
Here we have our favorite view of the sky 
in Athens again and in addition to the 
green celestial grid that I drew before 
centered on the north celestial pole, 
I've drawn in red, our local altitude in 
Azimuth grid, and so, we see our zenith 
directly overhead, we see lines of 
constant altitude in concentric circles 
around it and we see lines of constant 
azimuth. 
And as you see we are looking due south 
at the moment and so this is our prime 
meridian. 
The line that splits the sky in to an 
eastern half over here and a western half 
over there. 
And because the prime meridian goes 
through both the zenith and whichever 
celestial pole is visible because it in 
this case reaches the northern horizon it 
coincides with some specific celestial 
meridian for its long part, 
for the part including the zenith. 
At this point I think that I can tell 
that because this is zero hour celestial 
meridian this is the two hour celestial 
meridian. 
Our local meridian correspond with the 
celestial meridian for one hour. 
And when I allow time to elapse, then as 
we saw, the green grid on which the stars 
are fixed, will move from east to west, 
rotating about the celestial pole. 
The red grid, which is fixed to us, will 
appear to us to be stationary. 
So as time goes by, the celestial lines 
of, celestial meridians will move to the 
east. 
And, whereas the lines of fixed Asimoth 
will not. 
So that, if our local meridian 
corresponded previously to the meridian 
for one. 
hour of right ascension. 
Now we have two hours of right ascension 
overhead. 
Three hours. 
And so on. 
And so forth. 
This animation is truly wonderful. 
We have here two, the two views, 
simultaneously and synchronously. 
So, here we have our observer. 
I've put him in the vicinity of Athens, 
Greece. 
we see the celestial sphere inside it, 
the Earth, with an observer, position 
located upon it. 
And, as time goes by, the Earth will 
rotate inside the celestial sphere. 
Here, simultaneously, is the local view 
of the observer. 
We see, his horizon, his north, northern 
horizon over here, and, 
This, this is the half of the sky that 
the observer will see and as we begin our 
animation I've also drawn a few star 
patterns on the celestial sphere. 
We see here the big dipper near the North 
Pole Orion near the equator and the other 
southern celestial pole the southern 
cross and as our animation begins note 
that the observer looks to the east, 
Orion is just rising above his eastern 
horizon as it was exactly in our 
animation. 
in our simulation and our Orion will rise 
high in the sky, crosses meridian and set 
in the west. 
And this will happen as the celestial 
sphere rotates about the observer or as 
the Earth rotates, and you see Orion 
rising and setting. 
On the other hand, if we look at the 
motion of the Big Dipper we notice that 
as the Big Dipper moves, or as the Earth 
rotates, the Big Dipper's circle around 
the pole essentially never carries it 
below the horizon. 
It performs circular motion around the 
pole, and it never, 
ever since. 
And so, that makes the Big Dipper part of 
what we call As said the circumpolar star 
there is the whole region around the pole 
for each observer of stars that as they 
celestial sphere rotates or as the earth 
rotates never set below the horizon. 
And correspondingly if you look at the 
southern cross you see there are observer 
in Athens who will never be able to see 
it, it will never actually rise above his 
horizon so there is a, 
a similar region around the opposite pole 
that never rises. 
And so these stars are never visible. 
These stars are always visible. 
And then the stars in between rise in the 
east and set in the west. 
And it is this relation of the two 
systems of coordinates that we need to 
understand. 
Repeating as the sky looped, it's about 
the loop, first visible celestial pole. 
Stars close to whatever pole you can see 
never set. 
Stars near the opposite pole never rise. 
And stars in between, or near the equator 
rise in the east, move west across the 
sky, set in the west. 
Of course, for example if you're at the 
North Pole then you have half of the sky 
that is circumpolar and never sets. 
Half of it never rises. 
And stars can't rise in the east. 
And therefore at the North Pole their 
trajectories are just horizontal circles 
parallel to the horizon. 
And we see this in these collection of 
beautiful images. 
The two images on the left. 
Are star trails, in other words, somebody 
left the shutter open, around the South 
celestial pole on the left, and the North 
celestial pole on the right. 
So the stars in this image would have 
been moving counter-clockwise, and the 
stars in this image would have been 
moving clockwise. 
And then, on the left, we see a beautiful 
image of star trails near the horizon, 
this is Orion rising, and you can see 
from the fact that Orion is rising. 
And moving to the left you should be able 
to figure out that this means this image 
was taken in the Southern Hemisphere, and 
indeed, if you're a Northern Hemisphere 
denizen like me and you look at Orion, he 
seems to be doing a headstand. 
this is because in the Southern 
Hemisphere, one's head is pointing in a 
different direction in space. 
And so we have the beginnings of an 
understanding of what it is that we see 
in any given place and we're almost ready 
to make the calculation. 
And so the thing that we need is this 
description of the zenith. 
So I already described to you that our 
zenith at any point is a point on a 
celestial sphere whose declination is 
equal to our latitude. 
Now over the course of a 24 hour rotation 
of the celestial sphere 
which point along that celestial parallel 
we see changes. 
You see all of the stars over a 24-hour 
period which have that declination pass 
overhead. 
And so what we're going to is at any 
given instant we can look. 
Which celestial meridian contains our 
zenith? 
And 
We can look at the right extension of the 
celestial meridian and call that our 
local sidereal time and this will of 
course change with time as the celestial 
sphere rotates to the east or, to the 
west or the. 
Earth rotates to the east. 
Either way, your sidereal time will 
increase by about an hour each hour. 
And a complete rotation of the Earth or 
equivalent of the celestial sphere will 
advance your sidereal time by 24 hours. 
So, sidereal time is the name of the 
celestial meridian which coincides with 
the local meridian, the one that includes 
our zenith. 
And of course, over 24 sidereal hours. 
When this right ascension advances by 24 
hours, this is one full rotation of the 
Earth. 
So we can use the stars to measure time. 
The motion of the stars across the sky is 
a way to measure time, and indeed, in one 
sidereal hour the celestial spheres 
shifts by one hour of right ascension. 
One parallel giving way to the parallel 
moved by another hour. 
And until the 
20th century, this was, in fact, the 
definition of our units of time. 
A second was defined in terms of the rate 
at which the Earth rotates. 
this is a pretty stately and fixed rate 
and it's no coincidence, that, until the 
19th century, this was the most precise 
way to measure time that we had at our 
disposal. 
thing to note, of course, is that 
different locations on earth will measure 
different sidereal times, because 
depending on your longitude you will see 
overhead different celestial meridians. 
And as you move to the east you will see 
later and later sidereal time, because 
moving to the east, you see stars 
overhead that are farther and farther to 
the east along the celestial sphere. 
And so, fifteen degrees of longitude east 
advances your sidereal time by an hour 
but this varies continuously. 
So that moving east by twenty meters 
changes your sidereal time by some small 
amount. 
Returning to our view of Athens, we can 
now read off the sidereal time. 
In our picture, remember, the sidereal 
time is the time indicated by the 
celestial meridian that coincides with 
our local meridian, and we've seen that 
before, this is the one hour meridian. 
So the sidereal time in Athens at 900 
p.m. 
On November 27th was one a.m. 
and of course, as. 
We allow time to progress by an hour 
sidereal time will progress. 
We can see that by, allowing time to 
progress. 
And an hour later, it is 10:01.01. 
The sidereal time is changed. 
It is now two hours sidereal time. 
Now, 
we can use this, now that we understand 
the position of our zenith. 
And the 
location of our, your local meridian, we 
can use this to find a particular star. 
So let's suppose that at 900 p.m., we 
wanted to find a particular star. 
Let's imagine that we wanted to find, in 
the sky, this particular star, Alpheratz 
in the constellation Pegasus. 
So, if you look, Alpheratz you can look 
it up, has celestial coordinates. 
It has a declination of about 30 degrees, 
and a right ascension of about zero 
hours. 
It lies at the intersection of this line 
of, declination and the prime celestial 
meridian. 
That means that Alpheratz will rise in 
our East, 
become as high overhead as it's going to 
get. 
When it's on our local meridian and then 
start setting in the west. 
So the best time to view Alpheratz in 
fact, is when it is directly overhead. 
Because it is on the prime celestial 
meridian, best time to view the Alpheratz 
would have been at zero hours, at select, 
sidereal midnight. 
Which means at 9 p.m. we're an hour too 
late. 
But that's okay. 
We can fix this we can just though the 
wonders of technology go and observe 
Alpheratz at 8 p.m. 
Here we can move time backwards and 
indeed, as advertised, as a, at eight pm, 
Alpheratz is right on our celestial 
meridian, it has risen as high as it's 
going to rise and has not yet begun to 
set. 
So at that time, 
8 p.m. or more importantly. 
Midnight sidereal time, Alpheratz will be 
on our local meridian. 
Where will it be along the local 
meridian? 
Well, this is where Alpheratz declination 
comes into play. 
Alpheratz is at a declination of 30 
degrees. 
Our zenith, we said, because we are in 
Athens, is always at its declination of, 
about, 37 degrees. 
This means that the angle, the zenith 
angle of Alpheratz the distance between 
Alpheratz and our zenith at 
it's meridian crossing at the time when 
Alpheratz is as high in the sky as it 
gets. 
Well, its zenith angle is given by our 
latitude minus its declination. 
in general, the absolute value of this 
which in our case, is about, 30 minus 30 
or eight degrees. 
So, Alpheratz will attain a minimal 
zenith angle of eight degrees or a 
maximal altitude of 82 degrees above the 
horizon, quite high. 
Now, 
this tells us almost where it is. 
This defines it to within a circle but 
because it's on our local meridian, it 
can only be either north or south of the 
zenith. 
And because the declination of Alpheratz 
is less than our latitude, our zenith is 
at 38 degrees. 
Alpheratz only at, 30 degrees of 
declination. 
It's Azimuth will be 180. 
It'll be due south of our zenith at the 
time that it is as high as it can get. 
And so if we observe our Alpheratz at. 
sidereal midnight we know exactly where 
to find it. 
And now if we need to observe Alfrax 
later or earlier, we simply use the fact 
that we understand that over time the sky 
rotates about the celestial pole at the 
rate of fifteen degrees an hour so that 
if we wait an hour, and we indeed are 
observing at, 9 p.m. Alpheratz will have 
moved fifteen degrees to the next 
celestial meridian and indeed, that is 
what we will observe. 
So, we have a way of finding the 
approximate position of a star over the 
course of the night. 
Notice that there is an explicit formula 
that gives you altitude and azimuth as a 
function of sidereal time and declination 
and right ascension. 
The formula involves trigonometry, so we 
won't apply it. 
What I want us to understand is the 
principle from which it's derived. 
So, we've achieved what we wanted, we did 
not complete the mathematics because it 
would involve trigonometry. 
But I think we have the idea, we know, 
given the sidereal time and our latitude, 
how to figure out where in the sky, the 
pole is, which line of right ascension is 
directly over head, and then moving east 
or west we count an hour per fifteen 
degrees and we can find where the rest of 
the sky is, so we have an understanding 
which part of the sky will be visible to 
us at any given time at a position on 
earth. 
We've a sensually solved the problem. 
the one missing ingredient is that we 
still need to figure out what this 
sidereal time thing is because our 
watches do not measure sidereal time. 
It was one am sidereal in Athens when it 
was nine pm on November 27. 
If we could get the translation right 
we'd be set. 
So that's the next thing we have to take 
on.