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.