. 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.