Introduction to Astronomy, Week 1

Edited Lecture Transcriptsprovided by Derek Grainge during session 002, 2013-2014

1 Positional Astronomy Introduction

Today we're going to start our class, and the best way to start a class on astronomy, is to just go outside and take a look at the night sky. See some of the objects we're going to talk about in their actual natural habitat so to speak. There's far too many of us to fit in any one place and we're scattered too widely over the globe, but through the wonders of technology I am going to invite you, in a minute, to do a, an, virtual night sky tour with me, and I encourage all of you to take either a virtual tour or even better, if you can, to go outside and take a look up at the sky. Familiarity with the night sky is not the main object of this class. But it's a very pleasant side effect and I hope you'll enjoy it. So, without further ado let me invite you to join me virtually in field outside of Athens, Greece for definiteness, and we'll look up and see some of what we can see in the sky.

Welcome to our observatory. We've chosen to locate ourselves in the outskirts of Athens, Greece. It is November 27th at 9 p.m. And when our software is simulating the sky, and we're looking to the south, we have a very broad feel of the view of a full 180 degrees. So we can see all the way from the eastern horizon to our left, to the western horizon to our right. But because of the light pollution of a nearby city we see very few stars in the sky. This might be what can see, or it's certainly what I can see when I step outdoors, but the brightest stars in the sky are bright enough to shine against the glow. So we might see over here in the east, Betelgeuse and Rigel, and perhaps Capella and Aldebaran. And over here on the western horizon we'd see three very bright stars. Vega over here, Altair down there and Denad over here, and these make up what is normally called the summer triangle.

Let's step a little farther away from the city where more stars are visible, and try to get a better look. There that's better. With software it's much easier to turn off light pollution than it is in real life, and now we see the same view of the sky as it must have appeared before the invention of electric lights. And let's take a look and name some of the objects we see. Over here to the east we saw the two bright stars Betelgeuse and Rigel, which are now shown to be, seen to be, part of the famous constellation of Orion the Hunter. There's his chest and his legs, his belt from which hangs his dagger. His club or bow over here, his outstretched arm, he even has a head up there. And that would be Orion the Hunter, and Aldebaran will be part of Taurus the Bull. There's his head and his body and his long outstretched horns. Looking farther to the right we can identify some other famous groups. This w shaped object is associated with Cassiopeia the vain queen. And not far from here, this great square of Pegasus. And

from the great square Pegasus towards Cassiopeia, these two streaming lines of stars associated to Andromeda, Cassiopeia's daughter.

And up here Vega, the bright star we saw in the western sky, is part in fact of Orpheus Lyre. This is the constellation Lyra, and finally my favorite constellation - we talked about Deneb over here, and Deneb is just the tail of Cygnus the Swan. There's his tail and southern his heart. And a long, swan neck that ends at Albireo's head. And then his pretty, swept back, swan wings. You can certainly see a swan. There are many other constellations but I don't have time to show them all. Let's allow the software to do that for us.

Here I let the software designate for us the most famous of the constellations, and the brightest stars that comprise the shape or the asterism of the constellation. And we see Orion the Hunter and Taurus the Bull, and Pegasus the Flying Horse, and so on, as well as many others. And what we recognize is that many of them have their origin in Greek Mythology, at least in the northern hemisphere. The reason for this is. And the point I'm trying to make with this image is that if you show this picture, without the helpful blue lines, to someone who lived in Athens thousands of years ago, I don't know the, this, the philosopher Aristotle. He would not only recognize the stars he would name the constellations and he'd use the same names that we use. Because the pattern of the sky as Aristotle saw it on a fall day in Athens 2000 years ago would be extraordinarily similar, essentially identical to what we see today absent light pollution. There is this beautiful permanence in the sky. The stars that we see today are essentially the same as they've always been, and this is something that has made astronomy one of the oldest sciences. It's also something we can use to establish a geography of the sky. So that if I want to describe or if an astronomer wants to describe viewing an object that lies right here, even if it's not part of the asterism of Orion, he would say the star lies in the constellation Orion.

Astronomers have formalized this. By, in recent memory, actually dividing the sky into 88 regions. In the northern hemisphere they're associated ancient Greek constellations for the most part. And so now we precisely can distinguish a star that lies in the constellation, Orion from some nearby star. That lies in Taurus, and when we talk about stars lying in one constellation or the other, I will be referring to this treaty that divides the sky into the 88 constellations. The point that this again is trying to remind us of, is that a star that is in Orion now will be in Orion a century hence, and so there's this geography that we set up, and we can identify positions in the sky.

Removing for a moment all the distracting lines that the software was drawing, let's take a closer look at some of the objects we're seeing and try to pay attention again to what we're seeing. So we see large numbers of stars, we notice that they were different colors. Some of them are more reddish or orangeish or yellow, and some are blue. The colors of stars are in themselves an interesting topic, we'll talk about that. The distribution of stars is interesting. If you look you'll see that there are less stars here or here, than there are along this region of the sky. This stripe through the sky, and in fact, if we look closely we'll see that even in between the sky, stars, the sky here's brighter than elsewhere. This is of course, the famous stripe of The Milky Way. We'll want to take a close look at that.

Here's what part of The Milky Way looks when you take a slightly long exposure photograph. The exposure brings out many, many stars, and we see that this bright stripe in the sky that I call the Milky Way has a lot more structure. It's not just a white stripe. There are these darker areas and brighter areas, different colors, lots of stars. We'll take a close look at that and try to understand what the Milky Way is.

There are some other objects visible in the sky that we'll justify a closer look of the form that Aristotle would not have been able to take advantage of. If we look at Orion here, and if we look at stars, the three stars that form the dagger dangling from Orion's belt, and certainly the middle one. It’s pretty clear, it does not really look right. There's something fuzzy about that star. We’ll take out a telescope and get a closer look at that. The slightly fuzzy, suspicious star in the center of Orion's dagger that we mentioned, turns out in a moderate telescope, to be a lot more interesting than you'd think. In fact all of this stuff, all of these stars, all of this nebulous scattered light, blue and pink, these darker regions. All of this is what our eye sees as a slightly fuzzy star. And understanding what is going on in the Orion Nebula, is going to be something that'll be very exciting for us.

Slightly farther south over here just past Taurus, and in the Constellation Taurus in particular, is this weird collection of stars, a whole bunch of stars, clustered together very close, the collection of seven bright stars in the Constellation Taurus, call, traditionally called Pleiades in some cultures, The Seven Sisters different names for this obvious bright, collection. Here are the seven bright stars we see that there are many others near them. Just not bright enough for us to see, and even again, as in Orion, we see that there is this nebula stuff, there's this blue, shiny stuff that is not a star that is producing light. So some of the light we see is not coming from the stars at all.

And, if we look at this collection of stars then perhaps it's worth looking just below Pegasus over here. If we looked very closely there would be a faint and slightly fuzzy object, probably better seen with a telescope than the naked eye. In a moderate telescope the group, slightly faint fuzzy object in the constellation Pegasus that I mentioned turns out to be this beautiful globular cluster called M13. Again we see many, many, many stars far more in fact than in the Pleiades. We also see that they're all dimmer. The stars in the Pleiades were bright like this. These are dimmer objects, and they also appear to be very much more tightly clustered.

In the constellation of Andromeda up here in between Pegasus and Cassiopeia, if we look around here there's another one of those fuzzy objects, suspiciously not quite star-like. If you magnify it a little bit and collect light for a long time and process carefully, turns out to be this beautiful richly structured object, the Great Spiral Galaxy of Andromeda. I should point out that the fuzziness that we see is only the brightest central part of this. This object itself is about five times the size of the full moon in the sky. It's very large and very, very faint and requires careful image processing and light collection to get all this structure out. But nevertheless, our eyes were not misleading us, there's something interesting there, certainly worth investigating.

Farther to the west, remember the bright star Vega and the constellation Lyra. If you look over here in Lyra, we'll see with a telescope a faint fuzzy object. The slightly fuzzy object in the constellation Lyra that

we saw with a moderate telescope turns out to be this beautiful ring shaped nebulous light. lots of fun colors in this, and understanding what this ring structure is will be a lot of fun for us as we go along.

And finally, we'll take a close look at this moderately bright star Albireo, the head of Cygnus the swan. When you look at it more closely, it turns out to be not one, but two stars of differing brightnesses. There's a bright and a dimmer star, and of completely different colors. One of these is orange the other is described by various people as being either purple or blue or green, but certainly these two stars are very obviously different colors. Understanding why stars come in different colors, what these two have to do with each other will be another topic we'll spend some time on.

Hope you enjoyed the little preview of all the things that we can see that poor Aristotle could not, because he did not have telescopes. Let's go back to what it is that Aristotle did see. I've made a big deal of the fact that the sky is unchanging. But, in a sense I was cheating, all the time we've been talking it's still 9pm in our simulation. I'm now going to let time move at a somewhat accelerated pace. And then we will see that the sky is not in fact unchanging, things move. Orion is clearly moving from low in the Eastern horizon. To high in the south and eventually will set in the west, but things maintain their shape. The motion is not random, the motion is rigid. Of course pretty soon, the sun will rise and we might as well stop observing.

So let's collect ourselves after a good hot cup of strong Greek coffee, and summarize what it is we've learned. Back from our night under the stars, hopefully all energized and excited to understand all of things that we've looked at. Let's see what it is that I was trying to make clear from these observations.

We saw that we see many, many stars, we could see as many as 3,000 at a time on a clear night. And we saw that we named the groupings by ancient Greek names because the pattern that we see in the sky is unchanging. We also watched the sky move, but we noticed that the motion was a rigid motion. Betelgeuse and Rigel remain in the same relative position. They're both in Orion, in the same relative position throughout the motion of the sky. And in fact, have been in that position since humans started observing them. So we can set up geography of the sky. We can declare some part of the sky to be Orion, and in another part of the sky to be Cygnus. And this is a meaningful thing despite, or throughout, the motion of the sky. That said, the sky moves. Everything rises in the east and sets in the west. So what you see in the sky at any given time is not, of constant time. And our job for the rest of this week is going to be to understand and mathematically describe at some level of precision this motion and this permanence, and how we can predict what it is that we will see looking up from Athens at 9 p.m., in what direction in the sky.

2 The Celestial Sphere

What did we see out there? Well, we saw that the universe around us is, really fixed in its 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 a large three dimensional universe the stars are very very distant and that's why they appear to us 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 of 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 distances to stars, all stars appear to us to be essentially very far away well, how far will be a story that will have to be developed 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 the terrestrial globe. We call it the celestial sphere, and the positions of the stars are fixed on this celestial sphere because that is how they are.

So here is 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 the regular periodic motions. No, says Aristotle, there are laws of nature that compel the stars to do what 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, and comprehend them by making measurements down here on earth, because the laws are going to be universal. For Aristotle, there's a sense of laws governing terrestrial phenomenon, 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, throughout 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, whether the celestial sphere rotates, or the Earth rotates inside the stationary celestial sphere, is a matter of complete indifference, 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 could say a star is in Orion, but if you want to be more precise than that, you need to say where exactly in Orion it is. We need a way of specifying positions on a sphere, our 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 positions on this sphere all the time, we specify points on Earth by giving their latitude and longitude, those are the coordinates on the Earth, remind us what that means. Well, the Earth, because it spins, has an axis and 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 plus 90 and the South Pole is

latitude minus 90. And picking a latitude gives you a particular circle, a parallel, say the 38th parallel, if you live at latitude 38 North, you live somewhere along this circle. To specify where along this 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 what meridian you're on, that's a street address and then we know where you are.

Now picking those zero of latitude is very natural, it is the equator that split the distance to the poles. Picking the zero 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 zero on the meridian, you just pick one. Then you measure longitude east or west of that in degrees, so that say an 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 exactly we 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 centre 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. And you can see 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, is, you'd 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. In fact 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 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 we want as the celestial equates, sphere rotates from east to west, this star will move along this line of latitude. And just as on earth, we need to add to the specification of declination 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 zero longitude. Celestial longitude is called right ascension, so this is the zero right ascension circle. It is chosen so that it intersects the celestial equator and some particular star and in this case, this star is in the constellation Pisces. And so using the longitude of that particular star in Pisces, we call that zero 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 measure 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 at a star that is 90 degrees away from the prime meridian, instead of being at celestial longitude 90, will be set at right ascension 6 hours.

Here again, is our favourite view of the sky from Athens at 9 p.m on November 27. And when 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, 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 zero hours of right ascension is, and then to the 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 two hours go by, by magically moving the clock, notice that the two hour right ascension line, the sphere having rotated 30 degrees, is now exactly in the position where the zero hour right ascension previously was. Zero hours of right ascension has rotated over, and so hopefully this clarifies the way in which leg extension measures the rotation of the celestial sphere.

To summarize what we've learned in this quick first video, we learned that we can imagine the stars as fixed on a large celestial sphere concentric with earth and surrounding it. The motion of the stars was carried by the fact that the field 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 write ascension, measuring from some particular meridian in hours of right ascension will be to the east. And this corresponds to the rate of rotation of the celestial sphere.

3 The Local View

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 our 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 this 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 at 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 the 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. So the observer 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 in 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 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 our zenith, and intersects the horizon due south of us. So this is your local meridian. It splits the sky into an eastern half and a western half. And to designate the position of a star, we measure it's 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 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 the star by giving either its altitude above the horizon, or its angular distance from the zenith. These two add up to 90 degrees. And its 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 to 90 degrees or zenith angle equal to zero degrees. And the zenith has no azimuth, in the say 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 and 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 the two 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, their 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 on which hemisphere you're in. And you also know your latitude, because the higher your latitude, the higher the pole is in the sky. Clearly, if you're at the pole, then your latitude is 90, and the pole is directly overhead. 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, you will see different points on this line of 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 and 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 into 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 corresponds 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 Azimuth 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.

4 Sidereal Time

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’s 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 near the southern celestial pole, the Southern Cross. And as our animation begins, note that if 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 now Orion will rise high in the sky, cross his meridian, and set in the west. And this will happen as the Celestial sphere rotates about the observer, or as the earth rotates and we 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 sets. And so that makes the Big Dipper part of what we call a cir, the circumpolar star. There's a 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 that our observer in Athens will never be able to see it. It will never actually rise above his horizon. So there's 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 rotates above the 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 that 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 are a northern hemisphere denizen like me and you look at Orion, he seems to be a doing a headstand. This is because in the southern hemisphere, one's head is pointing at a different direction in space.

And so we have the beginnings of an understanding of what it is that we see at 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 the celestial sphere who

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 do is at any given instant we can look which celestial meridian contains our zenith. And we can look at the right ascension 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 length 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 sphere 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. The 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 15 degrees of longitude east advances your sidereal time by an hour. But this varies continuously, so that moving east by 20 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 9 p.m., on November 27th was 1 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. The sidereal time has changed, it's now two hours sidereal time. Now, we can use this, now that we understand the position of our zenith and the, location of our local meridian, we can use this to find a particular star. So let's suppose that at 9 a., 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 Alpharatz, 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 Alpharatz will rise in our east, become as high overhead as it's going to get when it's on our local meridian, and then starts setting in the west. So the best time to view Apharatz in fact, is when it is directly overhead. Because it is on the prime celestial meridian, best time to viewed Alpharatz would have been at zero hours, at 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 through the wonders of technology, go and observe Alpheratz at 8 p.m. Here we can move time backwards. And indeed, as advertised as, at 8 p.m., Alpharatz 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, Alpharatz will be on our local meridian.

Where will it be along the local meridian? Well, this is where Alpharatz's declination comes into play. Alpharatz is at a declination of 30 degrees. Our zenith, we said, because we are in Athens, is always at a declination of about 37 degrees. This means that the angle, the zenith angle of Alpharatz, the distance between Alpharatz and our zenith - ts meridian crossing at the time when Alpharatz is as high in the sky as it gets. Well it's Zenith angle is given by our latitude minus it's declination. In general the absolute value of this, which in our case is about 38 minus 30, or eight degrees. So Alpharatz 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 Alpharatz is less than our latitude, our zenith is at 38 degrees, Alpharatz only at 30 degrees of declination. Its 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 Alpharatz at sidereal midnight, we know exactly where to find it. And now if we need to observe Alpharatz 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 15 degrees an hour. So that if we wait an hour, and we indeed are observing it, 9 p.m., Alpharatz will have moved 15 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 tide 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 right line of right ascension is directly overhead. And then, moving east or west, we count, an hour per 15 degrees, and we can find where the rest of the sky is. So we have an understanding of which part of the sky will be visible to us at any given time at a given position on earth. We've essentially 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 1 a.m., sidereal in Athens when it was 9 p.m., on November 27th. If we could get the translation right, we'd be set. So that's the next thing we have to take on.

5 Where is the Sun?

We now understand pretty well how to tell which part of the sky is going to be visible at any given time at any given location on Earth. We have a missing ingredient which is this mysterious sidereal time. We need to understand sidereal time. Associated with this is another small omission. We've discussed the position of all and the stars on the Celestial Sphere, but we've forgotten one star, one that might have some impact on our lives here. Namely, our local star, the Sun. The Sun, like everything else, that is not on Earth, is somewhere on the Celestial Sphere in the sense that it rises in the east and sets in the west, as the Earth spins, or as the Celestial Sphere spins. And so, we can locate the Sun somewhere on the Celestial Sphere, but I haven't told you where. And where that is, is somewhat interesting because if you put the Sun somewhere on the Celestial Sphere, there's a whole section of sidereal time, a whole period

of the 24 hour rotation of the Earth, when the sidereal time is close to the right ascension of the Sun. When the stars overhead are, include the Sun and we do not do astronomical observations, at least not invisible wavelengths. And so it’s somewhat important to figure out where the Sun is, and this will turn out to be at the root of this sidereal time issue. And so let's see how that works.

This simulation might help us to explain what's going on. We know that in addition to spinning around its axis daily, the Earth also orbits the Sun once a year, equivalently if you prefer a stationery Earth the Sun orbits the Earth once a year from again from the point of view of who sees what when, the two are completely equivalent, The Earth orbits the Sun in the same sense in which it spins about its axis. And what that means is that as we see it, the Sun also orbits us in the same sense which the Earth spins about its axis. In other words, the Sun appears to orbit Earth once a year moving from west to east along the Celestial sphere. What that tells us is that the Sun can not be assigned a fixed right ascension, its right ascension changes over the year. As the Earth orbits, if we start at some point on the Earth's orbit, with the Sun in this direction in the sky, imagine the Celestial Sphere outside this figure, at celestial midnight. Then three months later, when the Earth has moved to this point in its orbit, the Sun has changed orientation relative to the Earth. The Sun's right ascension is now six hours and further more three months later, the Sun's right ascension is 12 and three months further on, it is 18 hours and after a complete rotation, the Sun will have made a complete circuit of the Celestial Sphere, moving to the east in the direction of increasing right ascension, so that the Sun's right ascension goes through a full 24 hours in the course of a year. You can't assign the Sun a fixed position on the Celestial Sphere. Rather, it moves along the Celestial Sphere to the east at the rate of one rotation per year. Now note, the Celestial Sphere is moving from east to west as it rotates around the Earth. The Sun is moving along the Celestial Sphere from west to east.

This is important, so let's think about it again. As the Earth spins about its axis once a day, it also orbits the Sun once a year moving in the same direction, which means as seen from Earth, the Sun orbits us once a year. So that means the Sun moves along the Celestial Sphere moving from west to east. In the direction of increasing right ascension to the east. And it completes one revolution per year around the Celestial Sphere. This means that which stars are invisible because they're only up at the same time as the Sun? Well, that changes over the course of a year, so all stars get their chance to shine, so to speak. That is good. We get to see the entire sky. It also means that since the Sun is moving across to sky from west to east while the Celestial Sphere is rotating from east to west. The Sun is carried by the Celestial Sphere, so it rises in the east and sets in the west. But its motion across the sky is a little bit slower than the motion of the stars. How much slower? Over the course of a year, the Sun moves backwards along the Celestial Sphere by one complete revolution. Let's see that extra day one more time.

This simulation will help us to understand the consequences of the fact that as the Earth spins, it's also orbiting the Sun, and therefore the Sun is moving along the Celestial Sphere. Yo make things a little more clear, we have pretended that the sidereal day, or a day, is not 24, but 240 hours long. This will make the effect much clearer. So imagine that we begin our simulation. With the Sun directly overhead for this tallest observer on the Earth. So, this observer will presumably think that it is noon, and now, let a sidereal day go by. The Earth will have completed a 360 degree rotation, notice it's oriented in exactly the same way it was before. However, it is not yet noon. The time it takes from noon back to noon again

is longer than a sidereal day because in the course of this rotation, exaggerated by a factor of ten, the Earth has moved along its orbit. There's this extra bit of rotation required to get to noon. From the point of view of Earth, this is a consequence of the fact that over the course of the day, as Earth spun about its axis but also moved, the Sun moved in this direction to the east along the Celestial Sphere. We can do that again. 360 Degrees rotation, and a little bit required to realign us to the sun.

Indeed. The time from noon to noon is longer than the time it takes Earth to rotate by 360 degrees by about 1 over 365 of a day because the Sun moves back along the Celestial Sphere by 1 over 365 of its complete annual rotation of the Celestial Sphere. That's about four minutes. So which of these is 24 hours, do you think? Is an hour going to be a 24th of the time it takes the Earth to rotate 360 degrees? Or would you define an hour to be a 24th of the time from noon to noon? Clearly you want it to be the time from noon to noon. Our clocks keep solar time. The reason for this is, because as we saw, while you, the difference between a sidereal day, 24 hours, and the time from noon to noon is only four minutes, these four minutes accumulate over the course of a year to a complete day. If you try to work with a sidereal clock then if you started it out such that noon fell at 12 hours sidereal, six months later noon the Sun overhead would fall at zero hours sidereal, and if you tried to operate by the sidereal clock you would want to have lunch when it was darkest. This does not work very well for agriculture, though it is very good for astronomy because sidereal time keeps track of the stars. The clocks that run our life, of course, keep not sidereal, but solar time. We call that local time. And so, what that means is that these solar clocks. Or local clocks that keep solar time, run slower than a sidereal clock. 24 Sidereal hours, one 360 degree rotation of the Earth, is less, than 24 solar hours, the time kept by our clocks, by about four minutes, a little less than four minutes. One over 365 of a day to be precise. So our sidereal clocks run faster than, the clocks that are used to measure time. So, how do you relate the two? Well, we have this issue of two clocks, one of which runs faster than the other. They're both 24 hour clocks, and so, at some point they agree. And then the faster clock runs ahead. And it will remain ahead until it has completed one full 24 hour rotation more than the other clock. And this takes, by definition, precisely a year. So once a year, there is a day when sidereal and solar clocks agree. And by convention, that day is, set to happen - we'll see why in our next lesson - on or about September 21st. So on or about September 21st wherever you are on Earth your sidereal time is approximately equal to your local time. Now, I should qualify this, remember local time varies from position to position continuously. Local time also varies from position to position. If you move about 15 degrees east in longitude, then local time is an hour later. That's what time zones are about. But for the convenience of arranging train schedules, we don't let the local time vary continuously so that each town has its own local time. Local time is fixed over an entire 15 degree slice of the Earth and then jumps by an hour, whereas sidereal time is defined locally so varies continuously. So, even when I say sidereal time equal to local time, I mean to within half an hour which of the precision that we are working with, is good enough. So, on September 21st to within this half an hour in precision, sidereal time is equal to local time and then if we know that, we can compare that to any time before or after September 21st because we know that the sidereal clock runs faster. A solar day is four minutes longer than a sidereal day. So, a day later, the sidereal clock, will have become run ahead, and be four minutes fast. And D days after September 21st, the sidereal clock is ahead of the local clock by D times 4 minutes. D days before September 21st. It was behind by four minutes and catching up. Now, of course, this is approximate. Four minutes is not

precise. We are, in any event, ignoring time zones. Beware of daylight savings time. This expression, of course, ignores the jump by an hour that we artificially introduce into our clock. So this is standard time, but if you want to, this is good for dates that are near September 21st. We can re-scale this at the four points of the compass, if you will, over Earth's orbit. So on December, respectively March, respectively June 21st, you can reset things so that sidereal time is local time plus 6, 12, 18 hours, and then on September 21st the difference will become 24 hours, which is the same as zero. So now, we know how to find sidereal time.

And, now that we know how to find sidereal time, let's use this to, solve an actual problem that might interest us. So suppose that we are interested in looking at the star, bright star Vega in the constellation Lyra. And we want to know when, Vega might be visible in the sky, as high as it can around midnight. Now, Vega, if we look it up, has a right ascension of 18 hours and 36 minutes. According to what we said, Vega is at highest in the sky, it is crossing our meridian, when our local sidereal time is 18 hours and 36 minutes, when our local meridian corresponds with the meridian on which Vega lies. I want to know when is this going to happen at midnight? Well, this will happen when local time of 24 hours, which is the same as zero hours, corresponds with sidereal time equals 86 hours and 36 minutes. Well, if you look back at the previous slide, you will realize that on June 21st, we had sidereal time, was approximately local time, plus 18 hours. And so, if we wait nine days later. Sidereal time will be local time plus 18 hours, plus four minutes times D. If you want four minutes times D to be 36 minutes, D is going to be nine days. And Vega will be high overhead at midnight, nine days after June 21st. On or about June 30th. This might explain to you, why the group of stars, the bright triangle of bright stars that included Vega, was something they called the summer triangle. And so now we finally have a way of understanding what the sidereal time is, predicting which stars will be overhead in which season. And, at any given time, we can map what the picture of the sky is that we'll see. So we have a complete solution of the mathematics problem we set out to solve describing which part of the sky will be visible where and when. Congratulations.

6 Tilt and Seasons

You know we have achieved our goal of understanding the motions of the sky and predicting which stars will be visible, which season, where on earth at what time of the night. There’re some other things we'd like to understand. we talked about the orbital motion of the Earth and how it affects which stars are visible in the sky when, of course there's another phenomenon on Earth that repeats with a periodicity of once a year mainly 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 sow 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 tilted by 23.5 degrees, rather than vertical with the North Celestial Pole therefore, off in 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 plane in which 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 now I've 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 talk about in a minute. And these, it turns out, are the positions, which by convention, we chose to define the Prime Celestial meridian of zero hours of zero hours of right ascension. So, when the sun is at the Vernal equinox, it is at zero hours of right ascension, this means that the sun is overhead at sidereal time zero. 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, six month later at 12 hours of right ascension 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 zero. 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 will explain to us what the tilt of the earth's axis or equivalent via 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's 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 on March 21st, the sun is at zero 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 twelve hours at the Vernal equinox. So this is the position on March 21st, 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

through the North Pole and equivalently through the South Pole, so over the course of a day every point on Earth spends a half of its time on the sun side in daylight. And a half of its 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 other words, on March 21st, when the sun is at the Vernal equinox, and again, twelve months, six 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. Its declination at this point it's right ascension is three hours so let's move it a little bit more. It's right ascension is four hours that means that about two 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 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 and light on Earth, no longer goes through the poles. But that in fact, a 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 point 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 northern most 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 which the Earth's North Pole is tilted which is always toward in the same direction, is now the direction of the Sun. So now points all the way to within a circle of 23.5 degrees latitude around the North Pole, are visible from the Sun and therefore have continual 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 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 most intense on Earth. And as the Earth continues its 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, line 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 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 pole have 24 hours of sunlight, and latitudes to within 23.5 of the North Terrestrial Pole have 24 hours the 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 axis points in the same direction towards the Celestial North Pole as the Earth rotates around the Sun and the changes in length of day and night and in terrestrial heating.

So let's summarize this, let's summarize what we've learned. Between 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 its declination is closer to the declination of your Zenith if you're in the Northern hemisphere. We saw that your Zenith is located at a declination equal to your latitude, the Sun will get 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 the climate is warming in the North and cooling in 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 if the sun comes so far north that it's close enough to the Celestial North Pole, never to set, for regions north of the Arctic circle for at least one day. Of course, at the precise north pole, the sun is circumpolar for six months For precisely six 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 equinox, March and September 21st Day and night are of equal length everywhere. The Sun's declination over the course of the year ranges from 23.5 degrees South to 23.5 degrees North. What this means is that if you live at a latitude between 23.5 degrees South and 23.5 degrees North, between what we call the tropics, then as the Sun's declination changes, there would be two days a year at which the sun's declination is precisely the same as your latitude. On those two days, the Sun will be precisely overhead, at your Zenith at noon. Of course, if you live on the equator, those two days are the equinoxes, where as if you live at the Tropics, this only happens once a year at the appropriate Solstice, June if you're at the Northern Tropic, December if you're at the Southern Tropic. Elsewhere on Earth the Sun will never be precisely overhead, the tropics are the regions where at least once a year the Sun is precisely overhead.

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 its altitude, or its Zenith angle, will be determined precisely by the Sun's declination. At the equinoxes March or September 21st the sun's declination is zero degrees. Those are the days when the Sun crosses the Celestial equator and for the course for the day we can imagine the Sun fixed along the Celestial sphere because it only moves by about four minutes along the Celestial sphere. If the sun is a declination zero, then at noon its Zenith angle is just the difference between its declination and the declination of our Zenith, which is given by our latitude. It's 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 mid-summer, the Sun in Athens will reach an altitude of 75.8 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 note that the Sun solar heating in Athens is minimal in December 21st. December 21st 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 a time, it takes time for the earth to respond to 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 two months later in August, the temperature has reached maximum, by that time. Solar heating is in decline, and with the, the same two month lag, the Earth starts cooling off, solar heating is minimal in December, all this in the Northern hemisphere, of course everything is reversed in the south, and two months later, the Earth is as cold as it's going to be.

7 The Age of Aquarius

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 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 that solar days are not all the 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 the sidereal day is very constant. The length of the 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 four minutes every day, and therefore there is a four 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. Then 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. Whereas, 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 in right ascension is slower near the equinox. 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 was 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 nearer the sun in January then it is at any other time of the year. At that time the sun's apparent motion in the sky is 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's orbit. So that it always points in this direction of the celestial north pole where sits some particular star, let’s 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 on 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 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 staying so far. Instead, its 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 procession, takes about 26,000 years. What that means is that the point that we defined as the celestial north pole moves 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 at different orientations. What this means is that the equinoxes, the hinges at which the points at which these two circles are hinged, move around the celestial equator. Hence this whole wobble leads to what is called the procession of the equinoxes. The vernal equinox shifts, then the coordinates of all stars shift, 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 celestial coordinates declination 90 north, which is the North Star, will not have those same coordinates for very long as the earth continues to wobble. And so when you look at right ascension and declination for a given star, they 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 27. What I've done is I've attempted to have the software keep us centered on the north celestial pole so that we will be following the north celestial pole as time goes on. And I will make time move by centuries. And what you will notice is that over time, the celestial pole moves. The celestial pole started out very near this pole star of Polaris, and the celestial pole 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 the pole star. And of course, in 26,000 years, the polar star will, the celestial pole 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 it's 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 AD. 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 2,600 and some, we will be in the Age of Aquarius. If you look at 22,000 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 the sort of arcane phenomenon of the wobble of the earth's axis. And the procession of the equinoxes and came up with an understanding of ways to date 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.

8 The Moon Moves Too

As I said there are more moving parts in our model of the cosmos. There are more things that do not remain fixed on the celestial sphere and just rotate with it but yet rise and sit so are off the earth. And the next most conspicuous one after the sun is our moon. Like the sun the moon moves along the celestial sphere; it's not fixed. For the same reason, the moon orbits the earth. It orbits the earth in the same sense that the earth orbits the sun and the same sense that therefore the sun appears to orbit the earth, moving towards the east along the celestial sphere. But the moon orbits the earth much faster.

The moon completes one orbit about the Earth in, what is called, the sidereal month. Which is about 27 and a third days. This means, its right ascension increases by 52 minutes per day. Now, as the moon orbits the Earth, it also rotates about its axis. We'll discuss next week, why this is. But this means that we are always seeing the same side of the moon as it rotates around us. There's the same side of the moon that faces us. This means that, while there is no such thing as the dark side of the moon, there certainly is such a thing as the far side of the moon. There are parts of the moon that are never visible from Earth, and, those were first, seen or indirectly seen by human eyes when the Soviet Luna spacecraft first orbited the moon, so this is why when we look up at the moon every time we see it, it has this familiar look. We're always seeing the same side of the moon.

The moon moves to the east along the celestial sphere by about 52 minutes per day. As we know the sun also moves to the east along the celestial sphere. The sun moves much more slowly. The sun moves by about four minutes per day. The difference, 52 minus 4, gives us 48 minutes per day. This is the difference between the the sun's motion on the celestial sphere and the moon's motion along the celestial sphere. But otherwise, it is the relative motion of the moon, relative to the sun. The moon is 48 minutes farther east, relative to the sun, each day, than it was the day before. Since our clocks are attuned to the sun, this tells us that the moon rises, on average, 48 minutes later each day than it did the day before, and sets 48 minutes later, than it set the day before. The relative orientation of earth, sun, and moon, therefore, repeats, but with a period longer than a sidereal month, because that change is not by 52 minutes per day, but only by 48 minutes per day. Make the calculation you see that moving at a rate of 48 minutes per day, the moon completes a full 24 hour rotation about the celestial sphere relative to the sun. Once every 29.5 days that is what is called a synodic month. What happens once every synod, what repeats every synodic month therefore, is the relative position of sun and moon on the celestial sphere. This controls, of course, the time of day at which the moon rises and sets. For

example, if the sun and the moon are 12 hours apart in right ascension, this means the moon rises about 12 hours after the sun rises, about at sunset, and sets 12 hours after the sun sets, about at sunrise, so the moon is up at night during that time of the month. And this repeats once every synodic month. In addition to controlling the times, that the moon rises and sets as we know this relative position of sun and moon on celestial sphere, also controls the appearance of the moon, or what we call the phases of the moon. And the way these two are related, is best captured by the following beautiful demonstration.

The best way to understand how the moon's position relative to the sun gives us the phases of the moon, is to just do it. To just do it, you need a source of light. You can use a light bulb. Or if you prefer, you can just use a sun, just go outside if you can't find a light bulb. Anything round, I have a white styrofoam ball here to play the role of the moon, and your head to play the role of earth; the view from your eyes will give you the view as seen by people on that side of earth facing the moon, of what the moon looks like. And then you simply turn around to perform a complete lunation to give you a sense of what this looks like, in case you're not going to do it yourself. Though, I strongly recommend it. What we have here is a, on the left side of the screen, you'll see a setup of, in the studio, of me, holding a moon and turning around in the relative configuration. Whereas, on the right side, you'll see the image of a GoPro camera that's mounted to the moon. So, it shows you the image of the moon as I see it. When the demonstration begins the moon is in between the sun and earth. It's on a line between sun and earth. And so the illuminated side of the moon as we can see in this beautiful picture faces the sun and therefore faces away from the earth. I see the dark side of the moon. And the moon will be nearly invisible. As I turn to my left, to the east. slowly, the western side of the moon will become illuminated, and we'll see a growing crescent, until after I've turned 90 degrees, we'll see a waxing quarter moon. Since the moon is now six hours or 90 degrees to the east of the sun, it will rise six hours after the sun. In other words, the waxing quarter moon rises about at noon, sets about at midnight. As I continue to turn to the east, the illuminated part of the moon will grow and become gibbous until, at last, when I'm 12 hours away from the sun, I will see a full, round moon illuminated. We're going to have to switch cameras at some point to give you the full view because of studio limitations. Don't get confused by that. 12 hours to the east of the sun, the full moon rises at sunset and sets at sunrise, and so the full moon is the only moon that is really up all night and only during the night. You'll note that to give us a view of the full moon, I had to tilt the moon's orbit. I'm holding it way above my head. We'll get back to that in a second. As I continue to turn to the east, now the western part of the moon is losing the sunlight. I see the eastern part of the moon illuminated. It is a waning gibbous moon and when I reach 90 degrees to the sun again, I have a waning quarter moon. The waning quarter moon, which is six hours to the west of the sun will therefore rise six hours before the sun. In other words, rise about at midnight, set about noon. This is the moon we see in the morning. Finally, as I continue turning to the east, the moon becomes closer to the sun in the sky only the eastern edge of the moon is illuminated, I find a waning crescent moon. And, after a full synodic orbit is passed, the moon is back in line with the sun, and again I see only the dark side of the moon, and we're back to new moon.

So, I, what I hope you saw, and I do encourage you to do it yourself. It's really fun, and you can show it to your friends and family, is that over the course of a synodic month as the moon orbits the earth

relative to the sun the shape of the visible part of the moon changes in the sky as well as because of the relative position of moon and sun, the rise time. So the new moon, when the sun and the moon are roughly at the same right ascension and the moon is completely dark in the sky, rises at sunrise and sets at sunset. The waxing quarter moon, when the moon is about 6 hours of right ascension to the east the sun, it rises six hours after the sun, so the waxing quarter moon rises at noon and sets around midnight and is visible all afternoon. The full moon, where the moon is 12 hours of right ascention ahead of the sun. In other words, on the opposite side of the sky. The full moon is the only time that the moon rises at sunset, and sets at sunrise. And the waning quarter moon, where the moon is six hours to the west of the sun, or 18 hours to the east, the moon being six hours to the west of the sun, rises six hours before the sun, in other words, rises about at midnight and sets about at noon, and the waning moon is visible all morning. So, when you see the moon the daytime, you should not be surprised, but should you ever see a full moon at noon something has gone terribly wrong. So both the phases and the periodic change in moon rise and set times are completely understood in terms of this model, where the moon reflects sun light and what we see depends on the angle between the moon, the sun, and the earth. You can simulate this, you can go to the simulation page of the University of Nebraska-Lincoln and get a less three dimensional version. But I encourage you to construct a, take a light bulb and a ball of some sort and make yourself a moon.

9 Eclipses

Our model explains nicely how the Moon's motion once a synodic month to the east relative to the Sun, controls moon rise and moon set times as well as the appearance of the moon in the sky, the phases. You may have noticed that when I was demonstrating new moon and full moon, I had to cheat a little bit. I had to move the moon vertically, up and down, to make the appearance of the full moon work out for example, the way we wanted it to work out. Does the moon move up and down? Well, if in our model, the motion horizontally to the left represents the moon's motion to the east. Up and down motion represents motion north and south along the celestial sphere and the question we're trying to get at is, what is the moon's declination.

So what is the moon's Declination? Well, remember that the sun's declination changes periodically over the course of the year. The sun does not move along the celestial equator, it moves along the tilted circle, tilted 23.5 degrees relative to the celestial equator. We call that circle the ecliptic. Now, for example when the moon is new, the moon's right ascension is the same as that of the sun. This means that the sun and the moon rise at about the same time, if the moons declination at that time were exactly the same as the declination of the sun, the sun and the moon would be in exactly the same point in the sky. And we would not be able to see both, we would be able to see probably only the moon since it's closer. And the sun would be invisible, it would be hidden beyond the moon. Of course this does not happen every new moon. The reason it doesn't happen every new moon is because the moon's orbit is about the Earth, is not in the plane of the ecliptic. The moon does not orbit the Earth exactly along the ecliptic. If you were a space monster observing the Earth orbiting the sun and as the moon orbits the Earth, you'd notice the two orbits are almost exactly coplanar but there's a tilt. And the tilt of the moon's orbit relative to the ecliptic is five degrees. Five degrees doesn't sound that much compared to

say, the 23.5 degree tilt of the ecliptic relative to the celestial equator. But remember that the moon and the sun are about the same size in the sky, and the angular diameter of each of them is about half a degree. This means that when the moon and the sun occupy the same right ascension, if their declinations differ by say two degrees, that's still four times the size of either sun or moon in the sky. And they certainly do not occupy the same point in the sky. We can see both the sun and the moon although we only see the sun, because the sun is much brighter. But the moon certainly does not obscure the sun if it is two degrees North or two degrees South of the sun. In order for the moon to obscure the sun, they have to be at the same right ascension, and at the same declination. Is that possible? Well, yes of course it is. Remember, what I said is that the moon's orbit is tilted by 5 degrees relative to the ecliptic. Just as the, ecliptic being tilted relative to the equator, means that there are two diametrically opposed points that are common at which these two circles are sort of hinged. Those were the equinoxes. When I have the moon's orbit tilted five degrees relative to the ecliptic, there are two diametrically opposed points at which these circles intersect. Those are called the nodes. What are these nodes? These are two diametrically opposed points that are both points around the moons orbit around the Earth, and points on the sun's orbit around the Earth on the ecliptic. This means that if the moon is at one of the nodes and the sun is at the node - remember, the node is a point both on the ecliptic and on the moon's orbit - both objects can be at the same point. And if they happen to do that at the same time, then the moon can and will obscure the sun, and interesting things will happen. Let's see in a demonstration how exactly this works.

In first our lunation, I carefully kept the moon way north, or up in this image, from the sun when it was full and way south of the sun when it was new. What we're going to do now is see what happens to the full moon when the line of nodes is aligned with the direction to the sun. In other words the moon when it is 12 hours away in right ascension from the sun, is at exactly the sun's declination. And, we start our demonstration with the waxing gibbous moon. And it waxes until it is becoming full. And then what we see is that from the eastern side of the moon fuzzy shadow covers the surface of the moon. Of course this is the shadow of my head. it would not be as shaggy were it the real Earth, but this is exactly the geometry for a total lunar eclipse. The moon enters the shadow from the west moving east, and therefore it moves out of the shadow, starting with its eastern edge first. So when you see a lunar eclipse take place and then the moon uncover, you're literally watching the moon orbit around the Earth. About two weeks later or two weeks earlier the moon is now at the sun's right, ascension. We're getting a new moon. And the moon is still at the same declination as the sun. And we'll see what happens, we pick up this story with a waning crescent moon. The moon moves in from of the sun, when it is at the sun's right ascension and declination. The moon in fact is obscuring the sun. And remember we're on the daytime side of the Earth. This is midday because it's a new moon. We shouldn't be able to see it at all. In fact we don't, but then we don't see the sun either. We get complete darkness in the middle of the day. And as the moon continues to move to the east, then we see this weird glow on the moon. This is an artifact of a shiny GoPro Camera, but even this is instructive. The lens of the camera is still eclipsed, but the bits of the camera on the right hand side are already shining in the light and there, you can see the reflection of that off the moon. This is what would happen if people living a few thousand kilometers to the west of you erected a huge shiny tower on Earth. During the total solar

eclipse, you might see the reflections from that shiny tower on the surface of the moon. And then as the moon continues to move east, the sun is again revealed and it's bright midday.

What did the model show us? The model showed us that we can get an eclipse, lunar eclipse at full moon, solar eclipse at new moon, if the moon is full or new at the same time that it is on one of the two nodes on the points where the moon's orbit intersects the ecliptic. Hence the name ecliptic. If the moon is on the ecliptic and new or full, then we can get an eclipse. Now, what does this mean? Well, for the moon to be new or full, means it is either at the same right ascension of the sun or 12 hours away diametrically opposed to the position of the sun. If the moon is on a node and either at the same right ascension or opposite right ascension to the sun, this means the sun, itself, is at one of those two nodes.

This gives us twice during an average year something called an eclipse season. And the alignment is, need not be absolutely perfect because neither the sun, nor the moon, are point-like objects, but there are two, sort of, one, one and a half month periods during each year, and in those one, one and a half month period, eclipses may occur. May occur that is, when the moon is either full or new. So that it is at the correct part of its orbit at the right ascension either of the sun or opposite that of the sun, and now at the right declination. So twice a year we have an eclipse season, and during an eclipse season you have usually between two or three eclipses. One of one kind, and flanking it, two weeks on either side two of the other kinds, you could have a solar eclipse flanked by two lunar eclipses. Or a lunar eclipse flanked by two solar eclipses, by the time the next eclipse of either type would have occurred the next new moon. For example the sun is now too far from the line of nodes, and you no longer get an eclipse. So typically no more than three per season, and this should happen twice per year. In fact, like everything else that is tilted the line of nodes wobbles. In other words, the tilt between the moon's orbit and the ecliptic is always 5 degrees. But its orientation wobbles, is the case with the Earth, it actually wobbles to the west, because it is orbiting to the east, and so the tilt processes to the west very slowly, once every 18.6 years or so. And this just means that the eclipse year, the year during which two eclipse seasons happen is a little bit shorter than the full orbital year of the Earth only 346.6 days. And so, during each of those two eclipse seasons a newer full moon might lead to a solar or lunar eclipse. So a new moon during the eclipse season means the moon is in front of the sun and at the right declination. So you can have a solar eclipse, a full moon means the Earth is between the moon and the sun, and the Earth shadow will obscure the moon, just as my shaggy head obscured the moon in our model. Now, we need to get our handle on the relative sizes and distances of things, to understand the difference of the two phenomenon. So, the moon and the sun are almost precisely the same size, in angular terms, in the sky. If you think about it, the sun is much larger than the moon, from our small angle approximation, we know that it is much farther. The ratio of their sizes is almost precisely the ratio of their distances. So that both the moon and the sun up here in the sky appear to be about the same size, an interesting coincidence. What this allows is when you have perfect alignment during new moon, the moon is able to completely obscure the sun. when you superimpose two discs of the exact same size in the sky. But this requires perfect alignment, so perfect that, in fact, it will only obtain for a small region on Earth. If you move a little bit away from that region on Earth, then the alignment is no longer perfect. The alignment is only perfect at one point on Earth. And when I say, one point, I mean a region on Earth whose size is up to about 250 kilometers. That's a very small part of a planet with a radius of 6400 kilometers. And in

that region where the moon's shadow completely obscures the sun, you get what is called a total eclipse. And we have here a beautiful time series of a total eclipse. Notice that as time progresses in this, the moon seems to obscure the sun from the left and then leave the sun to the right, whereas in my model this was happening the other way around. The reason is, that this eclipse in 2001 was observed near Zimbabwe. And near Zimbabwe, you're in the southern hemisphere. Which means east and west are still the same, but looking since the sun is now to your north, the moon entering the sun from the west, moving to the east, is now from the left to the right. So nod to our southern hemisphere viewers. This time it makes sense to you. The moon is coming from the west to the east across the surface of the sun. It obscures the sun completely for a few minutes, and then is seen to move away and what we're watching again is the moon orbiting the Earth. Although also this place Zimbabwe, is being moved along the surface of the earth by the Earth's west to east rotation, and moves out of the shadow as we'll see in a moment. And noteworthy when the moon completely obscures the sun, a region around the sun that appeared completely dark before is suddenly seen to be brilliantly luminous. This is called the corona, the crown of the sun. It is not visible when we see the the sun itself or the sun's disc, because the sun's disc is so bright that it blinds us to the brilliance of the corona. But once you obscure the sun's disc, you can see that the region around the sun's disc is illuminated when we talk about the sun we'll try to understand what this glowing crown is and how to observe it away from eclipses. A more common phenomenon in this beautiful totality is that when the new moon occurs the line of node is not exactly aligned with the sun and so the moon is a little bit south or a little bit north of the sun. And then the moon would have passed below or above the sun, above it if were south, since we're in the southern hemisphere, and below if north, or the other way in the northern hemisphere. And then, you will find a partial eclipse where not the full surface of the sun is obscure, but only some fraction thereof. And this is far more common because it requires a less sensitive alignment.

To give you a sense of what the, the shadow of the moon on the Earth looks like here's a beautiful image taken from the Mir Spacecraft on August 11th 1999. And, this is an image of earth and what we're seeing on Earth is the moon's shadow. So, the moon's shadow as I said the moon can obscure the entire sun in this inner circle of radius about 250 kilometers. Away we see this partial shadow these are regions where if you look up, you can see part of the sun, but some fraction of the sun is obscured by the moon. You can see a round it a little bit and so this is called Penumbra, or partial shadow people here see a total eclipse. People here see a partial eclipse. And now remember that underneath this shadow, this, the Earth is rotating so that this shadow is effectively moving along the earth at some 1,000 kilometers per hour. And so each individual location only gets a short period of totality. Now, this is what happens if alignment is perfect and if the moon and the sun are, indeed, exactly the same size in the sky. How can they change? Does the moon shrink? No, but the moon's orbit around the Earth is not completely circular. The moon is sometimes a little closer to Earth and sometimes a little farther. When complete alignment occurs, and the moon is on the farther part of its orbit, it's just a bit farther from Earth. Since its size didn't change, it's apparent size in the sky is a little bit smaller, it's then smaller than the sun in the sky and we get what is called an annular eclipse, because you see annulus of sun, a ring of sun surrounding the moon. This shadow here is the moon. We see some of the corona and even some of the chromosphere around the sun. But what we also see here is a little bit of the disc of the sun, the

photosphere as it's called. and the moon not completely obscuring it, in this beautiful image of an annular solar eclipse.

This is what happens when alignment during eclipse season occurs at new moon. During full moon what happens is that the Earth's shadow prevents sunlight from hitting the moon and the moon becomes dark. The moon as I said is moving to the east, it enters Earth's shadow from the west. Again, the eclipse can be total or partial and depending on the quality of the alignment, and we can get a penumbral eclipse when the moon is only in partial shadow. In other words, where from the moon you can see some part of the sun. But some of the sunlight is blocked by the earth just as pieces of the Earth were in partial moon shadow then the moon just slightly dims, it's kind of hard to even notice it. But when you get a total eclipse, when some part of the moon's surface is completely obscured from the sun, then this is the beautiful image you see. Now this image should surprise you. I mean perhaps some sunlight is reaching over here. Maybe I didn't time this photo to precise totality. But where is the light coming from that allows us to see the moon, this side of the moon at all, this side of the moon is in fact in total Earth shadow. You cannot the sun from this area of the moon. The Earth is bigger than the moon, so it handily obscures the sun. It's much easier to arrange a total lunar eclipse than a total solar eclipse and also a lunar eclipse when it occurs is visible, since the moon is dark, to anybody on earth who can see the moon. So whenever it occurs during your night time, you can see the moon eclipse. It's a far less delicate arrangement than the solar eclipse which only blocks out the sun for a small fraction of the Earth.

As indicated on the slide, the light that is reaching the moon despite the sun being obscured by the Earth has passed through the Earth's atmosphere, it has in fact been refracted - deflected a bit by the Earth's atmosphere. Why passing through the Earth's atmosphere makes the light red or causing this beautiful and famous wine red color of the moon during the total eclipse, is something we'll discuss in a week or two.

This is a picture actually taken at our observatory in the solstice eclipse of December 2010. And so, this is a, a pretty image. And lunar eclipses are easier to find and observe. I encourage you to enjoy them.

A few more fun facts about the moon, while we're discussing the moon. So, two well known things that people appear to observe. One is, that when you see the moon rising or setting it's near the horizon, it appears larger. This it turns out, is a psychological illusion. Taking pictures of the moon with a camera, you can measure its angular size and in fact, if anything, then strange optical effects make it appear a little bit smaller in the sky near the horizon. But there are various psychological theories nearer the comparison to other objects nearby, angular corrections. I'm not an expert on the psychology. But, it does appear to us, that the rising full moon seems huge, when it's on the horizon and small when it's high in the sky but this is completely in our head.

On the other hand, there is the other famous illusion where, which is called seeing the old moon in the new moon's arms. Which is that when we see a crescent moon where only a fraction of the moon is illuminated, then you can sometimes imagine to yourself that your mind completes the full disk of the moon, and you can see the part of the moon that is not illuminated by sunlight. And this one is a physical effect, here's a camera capturing it, here's the illuminated side of the moon. So the sun is down

that way. This side of the moon is dark, sunlight cannot reach it, and yet I see it. So there's sunlight hitting it. And there is, a reason of course, is that when the moon is a crescent. if you are on the moon, set up the configuration in your head, you would see that there will be a full Earth in the sky, and viewing the, the full Earth in the lunar sky means that there is bright Earth light illuminating the moon. So the light that is hitting the moon by which we are seeing this moon, is Earth light. In other words, light that was emitted by the sun, reflected by the Earth, hit the moon, reflected off the moon and came into the aperture of this camera. So, the fact that you can see the dark part of the moon when the moon is a crescent is not psychological it is true. You can only see this when the moon is a crescent basically because once the moon becomes too bright it blinds us, it dazzles us and we can't see this. Also, the larger illuminated part of the moon, the less of the face of the Earth is illuminated. Of course when we have a full moon then people on the moon would see a new Earth, the phases are complementary, and so for both of those reasons we only see this when the moon is a crescent.

10 Our Universe, so far

So, we've made some progress in understanding astronomical phenomenon, the way they change periodically. We've got the Sun, we've got the Moon. Let us close this discussion with some interesting relations between astronomy and timekeeping. I told you that units of time kept being defined in terms of the earth's rotation and its orbit. This is not a coincidence. We want our time to match what's going on. We want 6 a.m. on our clock to be solar sunrise because that's the time we go out and plant things, work in our fields. And so our 24 hour days are adjusted to be the mean solar day. Our months, the 12 months into which we traditionally divide the year, are approximately lunar. The synodic lunar month is 29 and a half days, our months our a bit longer than that but, and this allows for the rare phenomenon of two full moons falling in the same month, which is what colloquially is called a blue moon. It's a rare phenomenon. It requires a full moon right at the beginning of the month, but it does happen. Our definition of a year is designed to match the orbit. A year is 365 days. The Earth orbits the sun once every 365.2564 days. A little over a quarter of a day. This is a sidereal orbit. In other words, this is the time that it takes the Earth to return to the same position in the sky relative to the sun. Or, the time it takes the sun to return to the same position in our sky relative to the stars. Hence, a sidereal orbit. The first thing we observe is that the year is not, unfortunately, an integral number of days. This is a problem. It means that since our days turn over every 24 hours, that every four years your timekeeping, if you have a 365 day year, then every four years you are off by a day relative to the Earth's orbit. So who cares? Well, accumulate those days for 180 intervals of, of four years, and now you are off by 180 days, which is half a year. Which means that now January corresponds to northern summer. This is very inconvenient if you're trying to plan agriculture and we have solution to this. Right, this was discovered by Julius Caesar, or, in his time, and it was his legislation that added leap years. Once every four years, we add another day making that year 366 days long. The average year is now 365 and a quarter days long, and now we never drift more than a day off from having our orbit match our calendar. However, this is not completely precise enough, in fact what we want our calendar to match is the seasons. The seasons have to do with the relation between the Earth's position relative to the Sun, and not the stars,

but the direction of the tilt of the celestial North Pole, or the terrestrial North Pole. And remember that that wobbles to the west rather than to the east.

So complete rotation, where between solstice and solstice or equinox and equinox, is a little bit shorter than the sidereal year, rather than longer. This is called the tropical orbit. The mean time between solstice and solstice is 365.2422 days. Remember that the precession is very slow, takes 26,000 years, so it's not a big effect over a year, but it does make the mean tropical year a bit shorter than 365 and a quarter days. This was understood in the 16th century, and led to the correction from the Julian to the Gregorian calendar. Correcting for the deviation between .2422 and .25 required removing some of the leap years. That is why, on centuries that are not millennia, in other words years whose number divides 100 but not 1,000, we do not add a leap year. We do not add a 29th day to February. Those years are only 365 days long, the average year is a little bit less than 365 and a quarter days long. In fact, it's close enough to this number on average that it'll be millennia before we have to make another correction.

And so we definitely use astronomical phenomena to adjust our clocks and our calendars not for any silly reason, but because astronomical phenomena govern our life. And we need our time keeping to match that.

It's quite an elaborate universe we're starting to build around us. We have a celestial sphere where the stars are fixed. We have a solar sphere that rotates relative to the celestial sphere about this tilted axis, so the sun can move along the ecliptic. We have a lunar sphere tilted relative to the ecliptic around which the moon moves a little bit faster. Moreover, all of these tilted trajectories are also wobbling to the west, one very slowly every 26,000 years. The moon's a little bit faster, every 18.6 years. This is very elaborate, but it does explain everything we see, the alternations of day and night, the phases of the moon, eclipses, seasons, almost everything.

Let's go to Athens and see what you might have been missing. We're back to our favorite picture of the sky in Athens. And I've allowed the software to show us the ecliptic, and we see the 23.5 degree tilt relative to the celestial equator. We see the intersection of the ecliptic and the equator here at the prime meridian in Pisces. We still live in the Age of Pisces. And we see that the part of the ecliptic that's visible at night is mostly the part north of the equator. That's reasonable because, end of November, the sun is well south of the equator, the part of the ecliptic that lies south of the equator is what we see during the day. So far, so good. The other thing you'll notice, and I'm sure if you tried to run your own simulation. And certainly if you went outdoors, you will have realized by now that I fudged, I suppressed some things in the simulations I've been showing. In particular, the two brightest objects in the sky were omitted from my discussion up to now. One of these, the brightest of them, is this waning gibbous moon, which we see here. A waning gibbous moon, remember, is a moon that is past full. And so, it rises after sunset, and at 9pm we're still seeing it in the eastern sky. So we expected that. The moon happens to the lie very close to the ecliptic right now. It could have deviated remember, by as much as five degrees. We also have the next brightest object in the sky, is the beautiful planet Jupiter, and we have not brought up planets and once we allow that, we see that's scattered along this ecliptic are a few others non-star objects, Neptune, Uranus, and the asteroid Ceres. And, if we look at this image over time, we would notice that, like the moon and the sun, the planets move as well, which means we're

going to have to start next week by adding even more moving parts to our universe. And enriching what we know and eventually leading us to much, much deeper understandings.

11 The Small-Angle Approximation

I said earlier that stars were very far, the celestial sphere was very large and left for later an investigation of, what do you mean by very large? How far are the stars? Well, in one sense it's later now, because we're about to understand what we can say, or what it was that Aristotle knew about the distances to stars or how large we need to make the celestial sphere. In another sense, it will be later for the next few weeks, because as we'll see the effort to measure the distances to things in the heavens will be a central thrust of the effort to understand what it is that these things are that we see out there. So, our understanding of the nature of astronomical objects will be intimately tied with efforts to figure out how far they are.

As we go along, we will learn that humans have recruited into this effort a huge array of techniques and technologies for figuring out the distances to astronomical objects. Let's start at the beginning, and the beginning is direct Geomount geometry. We can make the measurement of distance to objects in the sky under some circumstances. The technology we use is called the small angle approximation. Let's see what the geometry is. And then we'll see how we use it in astronomy. So, as I said, this deals with the relation between angles and distances, so we're going to build a triangle. And we're going to denote two of the vertices of our triangle by A and B, and one by O. And the distinction is going to be, we're going to work with specific triangles. In which one of the sides of the triangle the one I denote by d is much shorter than both of the other sides. It's so much shorter that I could, assume to all, for all intents and purposes that A and B are equidistant from O. How can I assume that because if d is much smaller than the distance R from either, from O, d to A or B. Then the difference between the O A distance and the O B distance, which is at most d, is negligible compared to R. Does that hold in this triangle? Well, this brings us to the question what do we mean by very small? In this triangle, I would estimate that d over R is about a third. What that means is that if we use the small angle approximation, any calculation we make might be mis setting to zero. Or omitting things that are of order d over R. And so our answers might be off to about, by about 30, by about 30% of what we want them to be. If you want the small angle approximation to be more precise, perhaps you wish to work in a triangle like this one. In which d over R is of the over a tenth, and the small angle approximation will be good to about 10% accuracy. And many of our triangles, d over R will be a millionth and the small angle approximation will be even better. Let's erase all my scribbles and see what the small angle approximation allows us to actually compute.

So let's introduce the eponymous small angle, the angle that is small. When d is much smaller than R is of course the angle at the vertex O, I will call that angle alpha. And trigonometry allows us to construct a relations between R, d, and alpha. But we won't need trigonometry if d is much smaller than R. We can make the following calculation that simplifies that relation to something that is easier both to understand and to remember and does not require trigonometry. So, I'm going to construct an auxiliary geometric object here. I'm going to draw a great big circle around O, centered on O with radius R. And by my assumption that means it goes through points A as well as the point B. And I'm going to try to

imagine that instead of trying to figure out the length d of the interval between A and B. I want to under, figure out the length of this arc, the part of the circle, line between A and B. Well, that's denoted by AB with a little squiggle on top, reminding me that it's the arc length. If you imagine that the circle is a pie and the green line is the crust, and you want to know how much crust you get. when you take this slice of pie along with this slice of pie, comes a fraction alpha out of 360 degrees of the pie. And you get the corresponding fraction of the crust. The length of the entire crust is the circumference of the circle, which is 2 pi R. And we combine this into this fancy, well written relation.

This of course holds without any approximation, it's in the exact relation whether you're getting a small slice of pie or a large slice. What is true when you get a small slice is that if the slice is sufficiently small, if the angle alpha is a small angle, if d is much smaller than R. Then the difference between the arc length that we computed and the actual straight line distance between A and B is very small. So in the small angle approximation d, just the straight line distance between A and B, is approximately the same as this arc length that we've computed. This is what the small angle approximation buys us. Now again, what does that explicitly mean? What it means is that this is true to within, there is of, of the order of d over R. So in our case I would say that the difference between the arc length and the straight line length is no more than a third of either of those two lengths. This is the small angle approximation, so let's bring that in. The small angle approximation allows me to approximate the distance d by the arc length that we computed AB. And computing the math here, dividing 360 degrees by 2 pi, I can write that in this fancy formula. This is the small angle approximation, draw a box around it and remember it. It tells us that d, the distance between A and B, is related to R by, and to the angle that, that they subtend we say at O by this relation.

Now, in astronomy, we will have, build such triangles in several ways. So what realization of this did I have in mind? Well, when I was talking about the celestial sphere, I actually had the following idea in mind. I had in mind that O in fact was some distant star, some point along the celestial sphere. A and B, on the other hand, were two different points from which we can view the celestial sphere. Where do we get to the view the celestial field sphere from? Say, Athens and Rome. Two points on earth, and so the distance in the context that I was thinking about. D was of the order of the size of the earth at most the radius of the earth. Which we will denote this way is about 6,400 km. And R is, I don't know, R is the distance to a star, and in our approximation, it's the radius of the celestial sphere. So is the radius of the celestial sphere much larger than the radius of Earth? That's the question we're asking in this case, and the answer is yes. How do we know? Well, because we can measure the angle alpha. The angle made by the line from one point A on Earth to a star and a different point on Earth to the same star. if those angles are different, what that means is we actually obtain a different view of the sky. From the point A and B from two different points on earth, we call this angle in that context the parallax angle. We are now measuring the distance to stars in the same way that we estimate distances to things by observing them through our two eyes and comparing them to images. And Aristotle is well aware is that the sky looks almost exactly in tune within the precision of this measurement. Exactly the same as viewed from different points on earth. This parallax angle between two points on earth and a star has in fact not been measured. It's very, very small. So that tells us that indeed the celestial sphere is so large, that we can assume to any precision that we can measure, this angle alpha to be zero. Stars are indeed very much

farther from Earth than is size of Earth, and now this is what I meant by saying the celestial sphere is very large. Saying that it is very large is a meaningless statement. To say that some physical quantity is large is meaningless until you say large compared to what. This radius of the celestial sphere is much larger than the Earth's radius. We've settled the question of the size of the celestial sphere, but the small angle of approximation will go with us.

So as I promised, we are in no way done with estimating the distances to stars. But our first attempt to measure the distance, is if you want a failure. We measure that the distance is so large that we can't actually measure. So it's larger than any distance we can compute. We may as well assume it to be very, very large. That's what we've done. Now the angles that we measure in the context of astronomy are often going to be very small and measuring them in degrees will be very inconvenient. So we often will have recourse to smaller measures of angle. We will actually follow the ancient Babylonians and subdivide a degree into 60 arc minutes. And each arc minute into 60 arc seconds, and so a angle of one degree will correspond to an angle of 3600 arc seconds. And since we typically measure very small angles, arc second will be a convenient unit in which to measure angles, and so one can then compute that this angle to which I'm computing the ratio here of 57.3 degrees. Or a 2 pi th of 360 degrees is equivalent doing the math to 57.3 times 3,600. You find that the following is another equivalent way of writing the small angle approximation. the ratio between the short and long side of the triangle is given by the small angle divided by this angle. Notice that 206,265 arc seconds is just another way to write 57.3 degrees. This is the form in which we'll often use the small angle approximation. And this brings us to the last point I wanted to make, and I've sort of been hinting at it as we went along.

We are going to be doing physical science. We're going to be discussing physical properties of the universe, like the distance to a star, the size of a galaxy. And we're going to be trying to mathematically and precisely describe these and as Lord Kelvin taught us. To do that we need to assign them numbers, but what does that mean? So, assign a number to say, some important physical quantity such as my length. Well, depends who you ask. Some people will tell you that my length is about 167. And other people will tell you that my length is about five and a half, so who's right? Well, they're both of course right. The point is that assigning a number to the length of me is an irrelevant statement. What you, the number really tells you is how much longer I am or how much shorter I am than something else. The person who said that I'm 167 is my length, meant that I was 167 times longer than some standard they call a centimeter, and so on. We use units to detect what it is, what is the standard length, mass, whatever it is, to which I'm going to compare. This was very evident when we decided that we could measure angles in degrees or in arc minutes, or in arc seconds. So that one person could describe an angle as three degrees, another person will describe that very same angle as a 180 arc minutes. And a more ambitious person would measure that same angle in arc seconds. But I can't do that calculation in my head so I won't put that down. So, the number that you assign to, for example, an angle is meaningless until you say in what units are you measuring that angle. In what units are you measuring that distance. Making sure that when you, write, assign a number to a physical quantity, you remember what units you are measuring. What ratio it is that that number designates, is going to be very important as we go along. So in particular, when we derive or find all kinds of mathematical relations between physical quantities. For example, we found in the previous slide that d was equal to alpha divided by

206,265 arc seconds times R. Well, now, this is a relation, is an equality. It's an equality of two numbers. Thought of physically it relates this distance to that distance. But clearly say, this distance is the same as that distance means that they will correspond to same numbers if you measure them in same units. Now, in particular, the distance between A and B, remember that d was the distance between our points A and B. And R was the length the long the side of the triangle. If I measured this in millimeters and this in light years, this relation will not be true despite the fact that this is a correct relation between the distances. It will not be the relation between the numbers we associate to them. This will only hold if I measure d and R in the same units. What's the easiest way to assure that all of our equations are valid in whatever units we use them.?Well, one easy way to do it is to write our equation, write this equation in an equivalent but simpler way. Write it this way. This way, we see that on the left hand side we're writing the ratio of d over d to R. That's the ratio of two distances. That is actually a meaningful number. So, that is the ratio of d in meters to R in meters, or d in light years to R in light years. And as long as you use the same units to measure d and R, it's true, whatever units you use. Likewise, the right hand side is the ratio of two angles. The angle alpha, and the angle 206,265 arc seconds. The ratio of those two angles is a number, and so our equations are now relating two numbers. And whenever you can write your equations completely in terms of ratios, you will find that you will not have issues with unit conversions. And you will spend some of the time in this class converting the units in various calculations and making sure you're measuring everything in the right units. So consider this to have been your warning about this.