Hi gang, and welcome back to Analyzing the Universe. Today I want to talk to you about measurement of distances. Easy, right? You just take a ruler, and see how far it is to somewhere else. One ruler length, two ruler lengths, three ruler lengths, four ruler lengths. And so on and so forth. But what about if you can't even get to the other place? Or if the distances are so vast, that ordinary rulers are impractical? Such are the problems we have when we try to measure the distances to the stars. We need some sort of stellar bootstrap in order to extrapolate our earthly distance measurements into the realm of the cosmos. We begin, as we must, with our home, the Earth. And as usual, our story begins with the Greeks. And, in particular, Aristarchus of Samos around 250 BC. What you see on your screen right now is Aristarchus's working diagram by which he concluded that the Sun, on the left, was much bigger than the Earth in the middle. And hence, was most likely to be at the center of the Solar System rather than our planet. Let's dissect this diagram carefully, and see how he did it. Our dilemma is that all we can measure is the angular size of an object in the sky, and sometimes not even that. This means that the size of an object is ambiguous. Let's consider this drawing. Here, we have the Earth. And we imagine that we're looking out into space with a certain angular size. You see that the Sun, or the Moon, or any object with the same angular diameter can be either close and small, or far away and large, and still appear to be the same size in the sky. So our angle here is the same, but the object, depending on where it is, can be either large or small. Now, Aristarchus knew the angular sizes of the Sun and the Moon were the same. Otherwise we could never have a solar eclipse where the Moon almost exactly covers the surface of the Sun. He also knew that the distance to the Sun was much bigger than the distance to the Moon. How did he know this? By realizing that the times from first quarter to third quarter of the lunar cycle was almost the same as from third quarter to first quarter. Let's look at this carefully. We imagine that the Earth is here, and that the Moon, is going around the Earth in a circular orbit. Let's make that circle just a little bit better, huh? Now, over here we have the Sun, here's the Earth, and for the quarter Moons what has to happen is the Moon has to make a 90 degree angle between the Earth and the Sun. So, if we have the Moon over here, this will be a 90 degree angle and we will see the first quarter of the Moon. Now, when the Moon is down, oh about here, we'll have another right angle and we'll see that this is the third quarter. Now, it's pretty clear that for the Moon to go this way from first quarter to third quarter, is going to be more time than for it to go this way from third quarter to first quarter. Now, note that as the distance to the Sun increases, the differences between the two arcs of the circle, this arc and this arc become smaller. Let's see what happens. If we put the Sun much, much further away, somewhere off to the, outside of the blackboard, you can see that the rays of the Sun will come in like this. And the 90 degree angle will be formed, something like this, in a way that will start making this arc almost exactly the same as the other one. So, as the distance to the Sun increases, these arcs become more equal. This is so ingenious, no? Although Aristarchus's result was not particularly accurate, it was good enough to realize that the Sun must be much farther away from the Earth than the Moon. His value was 19 times further. Thus, the Sun must be 19 times larger since the angular extent was the same as the Moon in the sky. With the Sun much further away from the Earth than the Moon, the angle of the Earth's shadow is about the same as the angular size of the Sun and the Moon in the sky. Let's look at that part carefully. So we have the Sun and over here, we have the Earth. And consider the following situation. We're on the surface of the Earth, and we measure the angular size of the Sun. And now, we look and see the angular size of the Earth's shadow. Here is the shadow that must be cast behind the Earth due to the fact that the Earth has, more or less eclipsed the Sun, for any object that is in this region over here. So, this theta two is the angular size of the Earth's shadow. This is theta two, here's the shadow. And you can see that if the Sun is very far away, the angular size of the Sun, which is theta 1, is almost the same as this angle in here, the angular size of the Earth's shadow. Notice that these objects are not drawn to scale since the angles depicted are for clarity of understanding, much greater than the one half degree that the objects actually appear as in the sky. Now, as the final step, let's look at the Earth's shadow in detail during a lunar eclipse. Here we have the Earth, and here we have the Earth's shadow. Okay? So, the Sun is over here, all the way, a long way away. The Moon in the sky must be within the Earth's shadow. And how is that going to look? Well, we know that the angle that the Moon has in the sky is basically the same as the angle that the shadow makes. So, it'll look something like this. And the Moon can be anywhere in this region of the sky. But where do we put the Moon? And here is where the observation of the Moon during a lunar eclipse comes in. Because we observe that the time it takes for the Moon to traverse the Earth's shadow, let's get that shadow depicted like this. Is about 8 3rds of the time it takes for the Moon to move its own diameter in the sky. So the Moon's size must be about 3 8ths the size of the shadow. So the only place we can put the Moon in here to meet the requirements of the data, the requirements of the observation, is such that the Moon's size, right here. Is 3 8ths of the size of the entire diameter of the Earth's shadow, which we can delineate as the line A, A prime. Okay? We know these angles. They're about half a degree. We can put the Moon anywhere in this cone and still have it have the right angular size. But only when we put the Moon, well, I probably didn't put it in exactly the right spot. If we put it a little bit further on, it'll match a little bit better, but you get the idea. There's only one place that the Moon can fit so that it is in the proper proportion of the Earth's shadow in size. So now, we have the relative sizes of the Sun, the Earth, and the Moon but in terms of the Earth's diameter, we still don't know how big the Earth is. This problem was solved, also ingeniously, by Eratosthenes about 50 years later, around 200 BC. To understand how he did this, we have to realize that the Sun is so far away, that essentially, all the rays that arrive at the Earth are parallel. Let's imagine a light source near the Earth. If the Earth is here, and you put a light source over here, the rays from that light source will diverge like this to the top and bottom of the Earth. If you put the object a little further away, the rays, don't diverge quite as much. If you put the object over here, the rays diverge even less. And if you put the objects, such as the Sun, so far away that you can't even really tell the difference between these rays, all of the rays that will come in to the Earth are going to be essentially parallel. Now, Eratosthenes noted that at Syene, Egypt, which is now the modern city of Aswan, on the first day of summer, light at noon from the Sun struck the bottom of a vertical well. So that meant that Syene was on a direct line from the center of the Earth to the Sun. The picture looks like this. Here's the surface of the Earth. Here's the center of the Earth. And at this point, if this is the position of Syene on the Earth, this line represents not only the zenith direction at Syene, but also the position of the Sun in Syene. At the corresponding time and date in Alexandria, which was 5,000 stadia north of Syene, the Sun was slightly south of the zenith. So, its rays made an angle of about seven degrees to the vertical. So, here's the vertical in Alexandria pointing this way. So, this is the zenith in Alexandria. And that makes an angle of seven degrees to the Sun. Okay, here is an angle theta, that because we have gone along the surface of the Earth, about 5,000 stadia. The distance of Alexandria from Syene is 5000 stadia, and at that position, the angle that the Sun makes with the zenith direction in Alexandria is about seven degrees. So, here we have the center of the Earth. Here we have Syene. Here we have Alexandria. And since the Sun's rays are essentially parallel, the angle of seven degrees between the solar direction and the zenith is the same as if, this seven degrees was subtended from the center of the Earth. Now you see, ingeniously, that this 5,000 stadia can be extended to measure the circumference of the Earth, because we know this angle that is subtended by the circle, right over here. So, you can see that, this angle theta is to 360 degrees as the distance to Alexandria from Syene, is to the circumference of the Earth. Right? Here is a segment of a circle. D is to the whole circumference of the Earth, as seven degrees is to the whole 360 degrees, that makes the circle complete. Thus, the circumference of the Earth must be about 50 times 5,000 stadia or about 250,000 stadia. Seven into 360 is about 50, right? So, 250,000 stadia must be the circumference of the Earth. But what was a stadium? Was it a Fenway Park stadium? Was it a Yankees stadium or what? There is actually much debate about this. But there is no doubt that Eratosthenes got very close. Ranging from 80% to 99% of the true value for the Earth's circumference. So now we have a crude estimate of the distance to our nearest star, the Sun. A significantly better estimate was not forthcoming until the invention of the telescope almost 2,000 years later. The various ingenious experiments designed to measure this elusive number are fascinating to study, and more than just of academic interest. For our knowledge of the distances to the remote stars, which are so far away that we cannot even measure directly their angular diameters, depend crucially on our ability to perform measurements in our own backyard. Namely, to determine the distance to the closest star, our Sun. On the surface, it would appear that the situation seems hopeless for even greater distances. I mean, it's almost a miracle that we can determine the solar distance. How can we possibly extend our reach to the stars? Well, let's do a little experiment. Hold your finger up in front of your eyes, and blink your eyes alternately. Just like this. Notice that your finger moves relative to the objects in the background. This simple example of parallax can now be extended to the orbit of the Earth. This represents the orbit of the Earth. So the Earth in June might appear over here, relative to the Sun. And the Earth in December might appear here in its orbit. And a nearby star will definitely appear to be in a different direction relative to the background that might exist, populated by other more distant objects. The nearest stars then, should move relative to the backdrop of the further stars. But the distances involved are so great, relative to the diameter of the Earth's orbit, that changes over the six month span as the Earth traverses opposite sides of its path, are positively miniscule. Indeed, they are so small that many of the ancient Greeks used the lack of measurable parallax to conclude that the Earth was really at the center of the solar system. Aristarchus, himself, was forced to admit that if the Earth really did orbit the Sun, the distances to the stars must be vast, indeed. It wasn't until 1838 that the first stellar parallax was successfully measured. The displacement was less than 2 3rds of an arc second. To give you some idea of how small that angle is, let's imagine a golf ball. If you place this ball about six miles or ten kilometers away from you, it would subtend an angle in the sky of one arc second. No wonder it was so difficult to measure this. Note that the smaller the parallax, the greater the distance. Indeed, we can define a new unit of distance in terms of this. Let's look at this. Here's the Sun. Here's the Earth, and here is the parallax P, usually given in arc seconds. Here's our star. Okay? And here is the radius of the Earth's orbit. We define the parallax in terms of the radius of the Earth's orbit instead of the diameter. But the idea is basically the same. Notice that if the star gets further away, this angle P prime, I'd better not call it a prime, because you'll think that that's an arc minute. P1 and P2, P2 is definitely smaller than P1. So, the smaller the parallax, the greater the distance. Indeed, we can define a new unit of distance, by D equaling 1 over the parallax that is measured. So this distance here, in terms of the parallax is defined as 1 over P. And if P is measured in arc seconds, this distance defines a unit of distance called the parsec. If P is 1 arc second, the distance is 1 parsec. Unfortunately, only the nearest few hundred stars have measurable parallaxes, at least from the ground. Can we ever hope to get measurements of more distant objects? Amazingly, fortuitously, there are a class of very bright stars called Cepheid variables that pulsate with different periods depending on their intrinsic brightness or luminosity. What a stroke of good forture. This means that just by measuring how long it takes for the brightness of these stars to change, we get for free, a measurement of their intrinsic luminosity. What we see here, in the following picture, shows the light curve of delta- Cephei, and the period-luminosity relationship for many similar stars. And the fact that they are so bright, with some being over 10,000 times the luminosity of the Sun, means that they are visible out to very, very far distances. About 30 mega-parsecs, or 30 million parsecs, but wait a second, I hear you cry. Don't you need, at least initially, an independent measurement of the distances to these objects, in order to figure out what their luminosity is in the first place? And right you are. So the story, while fascinating, is not that simple. But we will touch upon this matter in the coming lectures. Which not only will allow us to use ordinary stars to determine distances, but also provide us with fundamental data concerning the nature of stellar evolution, and the role that this plays in our understanding of the incredibly hot X-ray sources in our galaxies and beyond.
