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Hello, this is Jason Kendall. Welcome to my next lecture on introductory astronomy. Today we're going to be talking about distances and why they're so important. Well, and how we get them. So distances in astronomy are both extremely important and really hard to get. So why are the important? Because if we can get to distances to stars, then we can learn more about them, such as their physical size or their luminosity, how bright they are.
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We can determine their masses indirectly, and then we can also determine their true motion through space. So distances are really important, and if we don't actually get distances, we can't actually get many other physical parameters about the stars out there. So how do we go about doing that? Because nobody's ever really been to a star and nobody's ever laid out a whole bunch of meter sticks or strong some string to another star.
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And even the Sun, which is extremely close at 93 million miles closer than any other star, we haven't actually sent any buddy to it because, well, they would get burned up. But it's also an extremely difficult project just to go into orbit around the sun, close in for study in any event. So we've never actually been to the stars that are distant in the sky that look like points of light in the sky at nighttime.
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So why not? Well, the problem is, is that we can't actually get there by anything. We can't touch it. So since we can't touch it, we have to use geometric geometry in order to get the distances to these stars. Well, let's actually take a first step and see how we use geometry to get distances. One of the more simple ways we can think about it is that is that it's done pretty much all the time by surveyors.
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Surveyors use the process of of out well, they use trigonometry and they use triangulation in order to get distances to these five stars. How do they do it? Well, let's pretend that we're trying to measure the distance across a river and maybe we're on one side and there's some significant tree on the other side. So we look at that tree and we say, okay, well, how far is that tree across the river?
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Well, let's see. We can then say let's we'll be on one side and let's say we find the place where it looks to be the closest a cross. So we'll call that our starting point. And so we'll look straight across the river so that it's 90 degrees to the shoreline. And so we'll start with that. And we look straight across the river and that's our distance.
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We want to measure one emission distance from that starting point to the tree across the river. Okay, So we plant a flag and we point a little arrow or make a mark on the ground, says this way to the tree, and then we make a 90 degree angle like that. And then we walk, say, 200 yards away along that 90 degree angle and look across again to the river.
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Now, here where we suspend, we're pretending that the river might be a couple of maybe a mile or so across. So the tree you have to a little telescope in order to see the tree, whatever. The point is, you've got to walk maybe 200 meters or 200 yards up the stream to create what we call a baseline. So when you get to the other end of the baseline, you put down your little telescope again and sightline all the way to the tree again, you find it and you find that the angle between the sightline at the second location and the baseline is not 90 degrees, it's less than that.
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So in the very simplest way we simplified it just for that because it's never this simple in real life, but we're trying to make it really simple. So the idea gets across. The idea is that you now have a 90 degree try a right triangle where it's 90 degrees here, less than nine degrees there, and some small angle between the two sightlines at the tree so that the sum of all the angles inside a triangle is 180 degrees.
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So let's say this is 90 Geller sightlines, maybe it's 85. And so the angle between the two sightlines at the tree is five degrees. So how do we get that distance? We use simple ninth grade geometry and trigonometry class stuff, and we say the baseline divided by the distance across the river is equal to the tangent of the little angle that attends the two sightlines.
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That's just the opposite over the adjacent, in the old in the old way of thinking about it, where the opposite is the baseline, which is opposite. The little angle that spends the two lines from the tree and the adjacent is the is the side of the triangle adjacent to the that that we're trying to measure. We ignore the hypotenuse because the hypotenuse is not the adjacent angle.
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It's called the hypotenuse. All right. So the opposite of the adjacent equals the tension of the angle. So therefore the distance across is simply the baseline divided by the tangent of that little angle, which might be five degrees. So that gets us our distances across a river. But that's the basic idea. But now let's pretend when we looked at this thing, we said, okay, so maybe when we were a cross, the way the tree was the reason we picked that tree across the river was because we're standing in the middle of a field and there were a whole bunch of things behind it.
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So when you went from one place on the baseline to another, the tree, well, the things that were looked like it shifted with respect to the background stuff across the river and that shift we call parallax. So what is parallax? A better way of thinking about parallax is let's make it really easy thing that you can do right here at home.
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What you do is you take your thumb, take your thumb at arm's length, and you bring it close to your face and then you blink your eyes left and right, back and forth, and you see that your thumb apparently jumps left and right know if it's close. The jump is big, but if it's far, the jump is smaller.
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And if it's really far with a stretch your arm out or you look at somebody else's thumb, then the jump is really small. So parallax is the shift that you see with respect to background and things. When you change the angle, the place from which you're observing. So stellar parallax will then be over. The parallax that we see with respect to background stars.
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So maybe there's some close stars, some close foreground stars, and some really far distant stars. And, you know, stars are apparently distributed in space. So why not? And that's the case. That is the case. There are some nearby stars and some of our stars. So what exactly is the parallax for stars? All right. So if we have some nearby stars, they will have a big parallax shift.
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And if they're smart, they'll have a smaller crack. The parallel lactic shift and the farther they are, the greater the pair lactic that the less the shift is until it gets so far that it's imperceptible or incredibly imperceptible. But what's our baseline? Okay, the baseline for looking at stellar parallax is the orbit of the earth around the sun.
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So as the Earth goes from, say, December around the sun to June, we have two different vantage points. And these vantage points are separated by a distance of 300,000 through 400 million kilometers or 180 million miles. So 93 million miles, 180 million miles, that's a date. The radius of the Earth's orbit is 93 million miles or about 150 million kilometers.
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So the diameter of Earth's orbit is about 300 million kilometers, which is a really big baseline, but it's just barely enough for us to get the distances to the stars. Okay, so how do we get that? So what do we what are we talking about? Okay, so this little angle, we're going to make a new definition. The angle that we're going to call pair Lactic shift isn't the full angle.
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It's half the angle. So we're going to make this angle with respect to the center in the solar system, which is the Sun kind of weird, but that's how we're going to define parallax. Half of the shift that we see from, say, December to June. Okay, So Parallax is our fundamental measurement. So how do we do it? Okay, so when we measure these things, how what, what is their size?
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The size of a typical parallax is very, very small for the stars. The size is on the order of an arc. Second, that's a really small measurement. So with the baseline of one astronomical unit, which is 93 million miles, the length, the triangle is extremely narrow and it goes out very, very far. It's all goes out to one arc second.
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All right. So instead of with our tree example, it might have been five degrees. So what's an arc second? All right, So an arc second is the following. We let's take your thumb and take it at arm's length again. We'll do that whole trick again. A lot of tricks like this that keep up a thumbs in hands. And now this is backyard astronomy.
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Okay, so your thumb held at arm's length is about a degree across. Now, if you make 60 even marks across it, those would each be one arc minute. So every degree can be broken up into 60 arc minutes. And guess what? Every arc minute can be broken up into 60 seconds. So imagine that you took 6060 marks across your thumb, little tiny marks like on a ruler.
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And then in between each of those marks, you put 60 more marks. So there would be 3600 little marks across the back of your thumb. That's a lot of little marks. And it sure would be a mess. But hey, it's all for science. Okay, so what's an angular shift of one arc second? Or how big is something that is one arc second?
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Well, take a dime and place it about two and a half miles away. That's the size of one arc. Second. Or take a football, you know, an NFL football and you push it 39 miles away. That's about one arc. Second, an arc second is a really small angular measurement. The typical things when you look at the moon, say when you look at the moon, the moon's about half the size of your thumb, the standard size of things that you can see on the moon without the aid of a telescope are about 5 to 10 arc minutes.
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So you're looking at things that are incredibly small angles. So ARC seconds are really, really tiny. All right. So what kind of parallax do we see for stars in fact, no star that we see in the sky has a parallel arctic shift that is greater than an arc. Second, every star has a parallel Arctic shift, less than an hour at second.
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And the biggest pair lactic shift is for Proxima Centauri, which is about three quarters of an arc. Second .77 arc seconds to be specific, and that makes it have a distance of approximately 1.3 parsecs. We'll get to that in a second. But really, let's actually kind of go back and see what how tough this thing is. In fact, it's parallax, as are so difficult to measure that the first one was only measured in 1837, 61 Cygni, which is a nearby star.
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All right. So how far are these things when we say let's let's see what we mean by the distances derived from parallax is if we look at Alpha Centauri and say, how far it is based on that triangulation method, if the distance between the sun and the earth was the width of your pinkie, then the distance to alpha Centauri is about 1.7 miles, and the distance to the brightest star in the sky.
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Sirius is about three and a third miles and Beetlejuice, which is that star in Orion that everybody says three times just for fun. Beetlejuice is 250 miles away. So where did that distance come from? Well, we derive a concept called the Parsec. If a star has a parallax of one arc second, then we define its distance to be one parsec.
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That's what it means. Parallax of one arc second a parsec. That's what Parsec stands for. So what's a parsec? A parsec is the same as 206,265 astronomical units. So one accurate and pretend one astronomical unit is the width of your pinkie. That's 93 million miles. Now. Now take 206,000 of those. That's a long ways out. So make 206,000 times your pinkie going left and right and that'll get you to the next star.
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Assuming that the sun is here and the earth is there. Now, on this size scale, Jupiter is about that far from the sun. Saturn's about that, far from the sun, and Uranus. And that Neptune's about this far from the sun. And the Voyager is about an hour about this, far from the sun. So getting to the next star is a long, long, long ways.
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All right, so how far is 26,265 astronomical units? Well, it's about 30 trillion kilometers or about 18 trillion miles. Or let's use another ring that you're probably familiar with. It's about three and a quarter light years. But what's a light year? A light year is the distance. Light travels in a year. Light has a distinct speed, 186,000 miles a second.
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So we can think how long, how far does light go in one year. Now that works out to be about 30 trillion miles. Right? So 30 well, it's about there. It's about 6 trillion miles anyway. So a light year, though, is a weird unit of distance because we're assuming that we can talk to the little photon that traveled all that light pack.
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It. The light pack. It doesn't carry a ticket with it saying this is when I left my place, my station. Now it doesn't have a clock on, it doesn't have a ticket, doesn't have any way of saying this is when I started. So a light years, a really bad unit of distance, because there's no way to say when the light started.
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However, a parsec is a fantastic unit of distance because it's based on geometry. So it's a real unit. So people use light years all the time because it sounds kind of cool, but light years are a unit of distance, not of time. And they're a really bad unit because you know what's a light year? It's like, okay, fine.
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And in fact, we'll find that that actually has some very strange meanings. We talk about relativity anyway, so the parsec is the fundamental unit of distance in astronomy, and it's the basis for which things are at great, great distance. All right, So how are we going to do these measurements of parallax? Well, Parallax, the best you can do on the ground is about a hundredth of an arc second.
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So that's pretty far that gets you out to about 100 parsecs or 300 light years. So on the ground you can see a parsec that goes out to about 300 light years. There's not a lot of stars within 300 light years. In fact, there's very few stars might be less and less than a few hundred within that time in the night in the early 1990s, 1980, 1993, the European Space Agency flew the Hipparchus mission and they got a stellar parallax is really good.
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Parallax is for about 100,000 stars or so, down to down to about 100th of an arc, second and two and a half million stars about ten times further, but lower precision. So that was our main catalog of Stellar Parallax is for a long time until the Gaia Mission very recently gave it and gave their their data set out.
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And they have a dataset of about a billion stars and they go out to about ten micro arcseconds that's really far or a 10th of one part in ten to the fifth of a part of a mile of an arc, second to the really far things. And that gets us almost across the galaxy. So we get about a fifth, almost half of the Milky Way is B can have its parallax done by Gaia, which is an amazing thing.
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And that data set will be used for a long, long period of time. Well, one of the key pieces of part of the parallax distance is we have to know the distance to the sun. So because the parallax angle, the parallax existence is so incredibly dependent upon the distance to the sun, how do we get that? Because without the distance of the sun, we'll never get the distance of the stars.
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So the distance to the sun is calculated in a very tricky way. If we look at the planet Venus and we see how it orbits the sun, will notice that at times in a telescope to Venus displays a quarter phase, just like we talked about with the moon. The moon has quarter faces which look like half illuminated. When Venus is in quarter phase, its exactly if we're seeing only one quarter of it.
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So that means that the angle that if we look at Venus, then Venus makes a 90 degree angle with respect to the sun. And so when we look at Venus, it's fourth roughly a little bit more than 45 degrees away from the sun in the sky. And then it'll be half illuminated, which means it's a quarter phase. And so therefore we have a right triangle and all we have to do is measure the distance between us and Venus at that time going to right triangle.
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And that gives us the distance of the sun by simple geometry and trigonometry again, so we can triangulate our way to the sun, not by actually measuring it out there, but by bouncing radar beams off of off of Venus, radar waves off of Venus. See how long it takes to get from there and back. And then measure the angular separation between the Sun and Venus when it's at Quadrature or when it's at when it's at quarter phase.
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And what do we get? We get the distance of the sun by trigonometry. Once we get the distance of the sun by trigonometry, then we get the distances to the stars. And this is the basis by which we start to get distances deep into the cosmos. It all starts with parallax, which is the distant, which is that which is derived as a result of the Earth going around the sun.
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And we see an angular shift of the stars in the sky as the earth goes around the sun from over the course of a year. All right. We'll see you next time.