Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 16 - Hubble's Law and the Big Bang

Chapter 1. Introduction to Cosmology [00:00:00]

Professor Charles Bailyn: Welcome to part three of the course. This is going to be about cosmology. One of the most amazing things that’s happened over the past half century or so is that cosmology, which is the study of the Universe as a whole, has become a scientific subject, and something that one can say something about in scientific terms, rather than merely philosophical terms.

In recent years, in the past ten years or so, there has been kind of a revolution in cosmology, which has come about because of the discovery that the vast majority of the Universe is made out of stuff that we have no idea what it is. So, the discovery is, we have no idea what the Universe is made of. And that actually, you know, it doesn’t sound so good. It sounds like this is a big failure of science, somehow.

But scientists are clever, so we describe it differently. What we claim is that what has happened is that we have discovered dark energy. Dark energy, what is that? We don’t have a clue, but it’s most of the Universe. And we’ve discovered that it exists. So, that’s kind of where we are right now.

What I want to do over the next four or five weeks is talk about the discovery of dark energy–how that was done. And now that we know that it’s there, but we don’t what it is, what people intend to do about it. And this is, I should say, only a relatively small fraction of modern cosmology. If you want to know about many of the other interesting things that are going in cosmology, you have to take a whole course, or several courses, on this topic. Such courses exist. I recommend Astro 170 or 220 if you find yourself interested in this kind of thing.

But nevertheless, this one particular discovery–and, in fact, what I’m going to be focusing on is not just dark energy, but one particular way that dark energy–the first way. There are now other indications that exist. One particular way in which dark energy has been discovered. And so, it’s a fairly narrow focus that I’ll be taking here, but that will enable us to get into some depth about how this discovery was made.

So, let me let you in on the secret before we even begin. Dark energy is something that you can’t see. That’s sort of its name. We’ve done this twice already, right? We’ve discovered planets around other stars–that you couldn’t see the planets. How do you do that? You look at the star–the motion of the star. We’ve discovered the existence of black holes which, of course, you can’t see. How do you do that? You look at the motion of some star that is influenced by the existence of the black hole. So, what are we going to do about dark energy? Well, this isn’t hard to extrapolate. What we’re going to do is we’re going to look at the motion of things that we can see, and infer the presence of dark energy.

And of course, how do we look at the motion? We look at Doppler shifts. That will be another recurring theme. And so, the plan of action here is actually not so different from something we’ve done a couple of times before, although the context and the implications, and the particular details, needless to say, are.

Chapter 2. Spiral Nebulae and Hubble’s Redshift Diagram [00:03:34]

Okay. So, let’s go back in time almost ninety years, back to 1920. So, supposing one were to give a course in Frontiers and Controversies in Astrophysics in the year 1920. What would one have been talking about? Frontiers and Controversies. The big issue, at least in terms of cosmology, was the question of the so-called spiral nebulae. These had been discovered over the previous few decades after the invention of photography, allowed pictures of the sky to be created. And as people took pictures of the sky, what they discovered was, scattered all over the sky are these little spiral clouds, the so-called spiral nebulae.

What are these things, they wondered. And, in particular, where are these things? And there were two basic hypotheses. One was that these are, sort of, clouds–shining clouds of gas. These were known to exist in other kinds of shapes. So, these are clouds of some kind of glowing gas, and that they are part of our so-called galaxy. Galaxy comes from the Greek word for milk. It’s basically Greek for Milky Way. The Milky Way is this band of stars across the sky. It looks like just a white streak. You can’t see it in the city with city lights. You know, if you’ve never seen this, go somewhere dark. Wait until the Moon goes down. Get your eyes adapted and then look for the Milky Way. It’s really very spectacular. If you really want to do this right, do it in the southern hemisphere, because the center of our galaxy is down there. It’s quite spectacular.

Anyway, it had been known since Galileo’s time that what the Milky Way really was, was a band of stars. And so–that there’s this huge bunch of stars out there, so faint, so packed together, we see it just as a continuous band of light. And so, all these stars–that’s our galaxy. And the question was whether these spiral nebulae might be little clouds of something or other scattered around our galaxy.

The alternative, less popular at the time, but perhaps more spectacular, is that the spiral nebulae are galaxies themselves–are whole galaxies of stars, different from our own galaxy. These were sometimes referred to as island universes. So, the Milky Way is our own galaxy, and each one of these tiny little spiral things is a galaxy itself, located much, much further away, obviously, than any of the stars that we can see.

And you can see that this is a question of some importance, in particular, to how big you think the Universe is. If all these spiral nebulae are part of our own galaxy, then maybe what the Universe is, is one galaxy. And at that time, it was already starting to be known how big the galaxy was, and so forth. But if each one of them is its own galaxy, then obviously, the entire Universe must be much, much bigger, because it contains thousands, perhaps millions, of individual galaxies, each of them more or less like our own.

So, we know what the answer’s going to turn out to be. This is correct [points to “island universes” on slide notes]. But they didn’t know that in 1920. And, in fact, they staged what was called the Great Debate in 1920, between a very famous astronomer named Harlow Shapley, who was at Harvard. And he maintained that the spiral nebulae must be part of our own galaxy. And so, he had many good reasons to think this. There was a lot of evidence that the spiral nebulae couldn’t be that far away. They must be really quite nearby. And so, Shapely had, in the true Harvard manner, all of the right arguments, and was completely wrong. And his opponent, a man named Curtis, educated at Yale, but at that point working somewhere else, turned out to be entirely right, but totally lost the debate because he just didn’t have that much evidence backing him up. So, the intuition of the Yale man comes through.

So, this is a famous incident in the history of science. It garnered enormous attention at the time. Remember, this is a year after the eclipse expedition has verified Einstein’s Theory of General Relativity, so people are pretty excited about modern science at the time. And this was reported widely in the press. So, this is another one of our, sort of, science fables, here, “The Great Debate.” And one version of the moral of this story might be, you can have all the right arguments and still be wrong–have many good arguments and still be wrong, as Mr. Shapley turned out to be.

Interestingly, at the time, in 1920, they thought, you know, it’s going to take us a long time to really get to the truth of this, and turned out not to be the case. Within just a few years after the debate had taken place, this problem was totally solved. And the man who solved it was perhaps the greatest observational astronomer of the twentieth century–a man named Edwin Hubble, after whom a telescope was subsequently named. And what helped solve this was simply better equipment. And, in fact, starting kind of a trend that has continued for a century now, what happened was–

So, Hubble was out in California, where there are nice, clear skies, and persuaded a rich man to give a lot of money to build a really big telescope. This was a 100” telescope. That’s the diameter of the–the diameter of the mirror. This is actually still in use, although the site is no good because it’s way too near to the lights of San Francisco. But he took this brand new telescope. He looked at the spiral nebulae. And he did the same thing to the spiral nebulae that Galileo had done to the Milky Way. He resolved it into individual stars. And he could see the individual stars in these spiral nebulae, and discovered in the nearest of them that they were made up of stars. And then, he noticed that the stars that were making them up were incredibly faint. And that even the brightest stars in these things were incredibly faint. And so, he inferred the distance to these things by assuming that, you know, they’re kind of ordinary stars, like any other kind of stars, except they’re way, way fainter, and therefore they must be much further away.

And this was compelling evidence that these spiral nebulae really were island universes–galaxies like our own. And the key thing to do–that is, to have a telescope powerful enough to resolve some of the nearby examples into individual stars. So, there are many galaxies. This was known shortly after the Great Debate took place. And that made the Universe very much larger than people had previously suspected, because there are all these galaxies around.

So, Hubble was the great expert at observing galaxies. And so, the next thing he decided he would look at is–okay, now we know these things are galaxies, let’s check out their– you know, he took Doppler shift measurements. He’s trying to figure out the motions of galaxies.

And so, he discovered a very strange thing. Namely, that they are all going away from us. Every galaxy he measured was redshifted. And so, by the usual Doppler shift interpretation of these things, they must all be moving away from us. And Hubble went further, and he created one of the most famous diagrams in all of astronomy, in which he plots redshift–which is to say Z, which is to say radial velocity divided by the speed of light, at least in the Newtonian approximation, which he was working in, versus distance. So, he figured out how far away these galaxies were. He measured their redshift. And what he discovered is that these things were very tightly correlated.

So, if you plot each galaxy, measure a distance and a redshift for each galaxy, for a large number of galaxies, what you find is that they line up like this. This is the so-called Hubble Diagram, and it’s basically the thing we’re going to be talking about for the next five weeks or so–is Hubble Diagrams, and what you can infer from them. And the Hubble Diagram can be, sort of, summed up in an equation. If this is a straight line, then it must be true that the velocity of a given galaxy is equal to a constant, which was given the letter H, for the Hubble Constant, times the distance. This is Hubble’s Constant. And that just basically says in algebra what this says pictorially, that these things line up in a straight line.

The Hubble Constant is an extremely important number. You measure it by creating these Hubble Diagrams and just measuring the slope of the line. And the primary scientific purpose of the Hubble Space Telescope, the largest and most expensive science project ever created, was to get an accurate measurement of the Hubble Constant. So, all these beautiful pictures you see are byproducts. The purpose of the thing was to measure the Hubble Constant accurately. There had been, for many years, a dispute over what the correct value was. This was resolved by the Space Telescope and other things within the past decade. And this was a big success. We now know what the Hubble Constant is. It’s measured in somewhat weird units. It’s seventy or so kilometers per second, per megaparsec [Mpc]. All right.

Now, let’s pause there for just a moment. You can see why they use this peculiar unit. Because you want to–they’re measuring velocity in kilometers per second, and they’re measuring distance in megaparsecs. Mega, of course, doesn’t just mean big. It’s a technical term. It means a million. It means 10⁶ parsecs. A parsec, you will remember, is about three light years, or 3 x 10¹⁶ meters, or 3 x 10¹³ kilometers.

And so, for each megaparsec that a given galaxy is away–and notice, now, we’ve changed our units from parsecs, which we used for stars, to megaparsecs. That’s basically the change in the scale of the Universe that came about when people realized that the spiral nebulae were galaxies. So, now you have to talk about megaparsecs. In fact, now we also talk, from time to time, about gigaparsecs, which are billions of parsecs.

For each megaparsec that a galaxy is away from us, it moves 70 kilometers per second more away from us. So, something that’s 10 megaparsecs away ought to be moving 700 kilometers per second. Something that’s 100 megaparsecs away would be moving 7,000 kilometers a second away from us. And that’s just an expression of what this relationships is.

Hubble, by the way, got this quite wrong. He correctly lined them all up, but his value for the Hubble Constant turned out to be about 500, in these units, which was quite wrong, and we’ll talk about that–why that was, later on. But, by now, with the Hubble Space Telescope, we totally know this, and that’s an important result.

Now, this diagram and this relationship are basically the key to what we’re going to be talking about. So, in fact, the fact that there’s this linear relationship tells you several things. It tells you that the Universe is expanding. And because of that, it tells you that the–indirectly, there’s other evidence that has to be brought to bear, as well, that the Big Bang exists. So, this is the basis of Big Bang cosmology.

Chapter 3. Measuring the Distance of a Star: The Parallax Method [00:17:35]

I don’t want to talk about that right now. We’ll talk about the Big Bang on Tuesday. We’ll have a whole big question-and-answer session, so, bring all your questions about the Big Bang and cosmology. We’ll do that all up on Tuesday. What I want to talk about now is much more mundane. Namely, how do you measure these points? How do you do this? How do you create this diagram? This diagram has these fabulous implications, but in order to understand what’s going on, you’d better know how you get the diagram in the first place.

Now, part of it’s clear. The x-axis we understand fairly well. Measuring radial velocity–that’s easy. That’s just the Doppler shift. So, that comes directly out of a particular kind of measurement that you can readily do. The y-axis is the biggest problem in all of astronomy; namely, how do you measure the distance to something?

And, if you think about it, that’s what this Great Debate was all about. Some people thought that the distances to these spiral nebulae were a few hundred, or maybe a few thousand parsecs. Other people thought they were millions of parsecs away, and it wasn’t clear which was right. And if you think about looking up into the sky, looking at stars in the sky at night, you can see where there’s a problem. You look up there. You see a bunch of stars, and suddenly one of them shoots across the sky. It’s a shooting star. Except for the motion across the sky, they don’t look that different, and yet, some of them–the fixed stars, are hundreds of light years away. The planets, which look much the same to the naked eye are, you know, a million times closer. And that shooting star is in the top of the atmosphere. And it’s very hard to tell what the distance is when you just kind of look up there. And so, distance is hard. And that’s basically the problem I want to talk about for the rest of today’s class. How do you measure distances? Yes?

Student: You said that redshift was easy, but how do we know what the original wavelength is?

Professor Charles Bailyn: Oh, how do you know what the original wavelength was? You make the assumption that atomic physics and chemistry are the same in the distant galaxies as they are here, and therefore, hydrogen should emit lines at the same frequency as they do in the atmosphere.

Now, one of the things you can do if you don’t like what the cosmological implications of things is, you can say, well, rather than interpreting this as cosmology, let’s just say that in distant portions of the Universe, physics is different. If you go down that road, you can get any answer you want. And this has been seriously discussed from time to time.

All right. So, how do you measure distance? That’s the key. Well, there’s one, and only one, kind of, direct way you could imagine doing this. Here’s the Sun. Here’s the Earth going around the Sun. Here’s a nearby star. And here are a bunch of other, much further away stars. They’re all scattered around the sky. And you observe this nearby star during the course of the year. As the year goes on, you observe it repeatedly. You observe it from here, for example. And when you observe it from here, it’s in this direction, and therefore, it appears to be in that position relative to the other stars. Whereas, when you observe it here, it’s in a little bit of a different position with respect to the other stars. It looks like it’s here. And so, if you observe this star repeatedly over the course of the year, it appears to move back and forth against the background of the other stars.

This is a triangle. We know what to do about triangles. You can measure this angle because that’s just the angular separation of the two apparent positions of this star in the sky. We know this distance. That’s 1 Astronomical Unit, because this is the Earth going around the Sun. And then, what we want to know is the distance to this star, D. Let’s call that D₁. And we know the equation for this already. We’ve done this before. This is α = D₂ / D₁, where D₂, in this case, is exactly 1 Astronomical Unit.

And you may recall that if you measure this in arc seconds, and you measure this in parsecs, and you measure this in AU, you get a consistent set of answers. And so, the way it works out is 1 / α, in arc seconds, equals the distance in parsecs. And the reason that works is because D₂, by this method, is always equal to 1 Astronomical Unit. If you try this on Jupiter, you have to account for the fact that Jupiter’s further away.

This is called the parallax method. It’s common in surveying. You know, you look at the same thing from two different places. You can figure out distances by basically trigonometry, here. This is called the parallax method. And a parsec–this is the definition of a parsec–is one parallax second. It’s a contraction. Because you measure an arc second and the distance is one parsec. And that is the definition of a parsec. That’s why we use parsec as a distance measurement, because it comes naturally out of this parallax method.

And so, that’s a fairly direct geometric method. This is a surveying technique. This is how–and this is a straightforward way of getting the distance. And the problem with this is that it only works on things that are really nearby. Because we can measure, you know, maybe, a hundredth of an arc second change in position, but no better than that. And so, you can only get measurements of distances in this way out to a few hundred parsecs. Works to a kind of maximum, given our current instrumentation of a few hundred parsecs.

But the center of our galaxy is 8,000 parsecs away. These other galaxies are megaparsecs away. We can’t be measuring 1 one-millionth of an arc second, at least not at the moment, in terms of parallax. So, this only works for the very nearest stars. And that’s why there was all this confusion about the spiral nebulae because, you know, if they were 500 parsecs away, you’d never be able to tell.

Chapter 4. Measuring Brightness: The Standard Candle Method [00:25:13]

So, most other distance measurements–methods of distance–are some form of what’s called the Standard Candle Method. So, here’s how the Standard Candle Method works. It’s a three-step process. Part one, you look at something. You know how bright something is–how bright something is. This is basically a version of what Hubble did when he figured out that the spiral nebulae must be very, very distant, and must, therefore, be island universes. You know how bright something is. In the case of Hubble, he’s looking at the brightest stars in the galaxy. He figures they’re about as bright as the brightest stars in our own galaxy, so, he knows how bright they are.

Part two, you measure how bright the object looks. You take a picture of it, or you count photons, or whatever it is that you do, and you figure out how bright it looks. And obviously, for something of a given brightness, the further away it is, the fainter it looks. And we know exactly how that works. It’s a distance-squared thing. If it’s twice as far away, it’s a fourth as bright. If it’s three times further away, it’s 1/9th as bright. It goes as one over the distance, squared. This is a well-known fact. You can try this out with light bulbs at home.

And so, if you know how bright it is, and you measure how bright it appears to be, then you can compute the distance. Okay. And this is why it’s called a standard candle–oh, so, the way this is–how do you know this? That’s the big question. How do you know how bright the thing is in the first place? And the answer is, usually, that you’re looking at something which is an example of a class of objects, like stars, or bright stars, whose brightness is known.

And that’s why you use the term, standard candle. Because here’s a bunch of things. They’re all the same brightness. They all have the same standard candle power. Some of them look fainter than others, but ones that look fainter obviously have to be further away. And if you know the true brightness of this class of objects, you can figure out how far away any example of it is. Hence, standard candles.

And the problem with this is this phrase here, as you can pretty clearly see. If you get it wrong–if you make the wrong assumption about how bright these objects are, you’re going to screw this up completely, and that’s what Hubble did. Hubble was looking at a particular kind of bright star, which he thought he knew how bright it was, and he was wrong. And so, he got the wrong Hubble Constant.

Now, because he used the same kind of star in all his galaxies, he got it the same amount wrong for all these different galaxies. So, they still lined up. They just lined up along the wrong track. So, it was still true that something that he thought was twice as far away as something else was in fact twice as far away as something else. He just got both of those numbers wrong by the same factor.

Okay. This brings us to the awkward question of, “How do you measure brightness?” And now, we have to talk about one of the great impediments to learning astronomy–namely, the magnitude scale. Astronomers count brightness upside down and logarithmically. And I am now obliged, by my membership in the astronomical community, to inflict this upon you. So, the magnitude scale–this is how we measure brightness. It’s upside down and logarithmic. And the key numerical relationship that works looks like this. If you subtract the magnitude of one object from another, that equals minus five halves times the log of the ratio of the brightnesses [⁵⁄₂ log (b₁ /b₂)]. Don’t panic. The magic word, here, is “help sheet,” which will be posted later this afternoon. So, this is a key equation.

So, let me just define the terms. This is the magnitude of two different objects and this is the brightness of the two objects. Whereby “brightness,” I mean something sensible, like how many photons per second do you get from them, or some other kind of true measure of how bright they are.

Now, this is a somewhat awkward equation, because it’s a relative equation. It doesn’t tell you what the magnitude of either one of these things is. What it tells you is the magnitude of one compared to another. So, in order to figure out the magnitude of something, you have to know the magnitude of something else. And so, you need one other piece of information to have this be useful, which is that they have defined a particular star to have magnitude 0. The star, Vega, is defined to have magnitude 0. So, if you start with Vega, you can figure out the magnitude of any other given object.

Actually, this is now causing trouble, because it turns out that in the far infrared, Vega is variable. And that’s kind of unfortunate for the basis of the whole magnitude system to turn out to be variable. But, they’ve coped with that in various ways.

All right. Let me pause and remind you of some things about logarithms, which you may have forgotten. What’s the definition here? The logarithm of 10^x = x. That’s the definition of a logarithm. So, for example, the logarithm of 3 x 10² is equal to the logarithm of 10^½ that’s 3 times 10², is equal to the logarithm of 10^2.5. Because when you multiply 10^x by 10^yyou get 10^x + y, which is equal to 2.5. That’s an example. More examples on the help sheet. Just, in general, log (10^x x 10^y) = x + y. Because when you add those together, that’s how it works. log (10^x / 10^y) = x - y. Again, because of the way logarithms multiply together. And the logarithm of–let’s see, 10^x, raised to the mth power, is equal to mx [log([10^x]^m) =mx].

This, by the way, is why logarithms are so incredibly useful. You should always do all your arithmetic in logarithms. You should just automatically convert everything in your head into logarithms. In fact, in the days before calculators, this is how people used to do arithmetic. This is how slide rules work, I should say. They mark the thing off logarithmically and then you move them back and forth. And people, you know–if you memorize, like, ten logarithms of a few convenient numbers, you can do all sorts of calculations in your head because multiplication turns into addition, which is much, much easier to do. And so, if you can convert things into logarithms in your head, then all you have to do is add numbers. That’s easy.

Similarly, taking something to the mth power, for example, taking the square root of a number, or taking the cube of a number, or something like that, reduces down to multiplication. It’s hard to take a square root of a number in your head, but it’s easy to divide a number by two, which is the equivalent of taking a square root. So, if you’re thinking in logarithms, all you got to do is divide it by two, and you can amaze your friends by doing square roots in your head–if your friends are the kinds of people who are amazed by that kind of thing. Mine are. And so, I recommend this to you. If you have to prove that you’ve learned something in college, spend a half hour memorizing ten different logarithms and then just blow people’s minds by taking square roots on bets. Okay.

Or you can do problem sets in Astronomy 160. For example, Sirius–the star Sirius, which is the brightest star in the sky, is about three times brighter than Vega. So, what’s its magnitude? What’s its magnitude? Well, let’s write down the equation. Magnitude of Sirius minus the magnitude of Vega is equal to -⁵⁄₂ log of the brightness of Sirius, divided by the brightness of Vega.

Now, any time you see three times brighter or twenty times fainter, or anything of that kind, what you’re really talking about is a ratio. If Sirius is three times brighter than Vega, that means the brightness of Sirius divided by the brightness of Vega is three. That’s what that statement means. And so, we can just plug it right in. -⁵⁄₂ log (3). Because Sirius is three times brighter than Vega, so that ratio is equal to 3. What’s the log of 3? One half. Thank you very much. This is equal to -⁵⁄₂ log (10^½). You will remember that the square root of 10 is 3. So, 3 is also equal to π, but for logarithms, the important thing is that 3 is equal to the square root of 10. And so, this is (-⁵⁄₂) x (½). -⁵⁄₄.

Now, the magnitude of Vega, we know. That’s 0. So, the magnitude of Sirius is equal to -⁵⁄₄. This is what I mean by the fact that the scale is upside down. The brighter the star is, the lower–or, if it goes through 0, the more negative the number becomes. Minus 1 is a brighter star than 0. 0 is a brighter star than 2; 2 is a brighter star than 5, and so forth. So, low numbers are bright. The magnitude of the Sun, obviously extremely bright when we look at it, turns out to be -26.5. And the magnitude of the faintest star that can be seen with the Hubble Telescope is about +30, which is incredibly faint.

And that, by the way, is why, again, why logarithms are such good news. Because the entire range of things we can see in the sky goes from -26 to +30. Those are numbers you can get your mind around. In fact, the difference in brightness between those things is a number that I can’t even pronounce. And so, much easier to deal with magnitudes. All right.

Chapter 5. Absolute and Apparent Magnitude [00:38:06]

Now, that has to do with how bright things are, how bright things appear to be, or how bright they are. But remember what we’re trying to do. We’re trying to compare how bright something is, intrinsically, to how bright it looks. And this gets you into the question of–there are actually two kinds of magnitude. The intrinsic brightness of an object is something called absolute magnitude, whereas the observed brightness is referred to as apparent magnitude. And now, the astronomers screw you up again, because we have different symbols for the apparent magnitude and the absolute magnitude. The absolute magnitude is given a symbol capital M, and the apparent magnitude is given the symbol smallm. And the problem is that you can’t actually tell the difference between those two in my handwriting, and probably yours as well. So, this leads to untold confusion, but that’s the way life is. And the relationship between these two has to relate to distance.

Okay. Before we get there, the definition of absolute magnitude. You’ll notice that in the previous comparison of Sirius to Vega, I was talking about apparent magnitude. How bright it looks in the sky. Absolute magnitude is defined as the apparent magnitude if the object in question were exactly 10 parsecs away. So, the Sun has an apparent magnitude, as I said before, -26.5, but it’s actually not that bright of a star. If you took the Sun out to a distance of 10 parsecs, its magnitude would be 4.7, which is among the fainter stars you could see in the sky. And so, the absolute magnitude of the Sun is 4.7, even though its apparent magnitude is - 26.5.

So, it turns, for example, that the star Sirius is about 3 parsecs away. And so–oh, I haven’t told you an important thing yet. We’ll come back to this in a second, though. The problem set problem obviously is going to be what is the absolute magnitude of Sirius, but I have to write down the formula first.

Little m, that’s the apparent magnitude, minus big M, that’s the absolute magnitude, is equal to 5 times the logarithm of the distance over 10 parsecs. Now, notice what happens. If the distance is equal to 10 parsecs, then you’ve got the log of 1. What’s the log of 1? Zero. Thank you very much. Because this is the log of 10⁰. log 10⁰ = 0. And so, if the log of this thing is 0, then this right-hand side is 0. So, at a distance of 10 parsecs, the apparent magnitude is equal to the absolute magnitude, because m - M = 0, which is exactly what the definition was. So, that works.

So, example. Sirius. So, what is the absolute magnitude of Sirius? All right. Here we go. The apparent magnitude, we figured out just a minute ago, is - ⁵⁄₄, minus the absolute magnitude, which is what we’re trying to figure out. 5 log (3 /10), which is ⅓, which is equal to 5 log(10^-½). 3 = 10^½; ⅓ = 10^-½, which is - ⁵⁄₂. That 5 comes from here. The -½ comes from here. So, M = ⁵⁄₂, let’s see, minus ⁵⁄₄, is equal to ⁵⁄₄. And that is the absolute magnitude. I’m sorry. Oh you want me to do–okay, sure. So, we okay to here?

Student: [Inaudible]

Professor Charles Bailyn: So oh, oh, okay, fine. Log of–this is actually an important point. Log(⅓), that’s what we’re going to try and do, is equal to log (1/10^½). 1/10ⁿ = 10^-n. That’s the key thing. And so, this is the log of 10 to the minus one half [log(10^-½)] That, then, means that 5 log (⅓) = 5 x (-½) = - ⁵⁄₂. Yes?

Student: [Inaudible]

Professor Charles Bailyn: There are no units–yeah, magnitudes are just numbers. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Ah, well, what I did–sorry, I skipped a step. To start from that. -⁵⁄₄ - M = -⁵⁄₂. So, I multiply both sides by negative 1, then I subtract ⁵⁄₄ from both sides.

You got to keep the minus signs straight. The easiest way to make mistakes in this is to get the thing upside down and lose track of where your minus signs are, which is really easy to do, because the whole scale is backwards. So, you know, -⁵⁄₄ is bright, whereas ⁵⁄₄ is faint. And so, you’re–one is constantly getting these things upside down. Be careful. Look at your answers to see it makes sense. If some incredibly faint galaxy turns out to have a magnitude of -50, that’s almost certainly wrong, because that’s much brighter than the Sun.

All right. So, now. Now, we can actually do the Standard Candle Method. Here’s the key problem. If you observe a star like Sirius and you–I don’t know, you take its spectrum, or some alien comes down and tells you this star is just like Sirius, or however you work this out. If you observe a star like Sirius and measure its apparent magnitude to be 8.75, how far away is it? And now, we’re back to where we were twenty minutes ago, before I started all this nonsense. Namely, we’re trying to measure the distance of something, which was the whole purpose, as you may recall. So, how far away is it? Let’s see.

m - M = 5 log (D /10 parsecs). The apparent magnitude is 8.75. The absolute magnitude, we just figured out, is ⁵⁄₄–1.25–is equal to 5 log (D / 10 pc). And you’ll notice I’ve chosen my numbers carefully, because this is going to work out well. This is 7.5, and I’m going to divide both sides by 5; 7.5 / 5 = log (D / 10 pc).

Or, 1.5 = log (D / 10 pc). Now what do I do? Yeah, exactly. You have to take 10 to the power of both sides. Whenever you’re stuck with log of something and you don’t want the log of the something, you want the actual something itself, what you got to do is 10^1.5 = 10^{log (D/10pc)}. Ten to the log of anything is equal to itself. So, this is D / 10 parsecs. And thus, what’s 10^1.5?

Student: Thirty.

Professor Charles Bailyn: Thirty, yes.

10^1.5 = 10¹ x 10^½ = 3 x 10¹ = 30.

So, 30 = D / 10 parsecs. And D is equal to 300 parsecs. And for the person who asked about units, which was a very good question, this is where the units come back. It’s because there’s this 10 parsecs embedded inside the equation. And so, the unit of length comes back here.

Chapter 6. Conclusion [00:48:04]

And so, that’s how you measure distance. You know how bright something is, probably by having looked at some other, more nearby example. You measure how bright it looks. You compare those two things, and out pops the distance. So, this is done, usually, in the form of what’s called the distance ladder, which we will talk about much more in section. So, you’ll get many more opportunities to do this.

Nearby stars, you measure the distance in parallax–from the parallax method. Then you find examples of similar stars. You measure the apparent magnitude. You assume the absolute magnitude to be the same as the absolute magnitude for things you already know the distance to. You assume the absolute magnitude. And you compute the distance. This, then, gets you new–this tells you that you learned from this, the absolute magnitudes of brighter things. Not just stars, but whole galaxies, supernovae, all sorts of things–brighter things, which you can then measure further away.

And just to conclude, you can see pretty clearly that this method is fraught with potential problems. Because every time you go through this–you know, that’s a swear word in science. You’re making an assumption, and that assumption can lead you astray. And so, the whole history of cosmology, since 1925, is in that word. Assuming the absolute magnitude of things for which you’re trying to measure the distance, so you can put them on the Hubble Diagram, so that you can deduce the rate of expansion of the Universe, and thus its age and ultimate fate. So, it all rests on this little point embedded inside the Standard Candle Method, which will be discussed at great length. Okay, that’s it for today.

[end of transcript]

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