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Professor Charles Bailyn: Okay, welcome back for more cosmology. What I want to do today is quickly review what we were doing about magnitudes and make a comment or two about the problem set, and then, go back and talk about the implications of the Hubble Law and the Hubble Diagram, which are formidable, to put it mildly.

Okay, magnitudes. There's a couple of these magnitude equations. I'm just going to write them down. The first of them looks like this. And this equation is used — okay. So, this equation is used to relate magnitudes of two different objects to each other. So, we've got two different objects. And it can be used for either kind of magnitude — either absolute or apparent magnitude, just so long as you don't mix them. So, it's two different objects, but only one of the magnitudes. One kind of magnitude. And depending on which kind of magnitude you use, this brightness ratio — it's either the ratio of how bright it looks or the ratio of how bright it is — whatever's appropriate.

Now, on the help sheet on the web, I have this equation in a somewhat different form, and it's important to realize that it's the exact same equation. Watch this. Let's see. Let's multiply both halves by - 2/5 which is - 0.4. So, this is -0.4 (M1 – M2) = log (b1 / b2). And then, let's take 10 to the power of that. That gets rid of the log. And this is now the form that it is on the help sheet on the web. So, it's exactly the same equation, just expressed differently. And you can use either form, whichever is more convenient.

Okay. The other equation looks like this. 5 log (D/10 parsecs). And this relates one object, but it relates both kinds of magnitude to each other. So, the first one is two different objects, but only one of the magnitudes. The other is one object and it relates the two different kinds of magnitudes to each other, and to the distance to the object. And as you can see, this — both of these equations, actually, have three unknowns. One, two, three. That means you've got to know two things in order to find out the third.

And that brings me to the comment I want to make about problem 2a on the current problem set. You are asked in this problem to determine the difference between the absolute magnitude of one kind of star, called Type 1 Cepheid, so I label it C1. And another kind of star, Type 2 Cepheids, which I label C2. And if you're asked to — and this difference is called, I don't know, delta MC or something like that.

And having been asked to do this, the logical thing that you might try to do is say, all right, I'm going to use one or the other of these equations — I'm not sure, in advance, which, to compute this one. Then I'm going to compute this one. And I'm going to subtract the two, and that's going to give me the answer. That approach will fail. Okay? That doesn't work in this particular case, because you don't actually have enough information to compute either one of these things. You do have enough information to compute the difference. And let me just give you a very brief hint on how you might go about doing that. Let's see. Let me take a new piece of paper here.

Write down mC1 - MC1 = 5 log (DC1 / 10parsecs). And now write the exact same equation down for C2, where the two different distances are the distances you get by assuming one or the other kinds of these magnitudes. This is equation one. This is equation two. Now, let's subtract.

1 – 2.

So, then, you get mC1 - mC2 - [MC1 - MC2] = 5 (log DC1 - log DC2).

Okay, magnitudes. There's a couple of these magnitude equations. I'm just going to write them down. The first of them looks like this. And this equation is used — okay. So, this equation is used to relate magnitudes of two different objects to each other. So, we've got two different objects. And it can be used for either kind of magnitude — either absolute or apparent magnitude, just so long as you don't mix them. So, it's two different objects, but only one of the magnitudes. One kind of magnitude. And depending on which kind of magnitude you use, this brightness ratio — it's either the ratio of how bright it looks or the ratio of how bright it is — whatever's appropriate.

Now, on the help sheet on the web, I have this equation in a somewhat different form, and it's important to realize that it's the exact same equation. Watch this. Let's see. Let's multiply both halves by - 2/5 which is - 0.4. So, this is -0.4 (M1 – M2) = log (b1 / b2). And then, let's take 10 to the power of that. That gets rid of the log. And this is now the form that it is on the help sheet on the web. So, it's exactly the same equation, just expressed differently. And you can use either form, whichever is more convenient.

Okay. The other equation looks like this. 5 log (D/10 parsecs). And this relates one object, but it relates both kinds of magnitude to each other. So, the first one is two different objects, but only one of the magnitudes. The other is one object and it relates the two different kinds of magnitudes to each other, and to the distance to the object. And as you can see, this — both of these equations, actually, have three unknowns. One, two, three. That means you've got to know two things in order to find out the third.

And that brings me to the comment I want to make about problem 2a on the current problem set. You are asked in this problem to determine the difference between the absolute magnitude of one kind of star, called Type 1 Cepheid, so I label it C1. And another kind of star, Type 2 Cepheids, which I label C2. And if you're asked to — and this difference is called, I don't know, delta MC or something like that.

And having been asked to do this, the logical thing that you might try to do is say, all right, I'm going to use one or the other of these equations — I'm not sure, in advance, which, to compute this one. Then I'm going to compute this one. And I'm going to subtract the two, and that's going to give me the answer. That approach will fail. Okay? That doesn't work in this particular case, because you don't actually have enough information to compute either one of these things. You do have enough information to compute the difference. And let me just give you a very brief hint on how you might go about doing that. Let's see. Let me take a new piece of paper here.

Write down mC1 - MC1 = 5 log (DC1 / 10parsecs). And now write the exact same equation down for C2, where the two different distances are the distances you get by assuming one or the other kinds of these magnitudes. This is equation one. This is equation two. Now, let's subtract.

1 – 2.

So, then, you get mC1 - mC2 - [MC1 - MC2] = 5 (log DC1 - log DC2).

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