Hi gang and welcome back to analyzing the universe. Last time we left GK Per in the precarious aftermath of an explosive state. But the story gets even more interesting, because during these infrequent minor outbursts, every three years or so. The object starts emitting copious amounts of x-rays, and does so in a very special way. So let's head back to DS9, and see what we can find out. So we open DS9 again. Were going to look now and see what those x-rays look like. Go to analysis, virtual observatory, connect using web proxy, Rutgers primary MOOK, observation number 3454, click on the title. There it is. Now, we're going to look at the light curve. So, we go to analysis. Let's cut off things here. Bring that over here. Go to the Chandra Ed analysis tools. Cut that off. Bring it over here. And click on F-Tools Light Curve. Plot the light curve data. And there it is. Boy that looks like it's bobbing up and down here. Maybe one peak here, two peaks there, another one there. But does it have a regular period? Let's zoom in. Let's take our cross here and if we click anywhere in there and hold things down, we can drag a smaller area and there it is and look at that, things are going up and down, and it looks like maybe one, two, three, four, five, six, seven peaks or so in about oh, I don't know, 2000 seconds. Man, this isn't very precise. Can we do better than just this eyeball method? Fortunately mathematics comes to our rescue with an immensely powerful tool called a Fourier Transform or Power Spectrum. Let's see how this works. You can imagine that it would become quite tedious to figure out by hand what the period of variability of an object might be, or indeed, whether it even exists at all. But let's see how mathematics can come to our rescue. First, imagine a light curve, which is a plot of intensity versus time, that looks like this. It's easy to see that a sine curve can come pretty close to duplicating the data. Now, what if our light curve looks like this? Still no problem. We can get a sine curve that matches pretty nicely. But what about a constant intensity source? Something that looks like this. Well, a moment's reflection will tell you that a sine wave of 0 frequency, or infinite period, will have precisely this property. Remember that f, a frequency, is one over the period. For example, if the sine wave has period of one-third of a second, then it will bob up and down, three times in one second. And if something happens, say, seven times a second, its period is one-seventh of a second, and so on. But now, what if our light curve looks erratic, as all real light curves will? The incredible thing is that any light curve, no matter how bizarre it looks, can be fit exactly by a bunch of sine curves of different frequencies added together with different amplitudes. It's a miracle, really! This transformation of an arbitrary curve, intrasinusoidal components, is called a Fourier Transform or a power spectrum. Entire books have been written on this topic and rightly so. Let's see what the Fourier Transform also known as a power spectrum, of our various examples might look like here. In the first one, you see we just have one frequency and therefore there will be zero amplitudes for all frequencies except that particular one. In this example the same thing is true, except the frequency is much higher. So in that situation, everything will be zero with the exception of a higher frequency. Now, it turns out, if you're really sharp, you'll see we made a slight error in that situation, because there is a constant amplitude upon which these two sine waves have been superimposed. So there really is going to be a zero frequency component in every single light curve that we examine. And that will represent the part of the light curve that is basically constant in intensity, upon which a certain amount of the emission will be periodic. In this case all we have is that constant intensity component and so our Fourier Transform would have a zero frequency component, and nothing more. Now, in this particular case, you can see that we're going to need a big mixture of different frequency sine waves, so our Fourier Transform might look something like this. And it might be very hard to tell whether or not there is any periodic component in this particular data. So now that we have an understanding of the basics of this type of analysis, let's return to DS9, and look at the power spectrum of real data, in the form of x-rays emanating from GK Per, in the year 2002. OK. Back at DS9. We look at the Chandra Ed analysis tools again, and now, after running our light curve and finding out what a power spectrum is all about, let's click on that power spectrum. Plot power spectrum data. And here it is. All of the amplitudes or power of all of the sine wave frequencies necessary to exactly reproduce our light curve. Let's zoom in and see it in detail. We are obviously going to be interested in this area of our power spectrum here, because that's where all the action is. So we click around here, and extend our box up over here. And now you see after zooming in, we have a frequency near zero, which is probably that constant part of an intensity source that might be just steady. And another one that is at about zero zero, oh, between zero zero two and zero zero three cycles per second. So, let's zoom in again here and see what it looks like. Now, you can see that we have a very, very strong peak at about 0.00284 Hertz, or cycles per second. Well, 0.00284 cycles per second translates into a period, and you should work this out, of about 352 seconds. Now we can do something really cool. We can fold all of our data, modulo this 352 second period, and put all the maxima and minima of all of the up and down parts of the light curve on top of one another, add them up, and see what the overall structure of the light curve looks like. Let's do that. We have a period fold. We enter the value of a period of 352 seconds, and we're going to divide that light curve into 25 little sections. So there will be 25 little piece of total length 352 seconds, and we going to fit the entire light curve into that particular display. So we say OK and look at that. There it is. The folded light curve of GK Per, bobbing up and down. You say, wow, it's just doing nothing but going up and down but now, notice the y-axis here. The y-axis has automatically been scaled to include the entire observation. It doesn't go from zero brightness all the way up but just from about 0.85 brightness units on up. So let's do this little bit more legitimately. Let's go to Graph and we click on Range, and when you do that it brings up a little box that allows you to change your x and y-axis. Let's change our y-axis so that it starts at zero, instead of 0.85, and goes up to oh, it looks like about 1.2. We'll keep that. We un-check automatic, so that we are actually doing things manually here. Click on OK, and now you see what the light curve of GK Per actually looks like in its average. OK! Pretty cool. Now let's do something interesting. Let's see what happens if we use the wrong period. A period that really doesn't match the actual periodic change in GK Per. By the way, before we do that, notice these little red tick marks here. That represents the error bars associated with each one of the assignments of these measurements, to the intensity of GK Per. You see, it's impossible to fit a, a straight line to this data. The error bars are too small to allow that. So we really have uncovered a truly periodic component to the x-ray emission of GK Per. So, now we're going to look at a period fold for a quote-unquote wrong period. Let's get rid off this. Let's do it again. And now instead of 352, were going to put in 360. We hit okay, and oh man does this look grotty. Doesn't it. But again, you see here we've got our Y-axis. It's scaled very strangely. So let's go to Range and once again, we'll start the range at 0 and go to 1.2, which I think we did before, unclick automatic, and OK, and look at that! With the wrong period, it looks like truly there is no periodic component, and you see, you ought to be able to fit a straight line fairly nicely through all of this data. What is happening in this case? Well, I want you to figure this one out on your own, and if you do, you will learn a lot about spectral, power spectrum analysis, and Fourier Transforms. It's really interesting that if you just make a small mistake, a small error in the assignment of your period, you can get something that is very very different from your periodic component. And, this is, in fact, why we think that there really is a periodic component to GK Per at 352 seconds. Now, we can ask a question, does GK Per do this all the time when it emits x-rays? What about some other times when it might be bright in the x-rays? Let's go to the discovery data, that was made with the EXOSAT Observatory, in 1983. So we click on DS9, click on analysis. Virtual Observatory, connect using web proxy, Rutgers MOOK, and now, if you look we have pre-loaded here, a non imaging observation of GK Per, and we click on that, and look at this. All you see is something just grey and the reason we display it this way is that there is really no image. Remember we talked about image formation a long time ago, a couple of weeks ago, and so our EXOSAT observation just looks like a grayish kind of blah. But there are x-rays in there and let's see what happens when we look at them. We can go to Chandra-Ed Analysis Tools. Let's cut this off here. Whoops, oh, we can't do it that way, can we? We gotta first cut it off over here under analysis. Then we can cut it off over here. Under Chandra Ed. And let's do the same deal. Let's look at the light curve. Plot light curve data, and now you see GK-per possibly going. It's definitely varying, but is it periodic? Well, let's zoom in a little bit over here. Oh, look at that. One, two, three, four, five, six, seven, eight, maybe eight in about 2,300 seconds or so. Hey, that's pretty close to 300 seconds, 350 seconds. But once again, we know we can do better than this. Let's do a power spectrum! We do our power spectrum and there it is! Let's zoom in. Oops, let's grab it this time and zoom in again. And you see there it is! At 0.00283 or 4 exactly the same frequency, and therefore same period as we fa-, we saw from the Chandra Satellite almost 20 years later. So, now that we've got the power spectrum, let's look at the period fold. We click on Period Fold, we enter the same 352 second period, 25 little segments. Let's do that and here we've got it. Let's go to Graph, Range. We're going to re-scale the y-axis from 0 to about 1.5. Uncheck Automatic. OK, and there we have it, gang. The x-ray periodicity of 352 seconds that was figured out, or discovered, by the EXOSAT Observatory in 1982. The same thing was happening then, as seemed to be happening in the year 2002 with Chandra. So what can this be? What is going on? Let's go back to the blackboard, and see what we can find out. [BLANK AUDIO]
