 Greetings and welcome to the Introduction to Astronomy. In this video, I will go over how to complete the calculations for the Solar Observing Project. So you've been making observations, hopefully all semester, and recording the height of your object and the length of the shadow around local noon. Now we want to go ahead and complete the calculations to find the altitude of the sun on those days, and its declination. So what we're looking at for this project, just as a couple of reminders. First of all, you must use the data set that I provide. That is the set for Lab 8. You need to use that in addition to yours for the final project. You'll need to include two separate data tables, the one you are finishing for this lab with my data, and the one with your own data. Do not mix them, do not try to combine them into a single data table. They need to be separate. And you're going to be making two graphs, which I will discuss in another video, that will show both sets of data on the same graphs. So that's why we can compare them, and you'll be able to use that when writing up your final project. Now, some optional things, and these are not required. But you are welcome to email me your calculations when you have completed them, if you would like me to review and make sure you're on the right track, and make sure there's no errors before you proceed on to the graphing. If you do that, make sure you're doing it early enough in the week, so that you can get them to me, and still be able to submit the lab on time. I will review them as quick as I can, but I do ask that you allow at least 24 hours for me to review and provide feedback. I'll try to do it faster, but I can't offer any guarantees that I will be able to get you something back on Sunday evening if you email it to me on late Sunday afternoon. And again, you still must use my data. You still have to complete this lab for the project. So don't skip this lab. You're going to have to do this work anyway for the final project. So if you don't, even if you're not using the advantage of getting some comments and suggestions on your calculations to make sure everything is right, you're still going to need to complete this data table for the final project. So let's look at the data table here and you will see a data table similar to this. This is a fall semester data table. So if you are taking the class in the winter, spring or summer, you will see slightly, you will see different dates and you will see different measurements for the shadow length and the object height. However, that's not a big deal. You just need to do the calculations that I'm showing will be exactly the same. And as I go through these calculations, I recommend that you do them as well and make sure that you can reproduce my answers. If you're not getting almost the same answers that I give, something is wrong and we want to get that straightened out quickly. Now in order to do this, you will need a scientific calculator. There's an example, image of one here. If you have a typical TI 30, we'll be able to work just fine for this. There's also an online scientific calculator that you can use. If you have your own of another kind, smartphones all have built in scientific calculators as well that you can use for this. So there are various different things that you can use. So I'm going to go through a couple of sample calculations. And step one is to find the tangent of theta. That is the next column in your data table. And to find this, all you need to do is to take your object height and divide it by the shadow length that you measured. In the case of my data, that will be 20 centimeters divided by 11.22 centimeters. And that will give you 1.79. So first step, that's all you need to do is that division. Now for the second step, we want to find theta. Theta is the altitude of the sun. And in order to find that, we need to use the inverse tangent function on the calculator. So you're going to take the number that you had here, this 1.79 goes right in here. And then you'll need to do inverse tangent or second function tangent on your calculator. And if you do that, you will get for this number 60.8 degrees. Now you do have to watch if you are using a graphing calculator, you have to make sure it is set in degree mode. They tend to default to radians, which for mathematical measurements of angles, is another way of measuring angles. In order to get the correct numbers here, you will need to set that in two degrees. If you're having difficulty with it, drop me an email or something and we will happily help you get that straightened out. So now we have the altitude of the sun, so that on this day at local noon, the sun was about 60.8 degrees above the horizon. Now our next step is to find the declination. So we want to find that declination and we do that by subtracting from the number we just determined. So we take that 60.8 degrees, that comes here. We subtract 49.75 degrees and that gives us the declination of 11 degrees. This 49.75 is specific to south central Pennsylvania. As long as you're in one of the hack regions around one of the hack campuses, this will work fine for you. If you're much further away, especially further north or south, then you may need to use a different number and you can contact me. I can help you figure that out. Now the last step is not applicable for this first one. We want to find the change in declination. However, there's nothing to compare it to, this is the first measurement. So we will look at the change in declination with the next calculation. So let's go ahead and do this again. Let's start off, we're going to again find, first thing is to find tangent of theta for the next row. So it's 20 centimeters divided by 12.4 centimeters, gives us 1.61. Now we want to find the altitude and that is done by taking the inverse tangent of this number, so this number here. And the inverse tangent of that then becomes 58.2 degrees. So the sun is a little bit lower. This is a week later, the sun is a little bit lower in the sky. So you'll notice that in the fall semester, for example, the sun will continually get lower and lower. If you're in the spring semester, it will get higher and higher over the course of the semester. And now we want to find, now that we've done that we want to find the declination. What's just as before, we take that 58.2 and we subtract 49.75 to get 8.45 degrees. So that will give us, now, the declination of the sun is 8 degrees as compared to 11 degrees the previous time. Again, the sun is getting lower and lower here. It's getting closer to the celestial equator. Now we can actually find a change in declination. In order to do that, you take the current declination, the number that we just got, and we subtract the previous declination. So in this case, we take the 8.45 that we just had, subtract the 11 from the previous one. That gives us negative 2.55. Then we divide that by the seven days between the observations. For the data I provided, I have them all seven days apart for convenience of calculation. When you do yours, this number will vary. Sometimes they might be three days apart, other times they might be 15 days apart. This depends on when you were able to get your observations. But that negative 2.55 degrees divided by seven days gives you negative .364 degrees per day. And we will see that as being negative in the fall semester. So negative, the sun is getting lower and lower, its declination will be decreasing. So for the fall semester in my data, you will get all negative values. And in the spring semester, you would get all positive values. Now let's go ahead and put this on our data table. So you can start looking at the data table, and you will have something like this. Again, for the fall semester, if you're in the spring semester, these numbers will be different. But the calculation method I've just given you is exactly the same. You will want to complete the rest of this data table. And then I recommend submitting it to me. Let me take a look at your calculations, see how your declination and your change in declination look. Make sure everything is on the right track before you proceed on to graphing, which will be covered in the next video. So let's go ahead and summarize here. Again, what I recommend you do is complete these early, submit them to me by email. Let me look at them. I will give you comments, suggestions, maybe some adjustments you need to make, and help you get everything right for this lab, because that will carry over into the final project, which is worth a much larger portion of your final grade. So that concludes this lecture on the solar observations calculations. We'll be back again next time for another topic in astronomy. So until then, have a great day everyone, and I will see you in class.