## How to calculate the angle of the sun

Listen to Dr. Colin Campbell, WSU environmental biophysics professor, as he discusses how to calculate the angle of the sun, or solar zenith angle.

## Transcription

Hi, I’m Dr. Colin Campbell. And this is a METER Chalk Talk. A couple of years ago, I was heading out into the backcountry and we wanted to figure out what kind of gear we should take along. A friend suggested we should just check the wind chill factor. But when I looked into it, we found out that it doesn’t even consider solar radiation in that calculation. Our exchange of energy in the environment is highly dependent on radiation, particularly solar radiation. And today, we’re going to talk a little bit more about that. Now the first thing to know about solar radiation is where the sun is in the sky. In fact, our absorbed radiation really depends on it. Interestingly, it’s one of the few things in life you can really count on.

With a few equations, we can figure out where the sun is in the sky at any time of the day. And I’m going to take you through some of these equations, one of the things I want you to know first is, they’re a little complicated, so don’t get stressed. In fact, if you just want to stop the video at a certain point. And check out these equations for a moment and write them down. That’s just fine. Now let’s just jump into it.

So here on my screen, I’m showing a graph of where the sun might be, at any point in a day if you were standing on the equator. Now in the middle, I’m going to draw this blue line across there, that is at the equinox. Now at the two solstices the sun might be here tracking across the sky, or here. And of course, this diagram is really showing kind of a fisheye picture of where that sun might be. There are two ways to describe where the sun is. One is a zenith angle. The zenith angle has a symbol, we call psi. In fact, the angle to the Earth’s surface from the perpendicular or normal, so this would be that zenith angle. Now there’s another angle we might be interested in, it’s called the Azmuth angle. But for our purposes of today, I just want to focus on this zenith angle because it’s the most important as we consider the radiation impact in an object that we’re interested in.

So to calculate the zenith angle, we’re going to go down and discuss the equation where this right here is zenith angle. And this here is the equation that we use to calculate that. Now you recognize the sines and cosines. And there’s just a couple other things in here. Of course, we’ve got t, which is time. And then a few other variables, phi. This is the latitude. Delta, this we call the solar declination, and finally, t zero, this is solar noon. Now before we get too crazy and worried about this equation, all we have to do is put in a few things into here, and we’ll be able to calculate that. So the first thing we need to know is the time of day.

Then we need to know the day of year. Now we actually call this a special name. This is called a Julian day. And it starts counting from January 1. The other things we need to know is of course, latitude, and longitude. And I’ll get to why in just a moment. The first parameter we’re going to try to find is called the solar declination. The solar declination equation looks pretty crazy. And anytime you see an equation like this in a book or something, the first assumption you should make is this is an empirical equation. As I look out on the internet and study other materials, I find that these equations actually are fairly common out there. And this isn’t exactly the way you see it in every piece of literature. But let me talk you through it here.

Really, there’s only one thing we need to know. It is the Julian day and we can go on the internet and calculate these a lot of programs just have those hard coded in like Excel. And all we need to do is just put that Julian day in for each of these values-here into here, and then we can eventually calculate the delta value. And then we can go put it back in this equation. So as long as we know the declination here, this is just the latitude. Let’s say my latitude is about 47 degrees. We just put that right here. All we need to know now is this t zero or solar noon. So what did we do for that?

Well, solar noon is calculated like this: t zero is equal to 12. That’s solar noon, and then we change it for wherever we are with respect to entered Meridian. And we call that the LC longitudinal correction, and then we also subtract off this equation of time t. We can start with the equation of time here. That’s this equation right here. And that’s not very small. In fact, not only is it not small, but it has a whole bunch of f’s in it. You can see f, here, this two times f, this is three times f, this is four times f. And now in the cosine or sines, then we have cosines here. So what is that?

Well, f is another one of these little bit long equations it is two point, or sorry, 279.575 plus 0.98565 times the Julian day. Now, if you get that, you just plug it back in here. And you can calculate your equation of time. And this is a number much smaller than one that you can plug in to this equation right here. Now, what about the longitudinal correction?

Well, the longitudinal correction Lc, that’s pretty straightforward. It’s essentially for every degree east of this of the standard meridian, you add 115. So for example, where I live, I’m at one 117.2 degrees, longitude, our standard meridian 120 degrees. And so the difference is, we’re east of that 2.8 degrees, and therefore the longitudinal correction, LC is just 2.8 over 15, or equal to 0.19h. So essentially, what I do is take that right there, and plug it in up here for the longitudinal correction. So essentially, we take 12, and we subtract off the longitudinal correction, and then with our equation of time, we get this value and eventually have t zero.

So what does all this mean? What does it sum up to? Well, there’s a lot of numbers in here. But if we go back to our initial equation, all we’re going to need to do now is simply this. We have our solar noon, we plug our time in. And then we use our solar declination here that we calculated on the first part of this discussion, our latitude here, and then suddenly, we’re able to calculate the Zenith Angle. And I’m going to try to link to a little calculation spreadsheet I did in Excel onto the sheet or onto the this video and then you can go ahead and look at that, how it’s done, and do your own calculations. For more content like this, check out our YouTube channel or head over to metergroup.com. Thanks for watching a METER Chalk Talk.