Explore the Total Solar Eclipse of 2024 with Wolfram Language

Hello again. We are ready to start this webinar where we are going to learn how to use Wolfram language to predict when the eclipse is going to happen and how it will look from any location in April 8, 2024. This webinar is going to be presented by Jose Martin Garcia, who is part of the research and development team here at Wolfram, and Jeffrey Bryant, who is part of the scientific content development team also at Wolfram. Before we begin, I'm going to open a poll so you can let us know if you will be traveling to view the eclipse. While you answer that, let me share a link to Stephen's Wolfram notebook predicting the Eclipse Multi Millennium Talc Tale of Computation. It is free if you have Kindle Unlimited. We have also a precision eclipse computational link that I would like to share where you can get a precision report on what to expect from your viewing location on that day. Okay, I see some responses are coming. I will let our presenters comment on those. So so without further delay, here is Jose Martin Garcia. Enjoy this webinar.

Okay, thank you. Welcome to another live stream with more from developers and motivated by the eclipse next week. We are Jeff, Brian and I are going to be talking today about solar eclipses in general in Wolfram language. So this is a summary of the talk. In the first part of the talk I will discuss the Astro functionality in Wolfram language. Then we will talk about the main function we use to compute about solar eclipses and then I will mention briefly the technique that is used behind those computations. And then Jeff will talk about the eclipse next week. He will compare it to the eclipse in 2017 and he will talk about other interesting things related to the eclipse that day. Okay, so let's start. So since late 2022 Wolfram language has a very powerful functionality to compute Astro data and related results like maps of the sky. So the main function we use is called astroposition, which is a function that uses internally the most precise and recent data provided by NASA. This data to compute positions of everything in the sky. So this is a function that computes for example the position of the sun and we see how we can do it in real time. So I will stop this one. Then we can change to other coordinates. For example, this was the position of the sun in horizontal coordinates as a mutant altitude. This is the position of the moon right now in equatorial coordinates or right ascension and declinations. And we can also see how it changes. So we saw before how fast this was changing mostly due to the rotation of the earth. This changes much more slowly. And this is the actual motion of the moon around the Earth. So here we have the second function that we use in the Astro functionality, which is astrographics. This is the function responsible for producing maps of the sky. And so what we are asking here is to have a map around the star Betelgeuse in equatorial coordinates. Again. So this will be right ascension, and this is declination in degrees. And we are going to have our astral range of 25 degrees. So that means that essentially from the center to the border of the sky, we have. Sorry, we have 25 degrees. And yeah, that's a map. We can add other things, like nice illustrations for the constellations, the 88 constellations in the sky. So this is the same thing with the same coordinates and range. But now we add this other background option specifying that we want this type of background with constellations. Or, for example, we can ask for a map of a different area of the sky. And so here we say, well, plot a point in white at these coordinates. So the 00 of the Galactic coordinate system. Again, 25 degrees of sky. Now we are using galactic coordinates. So this is galactic longitude, and this is a galactic latitude in which the galaxy, of course, looks flat, oriented horizontally. We have this background which is this nice image of the galaxy. And we are adding grid lines. Perhaps you don't see it via the zoom, but we do have a grid of galaxies, sorry, of lines in galactic coordinates. And you see how the zero latitude corresponds to this axis, which is aligned with the plane of the galaxy. So there are many other reference frames. There are many other options that you can use in astrographics to have a very detailed and nicely represented map of the sky. Okay, so those are the two main functions. There are other functions, but we will not go into them right now. Now, what is the main function to describe solar eclipses? So it's called solar eclipse. And by default, it reports the date of the maximum eclipse of the next eclipse, which is, of course, the eclipse next Monday. And this is computing the date, the exact moment, the exact second of the maximum eclipse, which is the moment in which the axis of the shadow produced by the moon gets closest to the center of the Earth. For example, we can now do again, anastrographics. Now we are centering at the sun. We have the reference frame being equatorial. And we specify that we want to look at that position at that instant. And so we move hours from minus one hour to three hours later after this instant. And this is what we see. So we See the moon coming here. This is computed from here from champagne and from champagne. We will not see totality, you see, we will have almost totality, but not quite. So this is enough to still require the use of glasses to look at the sun. Or we can now use ecliptic coordinates and look at the Earth from the sun. And now what we see is that we see here the Moon passing in front of the Earth as seen from the sun. So we understand from this map, why is it that we will see the eclipse only in the north hemisphere? Okay, so with these computations, you see how astrographics and astroposition can compute everything related to the eclipse. So something else we need is very nice control on the position and the moon of the Moon. So we have the new function in Wolfram language, 14, new moon, which reports the instant of the next new Moon. Interestingly, we can change this date, which you see, it's number day eight in April, to a Lunisolar calendar, for example, the Hindu or the Jewish calendar. The Jewish calendar is a Lunisolar calendar. And in those, those calendars are aligned with new moons. And so it will happen that new moons always happen in the very last or initial days of months. And full moons will happen in the middle of the month. So that's the date of the next full moon. So about two weeks after the eclipse, we have a function called lunation number that counts new moons from the first one in year 2000. And again, we can have the. Have it dynamic, and we see how it's changing continuously and how this number is progressing. So the eclipse that we will see on this date near this new moon will happen to be the new moon number 300 after the beginning of the year 2000. So here we have all new moons of this year, 20, 24. And if we see what happens for each one, for example, the next, if we go to the first one in January, we see how the moon travel far from the. From the Moon. However, if we go to the one next week, we see how the moon is going to pass in front of the sun as seen from somewhere on Earth. Okay, so, yeah, so that explains why for most new moons, we don't have a solar eclipse because the moon passes a few degrees from the sun, but twice a year, from two to five times a year, the moon will pass close enough that from somewhere on Earth, a solar eclipse will be visible. Okay, so the function that we have rewritten for version 14 to compute solar eclipses is called solar eclipse. This function has access to the full range of dates provided by NASA which is from minus year, from year minus 13 year plus 17,200, so more than 30,000 years. In that range of years there are more than 72,000 solar eclipses. And borfram language gives you access to all of them, all properties, about 100 properties, including polygons, dates, very precise, etc. You can compute global properties and also local properties if you specify an observation date or observation instant. So this looks a bit small, but it's important to know terminology. So that's the sun, of course this is not to scale. That's the moon passing in front of the earth. And we see how there is a cone here of shadow, that's called the umbra. And there is this vertex of the cone and the other side is called the antumbrance. And then there is this other cone with vertex here, which is the penumbra. So all those areas of the earth in which the umbra hits will see totality parts or annularity if the place of the earth is in the antumbra. And then the areas of the earth in the penumbra will see a partial eclipse. So this is up to scale. That's the sun with a radius of about 0.7 million kilometers. So that's the orbit of the moon, that little circle. And the earth is there in the middle. If we enlarge this by 400, then that's the moon, that's the orbit. So this circle we see here is that literacycle. And now we can see the earth. So you see how interesting it is that when we enlarge By 400, the sun and the moon look nearly identical in size. So that coincidence, that's amazing coincidence, is what allows us to have both annular and total eclipses and even a mixture of them, sometimes hybrid eclipses, as we will see in a moment. So let's discuss first global properties. So these properties are properties of the eclipse as a whole, not depending on a location on earth. So again, let's compute the next eclipse. That's the maximum eclipse we identify involved from language eclipses by their maximum date. And so for example, we can say what's the type? And it's going to report total. So the next eclipse is of total type. We can also do the computation for some other date. For example, what's the next eclipse after the beginning of the year 1900? And we get that date. We can also say, okay, look for the next eclipse in the opposite direction. So what's the previous eclipse before 1900? And we get that date, we can Also search by type. So for example, what's the next eclipse from now of type total? Of course it's going to be the next week. But what's the next annular eclipse from now? So that's going to happen later this year. Right. As I said, every year there are two to five total eclipses. So that's another eclipse this year. So if we look for the next hybrid eclipse, we will find it in year 31. And if we, if we look for the next partial eclipse, so it will be in year 25. So a nice property we have in solar eclipse is a map. If you don't specify a date, it will mean the map of the next eclipse, which is this map. And Jeff will talk about the next eclipse a lot more. So I will go through this very quickly, but this shows the path of totality, some level lines for different magnitudes of the eclipse. So this is where the eclipse start at sunrise. This is where the eclipse ends at sunset. Yes, as I said, the default property we report is what is called the maximum eclipse, which is when the shadow axis passes closest to Earth's center. Right. So you can explicitly ask for that property. And there are many other properties. For example, you can ask where is going to be the best place to observe the eclipse or where is this going to be, this maximum? And we get that position. So again, Jeff will explain this more. This is a manipulate just to explain what happens during the eclipse. So here we see the moon traversing in a straight line in front of the earth as seen from the sun. And we see how the combination of this motion together with the rotation of the earth produces this curve totality path. This point here in orange is the maximum eclipse. It's the moment in which the axis, which is this little circle, actually this is the umbra. So this is the penumbra. This is the umbra when it passes closest to the center of the earth. That is the maximum eclipse, which this time will be in Mexico. Okay. And then the eclipse proceeds and it ends there at sunset. So that was an example of a total eclipse. Now what happens in an annular eclipse? Well, we can go to the annular eclipse into section 30, we get this type of map. And now what we will see is that this is a similar situation as we had before. But now look at how the penumbra traverses the earth. And the thing is that when it reaches the center, you see how the moon is far enough that it cannot cover fully the sun. And so we see a nonular eclipse, we still see the Sun. And in these situations, we typically need glasses. We can have hybrid eclipses. For example, this was a hybrid eclipse in, in, in the year 48. And the geometry can be similar to that of an annular or to a total eclipse. But the thing is that when the eclipse starts, we see a little bit of the sun, so it's annular. Then as the eclipse proceeds and we are here in the middle now the moon covers fully the sun, so we are in totality. And then at the end, it may happen, and it did happen in this case that we see an annual annulus again. Right. So hybrid here means it's annular near the beginning and the end and total at the center of the, of the eclipse. Right. Okay. And then a partial eclipse. Partial eclipse is simply one in which the central part, the umbra, does not touch the earth. And so nowhere from the Earth we can see totality. And so, for example, if we look at the sun from a place like here, it almost covers the sun. But because the umbra doesn't touch that point, we don't see. And so when that happens, we have this type of situation on Earth in which the two lobes that we were seeing before gets merged. And so the maximum is here. This is, this place is, is exactly that place over there. Okay. Something else very interesting for eclipses is cycles. And, and so this is known from the times of, of, of the Greeks. And so each eclipse belongs to a family. It's called a Saros series. For next eclipse is going to be series 139. It has. And the, the interesting thing for the series is that eclipses are periodic. They are all like 19 years from each other. That's called asaros. And it's exactly that difference from one to another, you see, with very precise error. And so the magnitudes of a family typically are small. Then they reach one. So there are total and annular eclipses and hybrid eclipses here, and then they go down. And if we plot this, we see how the eclipses, this is evolving during the series. This is the totality paths. And they look kind of related to each other. Right. And so it is just a problem of geometry of, of the orbits of, of the sun and the moon. And then very quickly, I will also mention that all eclipses also belong to another cycle called the Annex. So there is the Saros property, giving you both numbers, the Saros numbers, the index number. And so every eclipse can be placed in this table, which can extend to both sides a lot. And each eclipse is a number, and it belongs to a family. So one of these vertical families is the source. Right. And this is 139 Saros for this eclipse. This is an index family. This is a Saros family. So each eclipse is a point on this diagram. Very interesting classification with lots of information. So yeah, I will go a bit faster now. So, local circumstances. So how do you compute local circumstances, local properties? So you have to specify where you are observing or when you are observing. For example, if I observe from champagne, when is the next eclipse going to have a maximum? So that will give me the second. So this doesn't mean it's the totality. It's just the best moment to watch the eclipse. If you want to specify some other date, you can ask for another property like local magnitude, which means what's the fraction of the diameter of the sun covered by the Moon. You can ask for contact dates. You can, for example, plot how the magnitude evolves with time. Right. So this is the maximum. This is one hour and a half or so before the eclipse. It goes up and it goes down again. It doesn't reach one because we don't have totality here. And for example, you can get polygons of what is the shadow. And here we are in Urbana Champaign. And so these are the shadow of the moon as it progresses minute by minute. And yes, I will finish saying that the key concept, the key technique used to do these computations was introduced by Bessel in 1829 and it's still used. It's very powerful and it consists in creating a reference frame. This is the Earth, exaggerated of course, to, to. To exaggerate the, the obliqueness of the ellipsoid of the Earth. And with this we can. Right, we have here the shadow of the Moon. We can change the position of the Sun. This is what the shadow does during an eclipse, is moving, etc. Etc. And so once we have this reference frame, it's easier to compute all properties that relate to the eclipse. For the Earth, you give a position in these coordinates and there are special techniques to do every computation you want. That's a reference frame that moves as the moon passes in front of the Earth. You see how this frame is going to be moving just to always align to make perpendicular the shadow of the axis of the shadow with respect to the reference frame to the Besselier reference frame. And yeah, so here I leave, as an exercise for those of you downloading the notebook, how to compute one of the properties, namely the moment of maximum eclipse.

Right.

So this is just the besselium element Functions of course reported by solar eclipse function. And then I will evaluate this quickly. We get these functions as a function of time. This is time in hours from the maximum. And then what we do is X and Y are just the coordinates in that plot in the Besselium frame. So if we compute the minimum of the distance from the position of the shadow axis to the center of the earth, which is this thing, we will find the moment of the maximum eclipse, we just convert it into hours and we get that number, which is exactly what solar eclipse would report, you see, 2 milliseconds. So once you have these functions, you can compute anything you want about eclipse. Okay, that's what I wanted to tell that's a general properties about eclipses. And now Jeff Bryant will tell us about lots of things about the next eclipse. Thank you.

Hello everyone. Just real quick, before I get started, I did want to share something. An interesting question did come in and Jose might be able to chime in more details about this, but I'll just share what I did. Somebody asked a question about how do you compute the rise time of the moon? And right now we don't have any built in system functions for computing that to high precision, but you can use entity value. I'm going to show that right here. Let's see here. I think, Jose, I think you need it. There we go. Thank you. Let me share my screen. And there we go. So this is essentially the code that you can use to compute the rise time of the moon. So here you can see we're using the entity value function. We've got the moon as the entity we're asking for the rise time property. But we have to give it qualifiers. So here we were giving it two qualifiers, the date of observation and the location we're observing from. So here you can see your latitude and longitude. And you can see here that at least in the local time zone that I'm in, it's going to rise at 6:20am so should rise around, well, close to the same time as the sun, but not exactly since the moon and the sun are going to be near each other in the sky obviously on the day of the eclipse. Anyway, I just wanted to do that real quick before I jumped in since that was a question that somebody asked. So hopefully that's useful. All right, let me close this notebook here, get that out of the way. So I'm going to talk more about specifics of this upcoming eclipse since everybody's going to be trying to observe it. Presumably everybody within this talk is interested in the upcoming eclipse. This is a picture that I took actually in 2017 from Grand Banks, Nebraska. That was the name of the town. Yeah. So here you can see it was just at the instant the sun disappeared. And you can see there's a little. This is called the diamond ring effect. So you can actually see that the. The sun is just barely peeking through some of the valleys there just before it disappeared. And you can see the atmosphere of the sun on the outside. This is called the corona. If you're in the path of totality, it's safe to look at the sun. Look at the sun when this is when this phase is happening, and you'll actually be able to see the outer atmosphere of the sun. So it's a very cool thing to look for. One of the things that most people want to know is, you know, where is the eclipse going to be visible. And of course, one of the easiest way to do that is visually. And here you can see a map. I'm going to shrink this just a little bit, since this is a bit on the big side here. Shrink it just a little bit here. So you can see that for this upcoming eclipse in 2024, the path of totality kind of runs diagonally. It comes up through Mexico, crosses into the United States, up through the Midwest, and then up through the. Eventually it ends up going up through Maine and Nova Scotia and even Newfoundland out there. So this is the general path that you're going to see. This particular map also shows you percent coverage. You're going to get 100% of the moon, of the sun covered by the moon if you're in the path of totality. But of course, these lines that run parallel to that, you can see the 0.8 here represents people along that line get 80% coverage, 60%, 40%, 20% and so on. And those are all. This is the property of solar eclipse. You can just ask for the eclipse map for a given eclipse, and it will generate a map for you with all those features in it, as well as some contact points that are a little technical, P2s and the P1s and such. So if we zoom into some of that and we want to look more locally, some additional maps here. Here we can see the area where it comes on land in Mexico. And you can see that the shadow first starts to come on land near Mazatlan in Mexico, and kind of continues on maximum eclipse, where the axis of the shadow is directly pointed very close to the center of the Earth, is going to be near Torreon, Mexico. And it's going to keep going, and then it'll eventually cross the Texas border here into the United States up here. Now, if we proceed on and we keep following the path of totality, once it crosses into Texas, you can see that it continues up through a little bit, little bit of Oklahoma. Not a lot of it continues up through Arkansas and eventually up into Missouri up here. And then I'm going to keep scrolling down, give you kind of a stages of the eclipse here. And notice that as we progress the shape of the umbra, which is these darker circles here, they start to become more elongated. The reason for that is that by the time the shadow of the umbra of the eclipse starts to hit this part of the earth, the shadow is making a glancing angle with the Earth. And so you're actually starting to get a little bit more of an ellipsoidal shape to the shadow. And so by the time you get up into Illinois and Indiana and things like that, it's really starting to look elliptical. Eventually crosses up here into Lake Erie, continues on into Maine and into far eastern Canada, southeastern. And then toward the very end, it's going to go into Newfoundland. And you can see how elongated the ellipse is here that represents the umbra. It's very distorted because at this point, the shadow is making a very glancing angle with the earth. So that's kind of the progression of the path that the shadow is going to make across the Earth. And Jose actually kind of touched this earlier, but I'm going to revisit it because it's just a nice example that you can see. Here's the textured Earth. Here you can see here's the shadow as observed from the sun scroll up here. I'm using machine that's a little slower than my usual one. So forgive me. As I click around here, you can see as it progresses, Shadow starts to make contact. Umbra hasn't touched yet. There's the umbra. So it's starting to touch the earth there. As we progress here, we're approaching Mexico. There's the umbra. That tiny little black dot on the very center is the umbra. Everything out here is the penumbra. So this. Everything in this gray area out here, we'll see a partial eclipse. Only people within that little black dot right there are going to see totality. So as we keep progressing, the earth rotates as the shadow moves across. And it continues on. So once again, if you have a better, faster machine than what I'm using for this talk, you can actually move this slider A little bit more continuously. But I'm kind of using a slow laptop right now. So anyway, that should be in the downloadable notebook that everybody there can evaluate, do it interactively for themselves. So here's an example of what you would actually see from the ground, not from space like we've seen up till now from the ground. What will the sky look like? So I've got a manipulate here where I've got hours before the eclipse for three different locations. Minneapolis, Champaign, and Carbondale. So if we're in Minneapolis, let's start here. Let me see if I can shrink this just a little bit, Just so that I can get it all on the screen here. I think I've got the size a little too big here. There we go. That's a little bit smaller. So here we can see. Sorry about that. Hard to get this all on the screen at the same time here. As we move through the eclipse, the moon does not cover all of the sun. I'm going to shrink this just a little bit more. I've got the magnification set pretty high here so that the text is readable, but it makes the graphics so big that it's hard to fit everything on the screen at once. It's a little bit better. So here you can see from Minneapolis. We're not going to see a total eclipse. You're going to see a partial eclipse. So the sun, you'll see the moon block part of the sun, but not all of us. If we click to champagne, which is where I'm observing from, well, that's where I'm at right now. It's not where I will be. But here you can see it gets very close to totality, but we're still partial. There's a little bit of the sun still visible. Here in Champaign, we're going to have something like 98% coverage. This is still not safe to look at with the naked eye. You have to be very careful when you're actually observing eclipses that the moon is completely covering the sun before you can look at it with the naked eye. This still requires eclipse glasses or some kind of a solar filter if you're looking with a telescope. So make sure you're very careful with that. If we go down to Carbondale, which is in southern Illinois, they're on the path of totality. And from there you can see the moon does cover the sun entirely. So from here we will see a total eclipse. So that shows that location, location, location is important. It drastically affects what you're going to see. Now, sometimes It's a little bit interesting to do a more of a quantitative approach rather than visual, you know, you know, how does. How does this eclipse compare numerically to other things? So, first of all, one of the things you can do with the new solar eclipse function is you can query for properties. By default, if you just feed in the date of an eclipse, you get the date of the maximum. So here you can see it's at 1300 hours on April 8, 2024, 17 minutes after the hour. That's when the maximum eclipse is, when the shadow axis is almost directly pointed at the center of the Earth. Where is that going to be? Right here near Torreon, Mexico. So here you can see I've asked for the maximum eclipse position that's down here in Mexico. So that's where you'll get the most circular shape to the umbra for this particular eclipse. From that location, you'll see that the maximum eclipse magnitude is actually bigger than one, which means the Moon is larger than the sun by a little bit. So it's about 5% above there. So the bigger that number is, the longer the eclipse will last. So you want that number to be as much above 1 as possible. From a given location, though, this is for. This is, remember, this is for the maximum location. That's that place near Torreon, Mexico. That's what it's there. From a given location, though, here we do geoposition of Champaign. You can see the local maximum eclipse magnitude is less than 1. So here you can see we're right about 98% from here in Champaign, but in Carbondale, it's over 1. So here you can see kind of a numerical result rather than a pictorial result of how things will shape out. And Jose already showed this plot, but I'll show it again Here you can see that from. From Carbondale, anyway, you can see where it does actually get up here. And then up here you get during totality, but you can see how the magnitude changes here with time. You can also calculate the times of the eclipse. So here we're talking about totality when you talk about local umbra contact date one. Once again, we're observing from Carbondale. This is a place that's in the path of totality, southern Illinois. We can see that it happens at about 1:59pm so almost 2pm is when totality will start, and then it will end at 14. So just a few minutes later, it only lasts. And you can. There's only about, you know, see, about four minutes, right around four minutes from Carbondale for the Totality. Beyond these individual queries that you can do for the solar eclipse function itself, we also have a Wolfram function repository item that is called the total solar eclipse 2024 explorer. And you can use natural language. So if you do control equals in Mathematica or If you use WolframAlpha Notebook Edition, you can just type this into your input fields. Explore the solar eclipse of April 8, 2024 and when you resolve that you can see that it becomes this WFR with no arguments and it generates a manipulate. And this manipulate allows you to put in a location that you want Champagne. By default you can specify what map you want to view, whether it's centered on your location, whether it's centered on the umbra, like where the umbra is at an instant in time or whether it's the global map. And you can actually change the slider here I'm going to decrease the magnification just temporarily so that I actually try to play with this. I'll enhance it again here in a second for the text, but it makes it easier for me to see the entire output. This is quite a bit of information presented in this output pane here. But here you can see if I move the slider takes a second to regenerate the maps here. Once again, this is not the fastest computer. My computer that I've got here is kind of slow.

So.

But this allows you to change the time and if you choose the time. Let me, let me switch to umber centered. Do a different map. Let's see the brackets yellow here. As it's thinking geographics is a little slow. So at the time that I've got the slider here you can see that it's still off the coast. When I move forward here. Come on, need to go about an hour later. Sorry, it updates. So. So there we go. So here we've. We've chosen a time that's about 2pm Central Time and you can see that it's moved into Indiana by that time. So here you can see the appearance from Champagne at that time. Champagne is actually right about here on the map. So that's Champaign. That's where I've set the location to view from. You can see the shadow is actually over here in Indiana at that time. You get all the properties, both the global properties, the eclipse at that time as well as the location dependent properties. You can see the width of the shadow path and the speed at that time instant. So I encourage everybody to play around with this. Like I said, if you have a faster computer, it'll respond a little bit better. But there's a lot of information here. So make sure you've got a nice, good sized screen so that you can see everything. All right, I'm going to switch magnification back to 200% here so that I can make sure that the text is visible. And now I am going to compare this. I mentioned comparing earlier. That was more of a numerical approach. But now we want to compare this to the 2024 eclipse, to the 2017 eclipse. How do they compare to each other? They were both total. I've got a little bit of code here where I initialize to get the dates. And then I've got little utility function that I'm going to use called property compare. This is all in the notebook that you guys can download. So you can see, you can analyze the code if you want. But the idea is that we want to compare the two eclipses to each other. So this one here is just a visual comparison. We'll get back to a numerical comparison here in a second. You can see the 2017 path was diagonal, but it started over here in Oregon, kind of came diagonally cut down through southern Illinois, down here in Carbondale, and then continued off into the Atlantic through the southeast coast of the United States. The eclipse in 2024 is also diagonal, but it kind of comes from the. Almost perpendicular to the. The previous path. You can see that it comes in from Mexico and comes up this way, once again passing through Carbondale. So Carbondale is kind of a bullseye for both eclipses. So people down there, it's normally a pretty obscure town, but they're pretty proud of the fact that they were at the bullseye for both eclipses. So, but it's going to go up through Indiana, Ohio, once again, up through the Great Lakes, the eastern Great Lakes anyway, and then up through northern New England and Maine and eastern Canada. Numerical comparison here you can see. I've got a comparison here where you can see the solar eclipse values for the 2017 eclipse, for the umbra path and for the 2024 you can see with property compare the function here. It gives you just the nice result here that the umbra path width in 2024 is wider. So it does last longer. And that's partially why there. Because the path width is wider, the umbra is larger on the earth than it was in 2017, and therefore the eclipse will last longer than it did. It was about two and a half minutes, if I remember right, for most locations in 2017 and in 2024, it's going to be four minutes, a little just slightly over. It's not the longest you can have. You can actually have them up to about seven minutes, but unfortunately not this year. We could also compare the maximum eclipse magnitudes between the two eclipses. So we can see that once again, they were both bigger than one. This is the maximum eclipse magnitude. It was a little bit over one here. And it's even more above one in 2024. Once again, the magnitude is bigger because the number is bigger. And so you get a longer lasting eclipse the more above 1. This is the apparent radius. You can do that numerical if you want. It's more useful to do this visually, which I've done below here. But here you can see the Sun's apparent radius in 2017 and 2024. They did differ a little bit. Just because the distance of the Earth away from the sun and the same thing with the moon's apparent radius, they differed between 2017 and 2024. So here you can see the Moon's radius is 0.2, almost 0.28 degrees. It was only 0.26 degrees in 2017. Or the sun, the sun was. So I'm sorry, I'm mixing dates up here. So it was. The sun was 0.26 in 2017 and 0.26 in 2024. So the difference wasn't that big. They're more distinct in 2024 here. So anyway, here's a visual representation of the moon and the sun. You can see that the, the moon is. The gray disk is larger than the yellow circle. That's the sun. So total eclipse, that's 2017. See how close they are. And in 2024, that difference is bigger. You can visually see that. So this is why we get a little bit of a longer eclipse in 2024. We can also compare the shadow speeds. Now, I will have a caveat to that. Here in a second, you can see the speed of the shadow. This is, this is in Carbondale. So this is at a particular location on Earth. You can see that in 2017, the speed was 650 meters per second, 838 meters per second. But keep in mind, Carbondale is not at the location of the center of the eclipse in 2024. So this speed depends on where you're measuring it at relative to the maximum eclipse. So visually, on a graph here, it looks like this was using. It was moving way faster. The reason why there's a little bit of a bias here. This curve you see here is for the 2017 eclipse, you can see how the speed changed as time went on. So here was maximum eclipse. And here you can see we're actually measuring the time. Time of totality in Carbondale was very close to maximum eclipse. The speed we're measuring there is about 650. That's the number you see right here. But if you look at the eclipse in 2024, here's where Carbondale is. But maximum eclipse is way out here. I'm sorry. I'm sorry. Carbondale is out here. Maximum eclipse is way back here. So you can see that we're measuring on the uptick by the time the shadow hits Carbondale, because the shadow is moving at a glancing angle across the Earth, it appears to be moving much faster. And so it's really hard to compare these velocities because you're not really measuring at the same relative position relative to the maximum eclipse. So you have to be careful in looking at shadow speed. They're neat to look at, but they're hard to compare. We can also do some ratios of the Sun Moon distance to see how that's changed from 2017 to 2024. So here we've done the ratio of the radius of the sun and the moon. It's about 400 times. And here we want this useful. We want to know how many days have passed from the beginning of the year until the eclipse. This will be useful for the calculations that I'm going to do here. So in 2017, there were 232.559 days that passed from the beginning of the year until that instant in 2017. So here you can see a plot that shows you basically the ratio of the distance between the sun and the Earth and the Moon and the Earth plotted. Now, this is a little bit cryptic here, but I'll explain what we're saying here. So points that are at the top are perigees. These are points where the Moon is closest to the Earth in its orbit. The line corresponds to a point where anything above this is a possible total eclipse if the Moon happens to move across the face, face of the Sun. So in order to see a total eclipse, it needs to be above this line. And preferably, you want it to be near one of these peaks, because that's perigee. That's when the Moon is going to be closest to the Earth. And so you can see in 2017, we were near a perigee, but not at the very tip. We were kind of off to the side a little bit. And it's not even, you know, Some of these others were. The Moon was much larger relative to the sun than it is in 2017, but this is where we were. So here I'm going to do the same thing for the 2024 eclipse. And here you can see we're very close to perigee. So we're very close for the current orbit of the Moon. But if you compare where the Earth is in its orbit to the sun, you can actually get peaks that are even higher than this location. So if the eclipse had happened out here, we would get an even longer eclipse. So this. This one, because it happens near perigee, it is total. It's above this line. We're closer to perigee. So you get a total eclipse that's larger, but it could be even bigger. Had the eclipse happened at, say, one of these peaks, you might start pushing six or seven minutes or something like that. If you're out here, here, we're going to get about four minutes. So a little bit of a cryptic thing. The code for all these is behind these closed cells. So feel free to look at this when you download the notebook and look. But it's kind of an interesting way of visually representing the eclipse differences.

Yeah.

Here's an example of an eclipse that. What was I doing here? I think this was an experimental one I forgot to delete. But I'm not going to spend too much time on this one. But yeah, well, you can see that it's possible for the moon to be like way down here if you pick it at a different date. And that would not be a total eclipse.

I think that's the eclipse by the end of the year and explaining why it is.

Oh, is that what that one is?

Yep.

Still, let's see. Oh, yes, that's right. Because I picked 410. So this is after the. So that's the next eclipse after the total one. So, yes, that's what that. Thank you, Jose. That's. This is one where I did this for the. The next eclipse that follows that there's one toward the end of the year, closer to the end of the year. And you can see that eclipse is. I guess that's what part that's gonna be partial would assume, right? Oh, that's. That's annular. I see. Yes, because we're below the line here. Okay. And then here, once again, I mentioned this verbally, but here you can see if you compare the local umbra duration once again, in Carbondale, you can see that in 2017 it was only 2.7 minutes long. But in 2024, it's 4.2 minutes long. So this eclipse should be a little bit nicer. I want to deviate a little bit here away from the eclipse itself and talk about a couple of other things that are a little bit slightly unrelated to the eclipse, but still relevant because there might, they might be. They might have an effect on things that you can see. Solar activity can be quite interesting. It affects how big the corona on the sun is. The outer atmosphere, if it's very active, the corona will be very large and kind of puffed out and expanded. And there's also sunspots. So if you have eclipse glasses or a filter on your telescope and you can look at the sun prior to the moon covering it, you might be able to see sunspots. And so we, if you, if we compare how things were in 2017 compared to 2024, you can see I'm plotting using this resource function here, space weather data to get the sunspot counts. And you can see that in 2017 we were near a solar minimum. So there weren't a whole lot of sunspots at that time. We were kind of approaching a solar minimum. But here in 2024, we're actually approaching a solar maximum. And so in theory, solar activity should be better in 2024 than it was in 2017. So there can be more things to see. This changes on a day to day basis. We also have another thing to show. I just evaluated that this morning. So you can see right now though, even though we're near solar maximum, unfortunately the sun is a little bit blank right now. This is an image taken this morning. Essentially you can see there's one sunspot starting to rotate into view. So between now and Monday, this will rotate a little bit more onto the face. But it doesn't look like we're going to get a really active face of the sun to look at, unfortunately. But it's something that you can pay attention to. There's another instrument you can use with this solar image Wolfram Function repository item. This is the SOHO satellite and its view. It can create artificial eclipses by putting a mask over the sun. And you can see what the corona looks like right now. And you can see right now there's a couple little outbursts and things like that, some streamers that are coming out. But in general, this is kind of an approximation of what you might see on eclipse day. During the eclipse, it won't be blue, of course, the sky will be kind of darker than that. Hopefully, assuming clouds aren't blocking your View. That's another concern that I'll get to here in a minute. But these are things that you can pay attention to during totality, though. You might be able to see the corona and you might be able to see some of these streamers and things outside the disk of the moon and the sun. So you might be interested to look for some solar sunspots and looking at the shape of the corona, some additional things. When the Moon covers the sun, the sky will get dark. It won't be pitch black, but it will be much darker than it is during the day. And you will be able to see brighter objects, not just the sun and the moon. So these are things you can look for during the eclipse. And so here I've got a few initialization things where I'm setting the position for Carbondale, Illinois. You can see that I'm setting the date for the time of maximum eclipse in Carbondale, computing the position of the sun for that date and position. I've got a utility function that allows me to accurately compute the position of the of a comet called Comet 12P Ponds Brooks, which is currently in the sky. It's not super bright, it's technically naked eye. I think you probably need binoculars if you want to see it. But this will allow us to compute its position. Here you can see where the comet's at in the solar system. Now I'm going to evaluate this real quick. Take a second to download this. I didn't have it pre installed here. And I'll be able to rotate this in 3D so you can see it. Get a vantage point. So here you can see a solar system map showing you where Ponds Brooks ignore the size scale. The scale is off here from the side. So the comet is not going to be a physical sphere this large. This is just a placeholder. But you can see as I rotate, you can see the comet is kind of diving down through the solar system. So you can see where Earth is. Over here you've got the sun. So you're going to be looking at the eclipse in this direction and the comet is going to be up here. So you can see as we look kind of along the plane here. And for a different vantage point, we can rely on astrographics. So once again I'm computing. I wanted to simulate not just the position of the comet, but its tail. So I've got some code here where I actually kind of create an artificial tail for the comet based on using the ecliptic coordinates of the comet and drawing lines out, drawing a line from the sun to the comet and extending them backwards. And what you end up getting is a graphic like this. So here you can see on the day of the eclipse, here's the moon and the sun about 24 degrees away along the eclipse. Hey Jeff. Yes.

Something is going on with your mic. I don't know, maybe like up and down really quick, but I'm just getting some like.

It kind of sounds like you're a robot. Interesting. Does it still sound messed up? All I can think of. I'm on wireless, so maybe that has some aspect. I don't have much control over it though, unfortunately. Yes, that's okay.

I can keep going for a minute and I have a wired headset I.

Can grab for you. I mean it is. What. I have a wired headset. It's just I'm on wireless broadcasting, so I'm streaming wireless. Let me see if turning your camera off.

Sorry, it just, it sounds very weird.

Okay, maybe it's a bandwidth issue. Do I sound any clearer if the camera's off?

No.

One sec. Put on mute. Just a second. Hang on just a second.

Okay, so we're not hearing you at the moment, Jeff.

Yes. Okay. Does that sound 100% okay? Maybe it was another bad headset then. I apologize. I don't know why it suddenly decided to go bad. But let me continue on here as we wrap up here. So here you can see the moon down here and the sun. They're eclipsed at this point in the astrographics, 24 degrees away. We've got the comet up here, Jupiter over here. Jupiter should be fairly easy to see. So if you're out there looking at the eclipse, the sky is dark enough, you'll definitely see Jupiter. The comet will be here, but it will not be this bright. You're probably going to need binoculars. I think it's technically probably about fourth or fifth magnitude for those of you that know the astronomy magnitude scale. So technically naked eye. But my guess is that you're not going to have much sky contrast. The sky is going to be still relatively illuminated, kind of a dark blue as opposed to a black. So. So you're probably going to need to have binoculars if you want to see the comet. But if you find Jupiter and then go up and to the right of Jupiter, you might be able to see this comet. So something else to try for. Here's an all sky view. Once again, I'm going to decrease the magnitude of my screen just so that it's. I can see the whole thing at once. So I think I can See the whole graphic now. So here's the horizon. This is looking at the whole sky looking south. The Moon and the sun will be pretty high in the sky. We're gonna have Venus over here and Mars. So two bright planets are gonna be over here to the right of the sun and the Moon. And then to the left you're gonna see the comet and Jupiter up here you'll have Orion over here. Some of the brighter stars, the brightest chart in the sky, Sirius, other than the sun, anyway, is gonna be low in the southeast. So these are some other things that you can look for during totality. And if the sky is clear and you're in the path of totality, you can try to look for some of these brighter things. The sky will not be black. Just keep that in mind. So only the brightest objects in the sky are going to be easily seen. Otherwise you may need binoculars. But if you do use binoculars, be very careful. Once the sun comes back out, do not ever point binoculars at the sun unless you have special equipment allowing that to be done safely. Something else that we have to be aware of is that April is about one of the worst months possible to have an eclipse. And it's spring in the Northern Hemisphere. And so here you can see if you look at the last 20 years and the trends based on that, this is kind of a mean cloud cover. So red here is good. That means more sun, the darker blue you get, the more obscuration, the more coverage of clouds you get. And so you see through most of the path of totality, we're in this blue areas. So there's very few places even historically. In fact, in Maine, historically it would be totally cloud covered, but parts of Texas wouldn't be too bad. But you get some blues and stuff in the Midwest here too. So overall, not great historically. But if you look at the forecast, they're just starting to issue forecasts. And we're getting close enough now that you can start to rely on the forecast more and more. What do we think it's going to do this year? Not just based on historical trends, but this year? So I downloaded this picture from the NOAA this morning and this is their current forecast for Monday. You can see in their legend, you want blue, blue is clear, the gray is cloudy. So here you can see if you're in Carbondale, it shouldn't be too bad. It's going to be relatively clear. If you're down here in the path of totality in Texas, you may be in for a bad time. They're currently Calling for clouds down here in Texas. So for those of you that are planning to travel down to Texas, hopefully you've got other things you can do when you're there because you might miss totality if the cloud cover is very thick down there. Also up here in Maine it's expected to be totally sunny, which is the opposite of what you see in this where here in this map, the color codes kind of reversed, but here it's like totally obscured. Historically this year it's going to be sunny. So this is a great place to be this year based on the current forecast, if it holds, I'm going to be in east central Indiana. So I'm kind of in this 50% mark here. So I'm kind of hoping for the best. But that's where my family is, so I'm going there either way. So hopefully you guys can start paying attention to the forecast and hopefully wherever you guys have planned to observe from, the weather will treat you well. Hopefully you're not going to this area of Texas because it's not looking promising. Hopefully you're not going to Northern Ohio. So some other resources though that I do want to point you out before we start wrapping up Here we have PrecisionEclipse.com this allows you to enter your location and find out the timing and the coverage of the eclipse from your location that you specified. There's a link here to Steven's book for predicting the eclipse. His kind of a background of all the science leading up to the study of eclipses. I've got a community post that goes more into detail about the comet that I told you about, so feel free to follow that link. And there's also a community post by a third party person that was actually quite nice. They generated a nice map showing you more information about Saros cycles that Jose talked about in his talk. Some additional things you might be interested in. They're astro, not necessarily eclipse related. We have a Wolfram blog entry that kind of gives you some additional links to astronomy resources for Mathematica. I have a community post. Recently they discovered a new, totally unrelated to the eclipse, a new volcano on Mars. So you might be interested to read about that. And later this year we could have an astronomical event. There's a star called T coronae Borealis. It's a name that just rolls off the tongue there. It may go nova this year. It it's showing signs based on historical trends that it's preparing to go nova. And if it does go nova, it could get as bright as the North Star, which is about the 48th brightest star in the sky. It's not the brightest star in the sky, but still it's a. It's a star that normally would be invisible to the naked eye by far and suddenly to become as bright as the North Star is pretty impressive if that happens. So feel free to read about that. With that, I'm going to wrap up my part of the talk. Stop sharing here. Assuming most of the questions were answered, let's see if there's some new questions. Looks like most of the questions were answered by others. Jose, was there anything else you wanted to mention? No.

There were questions about cloud coverage, but you already spoke about it, and perhaps you can mention where you got that map.

Let me share my screen again real quick. I actually went to the noaa. I unfortunately do not have. I threw it together very quickly this morning. Let me just share my screen again so you can see that map.

Okay.

And then I'm going to scroll up again. So I downloaded this from the National Digital Forecast Database. It's. I have to admit, there's a lot of NOAA websites. Unfortunately, I don't have the link handy and it would take me a few minutes to probably refine it. I ended up using Google to search for cloud cover.

But yeah, another thing that we could do is to go to the Precision Eclipse web.

Oh, yes.

And show an example of Carbon Daily.

Give me just a second.

There are some more questions now.

Okay. Oops, sorry, I didn't mean to close that. Let's see here. So this is what the website looks like. It's just got an input field and you can type your location in. So, for example, I can type in champagne, which is where I'm observing from. You hit enter and you can see it comes up. It gives you a couple maps, tells you the latitude and longitude that it assumes you're at. It says partial eclipse. Notice that it says partial eclipse begins there. You can see that we only get 98% coverage here in Champaign, but you can kind of see a timeline of events. Maximum Eclipse is at 205, but it's still partial. But you can see we get. It still get a tiny sliver of the sun still exposed here in Champaign. So we're not quite in the path of totality, but you can enter other locations like Carbondale, Illinois. And here you can see here we get totality. It tells you how long it lasts, what time it begins. So fairly easy to use. You just type in your location. All right. Have any other questions that haven't been answered?

There's a question about other interesting eclipses.

Jose's definitely done more study of that because he's done a lot of research when he was implementing this, so he might have some off the top of his head.

So, yes, if you know the date approximately when the eclipse happened, then of course you just tell solar eclipse which date you want to start looking for eclipses, and I'm sure you will find it. Now, something interesting that happens when looking at eclipses very far in the past or very far in the future is that they depend on something called delta T, which is essentially a measure of how much the Earth is changing its rotational speed. And because we don't know what's going to happen for that in the future, predicting when or where the eclipses are going to happen in the future is difficult. And for the past, it's actually easier because we have references, right? We have historical references.

And so just so you know, delta T changes because of things like Earth earthquakes and things like that, melting of the polar caps, things like that. These all have an effect on how fast the speed of the Earth is. And when you're predicting this, you have to know the exact speed of the rotation of the Earth in order to get this right. So, yeah, in the past we know what that was because it could be measured. But in the future, it's like they can only assume, you don't know if there's going to be an earthquake that's going to change the rotation speed of the earth or how much polar ice melting there's going to be, or what other effects might happen.

So here, for example, it was looking at an eclipse that is was recorded in, in year minus 136 that was visible from Babylon. And so if we ask solar eclipse where the eclipse happened, we get this band, that's the position of Babylon. And for example, if we had not corrected for delta T, then we would have had eclipse here. So you see how important it is that correction, right? So because we have that correction, then we get correctly the eclipse going through Babylon in year 136.

This is why when you go to, like NASA and a lot of other pages that talk about eclipses, they will often give you a bunch of numbers that may not make sense to you, but one of the more important ones is they give the value of delta T that they used for their computations so that if your values are different, it may be that you used a different value of delta T.

Exactly.

Because that has an effect on the exact precise timing and locations of where the path will be.

Okay. Okay.

Yeah.

Somebody shared a weather site down there that might be useful. Somebody wants to grab that on the Q and A. That might be a useful site for people to visit there to see for some, maybe some forecasts. As we get closer, I'm sure local weather forecasts will start talking about it more. But right now a lot of people are kind of afraid to put, put number, put numbers down yet because we're still a number of days out. And when it comes to cloud cover, that's hard to predict way in advance. By the time we get to Friday, I think a lot of the forecasts will get a little bit more solid. It'll be a little bit easier to say for sure what the cloud cover is going to be like. But.

Perfect. Guys. Well, thank you very much. I think we are ready to wrap this up. Any closing comments you may have?

No, I don't think so. Not from us.

Enjoy the eclipse and use Wolfram language to get lots of properties about lots of eclipses.

Perfect. Well, thank you Jose. Thank you Jeffrey, for leading this session today and thank you, Luke, for helping with the Q and A and all of the warframe you stick that made this webinar possible. That's all we have for you today. But please stick around. We have a survey where you can add your opinion on today's webinar. It was our pleasure to have you here. Enjoy the rest of your day. Thank you all.

Bye.