More on Astronomy: Eclipses

Hello, my name is Jose Martin Garcia. I'm manager of special projects in the Research and development department of Wolfram Research. Today I'm going to be talking about new functionality we have added in the Astronomy project. This is work in collaboration with Jeff Bryant, our astronomy developer in Ballfromalpha. And I will be giving the talk myself, but Jeff will be in the chat. So if you have questions, you can ask them at any moment. Jeff probably will answer, and if not, I will answer at the end. Okay. So the purpose of the talk, as I said, is to discuss new functionality in the ASTRO project, in particular about solar eclipses. To compute solar eclipses, we need to do two things. First, we need very good control on the position of the sun, the Earth and the Moon. And then once we have that very good position, then we need to do some computations to compute everything we want to know about the eclipse. And this is something that we do with a technique called Besselian elements. So the talk is structured in three parts. First, we will talk about how to get positions of the Earth, the Sun and the Moon. And this is something we will do with astroposition and astrographics functionality that was added a year ago or so. Then we will talk about solar eclipse, a function that existed before, but we have updated quite a lot. And then at the end, we will talk a bit about how to compute with Besselium elements. So astroposition is a function that allows us to compute the position in the sky in any coordinate system of any object. So in this case, for example, that's the position of the sun right now in azimuth and altitude coordinates. We can do it in real time. And we see how the position of the sun is changing. And the truth is that most of this change is due to the rotation of the Earth. So if we go to a different coordinate system, the so called equatorial coordinate system, which is a coordinate system on the celestial sphere that doesn't rotate with the Earth, then we see that the coordinates change very slowly. This is the true motion of the Moon through the sky. So these coordinates and these positions are computed with NASA data. This is the most recent set of data they have provided. This means development fmris. And this is a number that increases with time. And this is data that covers for the sun, the Earth, the Moon and the planets about 30,000 years, roughly from year minus 13,000 to year plus 17,000. With that data, we can also produce maps of the sky. So for example, this is the map of the sky around Betelgeuse. We are using equatorial Coordinates again, and it has a range of about 25 degrees. So the distance from the star to the border of the map is about 25 degrees in the sky. You see the constellations Orion, for example. Here there are some lines. So, so this line is the celestial equator. This is the equator of the Earth projected out on the celestial sphere. This is the ecliptic line. So this is the plane of the orbit of the Earth around the sun, or if you want, of the sun around the Earth. And this line is the plane of the galaxy. We can, for example, add illustrations for the constellations. Or for example, we can also add a different background image, in this case, an image of the galaxy. This is what we call the galactic sky astro background. It's interesting also here, for example, to see that we can say place a white large point at the of the galactic coordinates. So these are coordinates that align with the plane of the galaxy. And we are adding here some grid lines of about 5 degrees of space. And you see how this point appears here at the 00 of the Galactic coordinates. So with astroposition, you can control everything in space. With astrographics, we can draw maps of the sky. Okay, so now how can we use this for solar eclipses? So the function solar eclipse will report the date of the next eclipse. We will work a lot more with this function later. But for example, we can use astrographics to produce a map of the sky observing from here on that date. And I'm going to move the date by H hours. So what we see is that as time progresses, the Moon goes in front of the Sun. Because we are in champagne here and not exactly on the path of totality. The eclipse is not going to be total. It is going to be almost total, but still partial. And then there will be time. Notice how the sun is moving also along the ecliptic line. We can do the opposite and observe the Earth from the Sun. Now using ecliptic coordinates. And I'm going to use the same time. Now, the range here, it was just one angular degree because the diameter of the sun and the Moon are about half a degree. But if we observe from the sun, the Earth is much smaller. So we have to use a range that is much smaller. So now that is what will happen observed from the sun during the time of the eclipse. Okay, so before we go into solar eclipse, another piece of information we need is information about the Moon. And In Wolfram language 14, we have added several functions to the report. Information about the position and the timing of the Moon. So one of them is new Moon that reports the dates of new Moon. Notice that in some calendars, the lunisolar calendars or lunar calendars, the new moons happened to fall near the end of the month. So these are calendars that contain information about the phases of the moon. Our calendar is a solar calendar and contains information about the seasons, but not about the phases of the moon. We have also full moon and there is a general function called moon phase data to report arbitrary phases of the Moon. There is another one which is called lunation number, which is a counter of new moons starting from the first moon of year 2000. And as we did before, we can see how this is advancing. So we are getting closer to new moon number 295. Okay, so something important to know is that every solar eclipse is close to a new moon. But not every moon is a solar eclipse. So if we get here the new moons of year 2023, we have these 12. We put them in a manipulate and now we can select for example the first one. Notice where the moon was during that new moon. New moon means that they are aligned in in ecliptic longitude. But the moon was flying very far from the from this sun. If we choose another one later, it passes closer. And then in April, you see the moon actually passed in front of the sun and then it went through crossing this line. And this is the so called north nodal point of the Moon. This is where our ancestors believed there was a dragon eating the moon during an eclipse or the sun. Okay, so let's work with solar eclipse. So a solar eclipse happens when the moon passes in front of the sun and covers the sun. And notice that there are then two cones of shadow. There is the umbra cone, which is a cone that gets smaller and smaller and then hits the Earth. And then there is the penumbra cone, which is this larger one that gets bigger and bigger. So this is very much not to scale. And so in fact, if we do this to scale, let me do this a little bit smaller. So this is the sun and that's the orbit of the Moon. If we increase this by a factor of 400, we get this. We have the Earth there at the center of that little circle. And that's the circle and that's the Moon. So you see how very much not to scale. This is again the sun, the orbit of the moon, the moon, the Earth. This is an enlargement of this one by a factor 400. This is in millions of kilometers. So the amazing thing is that when we do this enlargement These and these look almost the same. The sun is 400 times farther than the Moon and it's also 400 times larger than moon. So we have this marvelous miraculous fact that we can have eclipses with both of them having near the same size. So with solar eclipse, we can compute the date of an eclipse. So again, by default it gives the date of the next eclipse. And then we can compute properties of that eclipse. For example, what's the type? It's going to be a total eclipse. We can search for eclipses starting some other date. For example, this was the eclipse, the first one after the 1st of January of year 1900. Or we want to, if we want to go back in time, we use the time direction option. And then we get the first eclipse before 1st of January of year 1900. We can search eclipses by type. So this means search for the next one of total type. Of course, that's the one in April. We can search for the next one that will be annular, the next one that will be hybrid, we will see in a moment what that means and the next one that will be partial. So these three types are very frequent. So there will be eclipses of those types in the next 12 years. But you see, hybrids are more rare. So we'll have to wait until year 2031 to get hybrid eclipse. So let's look at map at eclipse maps. So there is a new property called eclipse maps that will produce the map of an eclipse. And we can do this for any of the 72,000 eclipses that solar eclipses that there are in the time span covered by the NASA ephemeris. So we will stop a moment to think about what these various points and elements are. So something that happens in every eclipse is that there is so called maximum eclipse that happens when the shadow of, when the axis of the cone shadows, sorry, when the axis of the shadow cones passes the closest from the center of the earth, which in this case is that that point, you will see this in a moment. So that's the time we assign to the eclipse. Eclipse in solar eclipse are identified by dates and they happen to be exactly the maximum eclipse date, right? So that's the meaning of this date that we assign to every eclipse. So that is then the position from which we observe the maximum eclipse. And to understand that we are going to do the following. Here we have what we saw before the earth. The moon will be somewhere here. And this is the penumbra at the distance of the center of the earth. And so now when time passes we see the penumbra approaching the Earth. There is this first point where the penumbra touches for the first time. We will call those touching points contact points. And this is the first contact point, typically called P1. Then the penumbra is approaching again. And now we have here at the center, that little thing is the umbra. And at the center of the umbra is the shadow axis. So the first time that it touches the. We get the C1 contact point. Now notice how the penumbra and the umbra and the axis keep moving in a straight line. But from the point of view of the Earth, because the Earth is rotating, they are always on this path of totality. This is the point contact point, P2, the interior, the first interior contact point. Now we see how we arrive at the maximum eclipse location, which for this eclipse will happen in central North Mexico. And that's the point that is closest to the center of the Earth. Remember that we are looking from the sun. Now we continue. We have this additional point contact point, the first, sorry, the second interior contact point. That's called P3. Then we have C2. That's where the path of totality ends. And then finally we have this additional contact point here, which is the last point where the penumbra touches the Earth. So we have seen the meaning. Now we can go back to this map. And we see P1, P2, P3, P4. That's the maximum eclipse. That's the path of totality. These ellipses are the shape of the umbra at various times during the totality. And these are lines of constant magnitude. Essentially, for example, here on this line, you will see the sun covered 60%. Here you will see the sun covered 40%, etc. Etc. So that's the end of the eclipse there. And this is. So this is where the eclipse starts. This is where the eclipse ends. This is sunrise. This is sunset. Okay, so in various parts of the talk later, I'm going to use an approximated version of that. And here we see, we start at maximum eclipse. This is an enlargement of this area here so that you see, you get a feeling that's the umbra and that's the. This point is the shadow axis. So again, this is moving here. And here you see how it looks from the maximum eclipse location. Okay, so that was an example of a total eclipse. So what is the difference with an annular eclipse? Now, I've selected the first annular eclipse, the first solar eclipse of year 2030. And the map looks like this right now. So, so the first thing you notice is that now the two lobes that we had at sunrise and sunset, now they have merged. There is another point there near a pole. And then we have the path of totality, same thing as before. That's maximum eclipse. We have P1 and P4, but there are no P2, P3. Now, to understand why this happens, we go back to our manipulate here and what we see. First, that from maximum eclipse we see an annular eclipse. So of course astrographics knows the distances to both the sun and the moon. And so we can get all these angular sizes correctly. And second, notice that there is never an interior contact point because there are parts of the penumbra that never gets inside the earth. That's why we don't have P2 and P3 contact points. Here we will look at a hybrid eclipse. So we look for the first one after 1940. We get one in 1948. So that's the eclipse map. And what we see is that the path of totality is very thin. So what happens here. So what happens here is the following. The magnitude of the eclipse is computed by this property, local magnitude. And what I'm going to do here is to compute the magnitude of the eclipse at a given date and as observed from a given position. That's going to happen to that is going to be the position of the shadow axis at that time. So that's what this is doing. So when we compute that function, we get this result. And what we see is that this is the beginning of the eclipse. This is the end of the eclipse, like three hours later. For some time, the magnitude is greater than one. That phase is total phase. But before that, and after that, we have a magnitude at this smaller than one. And so not the full sun is covered by the moon. So during the central period, we will see a total eclipse. In the borders, we will see an annular eclipse. That's the definition of a hybrid eclipse. So again, here we are observing from maximum eclipse and we see a total eclipse. But if we now move towards the border and we place ourselves more or less there, I don't know whether it's clear, but there is a yellow border around here, you see? And so from those positions of the earth, the eclipse looks angular, while here in the central part, it looked total. That's a hybrid eclipse. And then the last. No, there are two more types. I want to discuss the partial eclipse. That's a typical map of a partial eclipse. Again, the two lobes have emerged. And what happens here in a partial eclipse we see it in this plot, is that the umbra never touches the Earth. You see here we see that it gets closer to the surface at some point, but it never touches the earth. And so there is no point on the surface of the Earth from which it's possible to see the moon fully covering the sun. The final type of eclipse I want to show is something that happens very, very rarely. The last time that this happens was in 2014, and this was an annular eclipse, but it's a grazing eclipse. What this means is that it is an eclipse in which the shadow axis does not intersect with the earth. So it's not a central eclipse. And we cannot compute the C contact points. These are the contact points of the shadow axis. But it is an umbral eclipse, meaning that the umbra is touching the Earth. And so we can compute contact points of the umbra that are typically denoted with the letter U. So that's a form of one of these eclipses. And, and what we will do is observing from maximum eclipse. Now you see what's happening here. The shadow axis, which is the black point, never touches the Earth. But there is a moment here in which there are very, very small area of the Earth intersecting with the umbra. So those people, which in this case, it happens to be here in Antarctica, we'll see an annular eclipse. It could be total, but in this case it's annular. So this is a grazing eclipse in which the shadow axis does not touch here, but the umbrella does touch the Earth. Okay, so another thing that I want to discuss a bit is cycles in eclipses. So one famous cycle is the so called Saros cycle, which happens every 18 years and 11 days, more or less. And so every eclipse belongs to a Saros series or Saros family. Our eclipse next year belongs to family139. So with this, we compute all the eclipses of the Saros family139. This is a family that has been active since year 1501. So here we come. It contains 171 eclipses. And you see it runs for more than a thousand years. We are right here, more or less in the middle. So, yes, so this is the number of days. And for example, if we compute the differences of all these eclipses and compute an estimate of the mean, then you see it's very precise, that number. And this is a very interesting number because it happens to be an exact multiple or near exact multiple of the synodic month, the times between New Moon and New Moon, and the draconic month, the time that it takes the moon to pass from the two nodes. Remember that there were dragons in those nodes. So this is the time between visiting those dragons, the draconic month. So it's in both cases. So these two numbers being so close to one is what makes these numbers so very special. So our Eclipse is the 30th in that series of 71. We can compute the magnitudes of all those eclipses. And we see this, we see, we start with low magnitude, these are all partial eclipses. And then we have a series of similar annular or total or hybrid eclipses. And then at the end we have again partial eclipses. These eclipses are in the same family because they are very similar. So let me show you what happens with all these eclipses. We have first the partial ones, and now we have 50 of these eclipses that all look similar. And then we have again a few partial ones with no path of totality. And so if we had the continents and the background map before, you would see the earth moving very fast. So that's why I'm not drawing it here. But you see how these eclipses look very similar, related to each other. They form a family. They form a SARS family. So every eclipse is in a Saros family. So then another important cycle is the so called inex cycle. And our eclipse is in the index family 55. So we have a family, sorry, a property to get both numbers together. And this is very useful because here I'm going to get all the Satos index pairs for all the eclipses between year 1000 and 3000, which are these number of eclipses. We put them together and every point here is an eclipse. This is the Saros number, this is the index number. This is our eclipse. If you move one up, it's 18 years. If you move one to the right is 29 years. So every eclipse is here. Every eclipse belongs to a family. We can do this for a much bigger collection. So let's go 13,000 years, 17,000 years. So now we have that and now we have that. So you see, we have here like a thousand families design complete families. But our eclipse is there. Every eclipse is one of these points. So you have here a lot of data to the statistics, much more than you can find it in other places. Okay. And then I'm going to finalize by computing things from a local place. Imagine that we are observing our eclipse from champagne. So when will be the best moment to observe from champagne? It will be that instant. Very close to that, but different. So for example, what's the magnitude of the eclipse from this position at the date, it's this number meaning that the sun is covered in diameter by about 98%. So it's not total, it's not 100%. So because not total, we will still need glasses if observing from here, we can compute contact points. When is the penumbra touching us for the first time? And then at the end, what are those two moments? And this is what Stephen was computing earlier today. Here we see the evolution of magnitude again, remember, fraction of the diameter of the sun covered. And then here I'm going to draw where will be the umbra in three consecutive minutes from our location. Okay, so we have only two more minutes. Here I have some manipulates to explain this technique of besselion elements, which is a way of computing the intersection of a cone with the ellipsoid of the earth. And here you see how the cone moves. We can change where is the vertex of the cone, etc. Right. Or the aperture of the cone. So this is what is involved in the computation of Besselian elements. The key idea of besseling elements is constructing a clever frame that moves. It's centered on the earth, but it's always perpendicular to the axis of the cone. And with these, you can get these Besselian element functions that allow you to compute anything. Here I have an example of how to get various geometrical elements. These are standard. And then by doing this computation, which you see here done with simple functions, you get exactly the same result here. Okay, you have access to this notebook. If you have questions about this computation, let me know. But this is the key technique that is used for computations in solar eclipse. Okay, so this is the concluding slide and let me see if there is any questions I should address. I think Jeff has. Address them. So. Okay, so I think we are in the last minute, so thank you for your interest. And as I said, if you have any question about solar eclipses, please contact me or Jeff and we'll be happy to address them. Let's see. Yes, donation numbers can be negative. Indeed. Yes, indeed. The shadow axis is the center of the umbra. Any other question? It's already five, perhaps because I am the last one. Click share. Oh, oh, I see. Okay. Share. Yes. Sorry, I hadn't clicked the share button on the. On the talk notebook. Yes, so this is functionality in version 14. And if you have questions, contact me@joseborfram.com or, or jeff. Okay, thank you, bye.