Before we can get to the topic of eclipse maps, I want to talk about the calculations that are necessary, in order for any of us to be able to have a chance at seeing totality! This is a very technical topic, and we won't even scratch the surface here. I promise there won't be any math, but I want to give everyone a sense for just how much work goes into these calculations. If we didn't have them, we wouldn't have any maps at all - and we wouldn't know where the path of totality was going to be!
Fred Espenak
You don't have to understand all the math of this to appreciate the eclipse - it's just a spectacular visual phenomenon.
Xavier Jubier
Anything you want to do correctly, I mean, you need to spend some time on it, and study what's behind [it]. So this is, of course, well connected to people that have some motivation, and always strive to know more. That's the most obvious connection: Anything you do, you try to do it the best you can. When you know that, for an eclipse, I mean - what you want is to be able to see it. Of course, what you're looking for is to try to optimize the chances. And so, one way to do it is of course to know exactly what you are doing, and how things happen, and be able to adjust your location in the last few hours if you need to, so what best way to do it, [than] if you can compute everything?
Fred Espenak
The trickiest thing is to nail down a very very good, high quality ephemeris. You have to use something like the Jet Propulsion explanatory supplement, the Jet Propulsion developmental ephemeris, for doing that - and there are a number of different versions of it. The most recent ones in the past 5-10 years are all plenty accurate enough for calculating solar eclipses.
Xavier Jubier
The other problem we have nowadays is that Google is modifying the API quite often. And every couple of years, they completely change the engine that is behind [it]. And usually, you have to make huge adjustments to make it work, in fact, with the new version. So right now, I mean, most of the time usually is spent on this, trying to make the necessary adaptations, and to be able to do it enough in advance so that when it is released, it's already ready. The last time they had to make this transition, it was supposed to occur [in] late 2012. And I told, in fact, Sergey [Brin, of Google], OK, Sergey, look - I don't have the time to, you know, make all the updates. I know I don't. Clearly, I cannot make it before the deadline. I have too much stuff elsewhere to take care of - I can't do it. And he said, OK, if I give you six more months, is it OK with you? So I said, OK - six more months, it's not perfect, but OK, let's do it with six more months. So, they do listen.
Fred Espenak
Well, it's not magic, this stuff is published in books. I mean, Jean Meeus has several books that describe how to calculate the Besselian elements. The US Naval Observatory has a book called the Explanatory Supplement, that's been my bible really for many years on how to do this stuff. And the Besselian elements? Well, basically you need to have a good astronomical ephemeris for both the Sun and the Moon, that has the right ascension, the declination, the celestial coordinates of each of those objects - you have to know the apparent diameter of each of these objects, and how they are changing as a function of time. And all of that stuff gets transformed into Besselian elements through a series of equations. It's really a coordinate transformation that transforms those [apparent] motions of the Moon and Sun, and converts them into their relative motions with respect to this fundamental plane through the Earth. That simplifies things dramatically.
Fred Espenak
Basically, he simplified the geometry of eclipse calculations immensely by taking the motions of the Sun and the Moon with respect to the Earth, and simplifying them. He said, OK, let's look at the axis of the moon's shadow. That's pointing directly opposite the Sun. And then, as that moves across the Earth, let's not imagine the Earth as a sphere, let's just imagine that the Earth has an x and y coordinate system - a flat plane - passing through the center of it, that's perpendicular to that shadow. And now, the motion of that shadow on that plane is quite simple to describe. It almost moves in a constant direction, at a constant rate so you can describe it very simply. Those are called "Besselian elements", the numbers that describe where that shadow is as a function of time. There are a couple of other parts of the Besselian elements, that tell you things about how big the Sun is, how big the Moon is, and what rate these things are moving at. But once you've got that kind of information, you're no longer concerned with the details of the Moon's orbit, or the Sun's orbit or those complexities. You've got enough information that now - with some spherical trigonometry - you can convert that information to the position of the Moon's shadow on the spherical surface of the Earth, and predict exactly where the shadow is at any instant in time, and how rapidly it's moving, and how long the eclipse lasts from any point on the Earth's surface. It's a great simplification. You have to remember that they were doing these calculations back in the 1800s. There were no computers. I should say, back in those days a "computer" was a graduate student who did all this stuff with pencil and paper.
Fred Espenak
There are a couple of things you need to know in calculating an eclipse. And of course you need to know where the Moon is with respect to the Earth, you have to know where the Sun is with respect to the Earth, and we can calculate both of those pieces of information quite accurately - even several thousand years into the future or into the past. The other thing you need to know is which side of the Earth is facing the Sun and the Moon during the eclipse. And at first glance you might think, well, that's a simple answer, the Earth rotates once on its axis in 24 hours. But not exactly, because the Earth is slowing down. And it's slowing down because of the gravitational interaction between the Earth and the Moon, and the Earth's oceans primarily - the tides. And as the Earth rotates, the Moon is pulling on those oceans, and it's slowing the Earth down. And it's not slowing down at a constant rate - it depends on how deep the oceans are, how much ice is locked up in the polar caps. So, the rotation of the Earth is slowing, but it's not slowing at a rate that is predictable. You can look at what it's been doing in the last 10 or 20 years, and you can extrapolate into the future, but it's not real physics. You're really not understanding what's going on - you're just looking at the current rate of how it's changing, and there's no guarantee that it's gonna change at that same rate 5, 10, or 20 years from now. If you look back historically, the rate of change that the Earth has slowed down, changes from time to time. It's not predictable, and we've got measurements of this by looking at occultations of stars and planets over the past couple of hundred years by the Moon. Because that allows you to nail down exactly which face of the Earth was pointing toward the Moon at that point, and be able to evaluate this parameter - delta T.
Fred Espenak
What is delta-T? Well, you have to start with a standard. And what astronomers have done for many years is they look at the rate that the Earth was rotating on January 1st, 1900. And that's the reference that you use. And you find that looking at the rotation of the Earth now, it's rotating slower than it did back in 1900. And the accumulated difference, between the Earth's rotation rate in 1900 and now, is the parameter delta-T. And it's about 66, 67 seconds right now [ed: in 2014. As of April 2017, delta-T is about 68.8 seconds]. That means we have lost 67 seconds over the course of 114 years. And that plays an important part on exactly which part of the Earth is facing the Sun, especially when you go hundreds or thousands of years into the future or the past. These differences can accumulate so that delta-T grows to values of tens of minutes, even tens of hours. Now, if you've got a difference of an hour in delta-T, that means the Earth is 15 degrees ahead or behind where you thought it was. And that's an enormous amount of real estate as to where an eclipse path can pass.
Fred Espenak
Yeah, because now, I mean, we know what the value of delta-T is. Just looking at the historic values, we know that it can't vary [from the value in 2014] by more than a second or two between now and 2017. So we've got a pretty good handle on what value delta-T is, and even if we're off by a couple tenths of a second, it's only gonna move the eclipse path, around, oh, less than a kilometer. And when you're talking about an eclipse path that's over 100 kilometers wide, that's a very small amount. The only point where that would play an important role, is if you were planning on observing the eclipse from the very edge of the path. Then you'd need to factor that in.
Xavier Jubier
The good thing about this is that sometimes, you can even choose a location because of the way the beads are going to look like. For me, I mean, the good thing about this is that some people say, OK, the prediction cannot be, you know, that accurate. And now I can tell them, look - make a simulation of the beads from a location, and then go to that location, take pictures, videos, whatever - and then compare the two. If both match, then it means that of course the computations are correct - and even to a tenth of a second. Because otherwise, you know, the geometry would not work out.
Michael Zeiler
We're taking into account refraction. The path of an eclipse is stretched by roughly 100km at sunset and sunrise, due to the refraction of the atmosphere. So I can take that into account, and I can tell you exactly where the extreme limits of where you might see the eclipse [are]. And speaking of which, Dan, I know that you had a very special experience a couple of years ago for the annular eclipse in 2012. You told me your position, and I looked on my map to see where you were - and as I recall, you were actually outside the classical sunset limit of the eclipse. You actually saw it from outside the classical limit, and so you were really pushing the limit, and saw the annular eclipse right on the horizon.
That's right! Thanks for helping to make it personal, Michael. That location was selected very carefully, and as a result the 2012 annular was indeed a special experience for me - just like all eclipses are!
Next, we'll be looking at some of the wonderful mapmaking that Michael Zeiler and Xavier Jubier have been doing - to let us know just exactly where to be, to make sure we are IN the shadow!"