 So to describe this new result to you, I think the best way is to bring you on a history lesson and give you the impression just how old this problem actually is. And there was part of the reason I was attracted to working on the problem. So you can go back all the way to a paper written by an American astronomer called George Born in 1861. So just for fun, I'll show you a screenshot from this paper. This comes from a time where papers were so old that you had to use typewriters and they had to be in physical form. And George Born, who was the director of the Harvard Observatory, he said many things in this paper. But one of the things he noticed, which I'm sure people noticed centuries before him in other parts of the world, was something about the phasers of the moon. This is something we can all easily understand, right? We understand the phasers of the moon by looking out in the sky. And he noticed that the moon gets really bright when you look at it at the full phase. When you see the whole disk of the moon, there's a sudden brightening of the moon. And because you're looking at the phasers of the moon, if you record the light of the moon at the different phases, this is called a phase curve. So I'm going to use this term a lot in this video explaining this to you. And it turns out that you can get phase curves of not just the moon, but you can get phase curves of the different planets and other moons in our solar system. And you can get phase curves of exoplanets as well in recent, in the last 10 years or so. So I'm going to bring you on a tour of this. So this is, again, just to continue the history lesson, here's a paper from 1963. I know we are very interested in the moon because it has historical significance for the University of Berlin. And this goes back to a researcher called Bruce Hepke. And he's famous for analyzing the phase curves of the moon. And here's one of the plots that I screenshot from his paper from 1963. And you can see he was comparing his calculations with the actual measurements of the phase curves of the moon. And indeed, you can see that when it approaches angle zero, which is when you see the full moon, you can see that there's a sudden peak. So George Born in 1861 was correct. And this research continued for many decades. And now the question is, sure, it's nice to measure this and notice this phenomena. But as astrophysicists, we are not satisfied with this. We would like to use this to understand physics. And we would like to use this to understand more about the moon and the object for which the phase curve is being measured. And so you would like to make meaningful calculations to compare the data. But before I show you more history, here's something more modern. This comes from a review I wrote in 2015 with the late Adam Showman. And this is data measured by other astronomers, not by me. And this is measured using the Spitzer infrared space telescope. And here you can see we put the cartoon on the top to remind you that it's not very different from phases of the moon. You can see that if your telescope is really powerful and your data is really precise, if you zoom in on the data in the bottom panels, you can see this up and down pattern that is very similar to the phases of the moon. As I said, maybe it's not so short. But then on top of this up and down, you see when the planet goes in front and behind the star, right, from the transits and the eclipses. The point is you can also do this for exoplanets. It's very general. So back to the subject of how do we make meaningful calculations to compare the data. And again, just to give you a sense of how old this is, the most famous mathematical solution for these so-called phase curves, which I explained to you, the oldest one is called the Lambertian sphere. Or you can just call it Lambert's law of reflection. And coincidentally, you can trace this back to Johann Heinrich Lambert, who is a Swiss mathematician, physicist, and astronomer who lived in the 18th century. So a long time ago. So you see his, I published this on Wikipedia on the left. And then on the right, you can see data of Jupiter in blue. And Callisto is a moon of Jupiter and Saladus is a moon of Saturn. And in black, you can see what the phase curve would look like if, according to Lambert's law. And the thing that's very striking is that real planets and moons don't behave like Lambert's law, right, because Lambert's law is a very idealized situation where light is reflected equally in all directions, regardless of which direction you look at the object. So this is very idealized. And you can see that this doesn't match the data at all. So this was realized, you know, this has been realized many times by many different people. And so of course, the then the logical step is to figure out what other kinds of reflection laws do you need to consider to analyze data. Now on this same subject, I bring you back to 1916. This is again, a very old paper by a famous American astronomer called Henry Norris Russell. And here what he was trying, he did many things again, by what he was trying to demonstrate was again, he took measurements of Venus, moon and Mercury, the phase curves, he doesn't have a nice plot I can show you. And he again asked the question, can this data be explained by, you know, where I establish reflection laws? And the short answer is no. And again, like these reflection laws by by Euler, who is, I believe also Swiss, lived at least 100 years ago, Lambert and Siligar. And none of these idealized reflection laws look like what the moons and planets of our solar system look like in reality. So as part of my digging into this literature, I'm very fascinated by how scientists centuries ago think about problems. So I was fascinated to read this historical literature. And in my digging, I found this as well. I was curious about who this Siligar person was. And most of his almost all of his papers are in German. And my written German is not it's not fantastic. So it took me a long time to find this. But if I dug back to I had to dig back to a meeting report of the scientific Academy of Barbaria in 1888 to find this, where he actually spends more than 40 pages talking about different laws of reflection. And this eventually came to be named after him. This was this is called the Lomo Siligar law of reflection. As you saw in the previous slide, I showed you it doesn't seem to match data as well. So the other things that I the other things I experienced on this history tour where to read all these historical papers was and I learned something from this, which is is interesting that there are these historical works. And most people don't spend the time to read all of them. And if you read all of them, you can combine the insights from this scientist will know some most of whom are no longer alive. And you can actually discover new insights. So one of the classical classic references that I really value is this textbook by a Russian master of radiation, radiation physics, his name is so polite. It's a translator textbook in 75. So I spent time reading this. And he taught me about coordinate systems, which I'll tell you about very shortly. And then there's a very famous paper in 81 by Bruce Hepke, who I already mentioned, he's a American lunar scientist who studies the moon. And his work builds on classic work of an Indian American astrophysicist is known as Chandraseka. And the point of telling you this is to tell you that science is never really done in isolation where you lock yourself in a room, you're often building on the legacy of generations and generations of scientists with thought about problems very deeply. And one of the things I really enjoy is to be able to understand these scientists and build on their insights and legacy. So I'm not going to show anything very technical in this explanation video, but I would like to share with you, like to give you a sense of maybe the aha the breakthrough moment. And the breakthrough moment was I was reading the textbook of the Russian researcher Sobolev and several other papers. And I realized that if I had was to come up with a mathematical solution for face curves, that I needed to understand the coordinate system, meaning that you cannot do calculations unless you can, you know, place the different things that you're interested in calculating in some kind of mathematical coordinate as you see in this picture. And it looks horribly complicated. And it took a long time to understand this because you need to deal with something called spherical trigonometry, which is a little bit messy. And I had to think about it for a little bit. But the breakthrough was I realized that different parts of the mathematical derivation become easier if you choose the correct coordinate system. So for example, certain steps were easier if you chose the perspective of a person standing on earth. And certain parts were easier if you chose the perspective of a person standing on the planet. And so on. And it's and it's because of this insight that I realized that you could actually do every order all of the calculations on on paper without touching a computer. But again, this was because of inspiration from all of this this long history of scientists and their legacy that I described to you. So here's the the elevated pitch of summary of what I found. We because it's not just me is also Brad Morris and Daniel Kitzman who worked together with me on this, even though I'm the lead author. And we realized that one can write down a very general mathematical solution for face curves that apply to any law of reflection. And that's the key point. And that's the reason why I spend so much time talking about these historical reflection laws. The insight is that one actually doesn't need to choose. You can let the scientists using the mathematical solution choose whatever law of reflection they want. And this is what is new. And I in a minute I will show you some real data what this means and why this is significant. And this generality means that one can actually use these solutions to easily analyze data. And instead of telling this to you, I'm going to show it to you. So here's an example of a parallel from a parallel study to the the main study. This is a study where I collaborated with a scientist in Houston, Texas, who is a specialist in analyzing data from the Cassini spacecraft. So in the early 2000, there was a spacecraft called Cassini. You can see a cartoon there. And the spacecraft was on its way to Saturn. And on its way to Saturn, it passed by Jupiter and it took data in the early 2000s. And this data can be analyzed and constructed into face curves just like of the moon and exoplanets. And for reasons that I don't fully understand, these face curves were never fully analyzed and interpreted. So after we finished the main work for the theory, I took an interest in this. And this led to this second parallel study that you see on the on this slide. And now you will see why some of the things I spent so much time talking about make sense. Because on the left, you will see one example of the data. And you will see in blue, green, and yellow are the three examples of so-called classic reflection laws. Yellow is the famous solution of Lambert from the 18th century. And the blue is the famous solution of Rayleigh, again, a solution that's very old. And the green is a solution called a Hengen-Greenstein. Details are not important. But the point is that you can actually prove, I can reproduce the idea that these classic reflection laws don't fit the data. You don't need to be a scientist to look at this and know that these three curves in blue, yellow, and green don't fit well the data in black. I think this is very obvious. But because again, remember what I told you, these solutions are very general. And you can choose any reflection law you like. And this leads me to the diagram on the right, where you can see that if you choose the appropriate reflection law, you can choose something that not only matches the data, as you can see, the red curve matches the data quite well. But you can use the outcome of this analysis to learn some physics. So I'm going to show you what this means. Now, it turns out that they are 61 phase curves of Jupiter in this study that we considered. So we analyzed 61 phase curves one by one. And this produced a very complicated looking plot on the left, which probably doesn't mean so much to you. But this plot on the left of different numbers can be translated into geometry on the right. And then this geometry starts to tell you something interesting. This geometry tells you the pattern of how light is scattered by clouds in Jupiter's atmosphere. So if these were not clouds, but atoms or molecules, the scattering pattern would be called Rayleigh scattering, which is the pattern in the blue. You can see in the pattern on the blue that about the light being scattered forwards and backwards is equal in both directions, which with a little bit less sideways, that's Rayleigh scattering. But the clouds, according to our data analysis, using these new solutions, there was this analysis, which was previously not possible before. It's not possible. And this analysis tells you that the clouds in Jupiter's atmosphere don't scatter radiation like that at all. They scatter like what you see in the red. So they scatter radiation mostly forward. You can see this big loop forward. And they scatter radiation a little bit backwards. And without going into the details of the physics that will bore you, this actually tells you that the particles in Jupiter's atmosphere are big and they're irregular. And as I was digging through the papers on Jupiter's atmosphere, I was surprised to find out that actually after decades of studying the clouds in Jupiter's atmosphere, we still do not have a complete understanding of them. And we don't have a clear idea of what they are made of, what their chemical composition is. So my hope is that this kind of approach, which is new, will give us some clues on the way forward. So I would like to end this explanation with talking about some long-term implications. Since I talked about, this is a difficult topic and I talked about things where rather technical and rather obscure. And I tried to give you a sense of why you should care. So first of all, future textbooks now have a new entirely new family of mathematical solutions for phase curves. I find this exciting. It's a bit nerdy. This opens up, I would say, new ways of analyzing data. I'll show you one example where I applied this to Jupiter. But it's very general. It's a new way of analyzing data, both of planets in the solar system and also of exoplanets. For example, I believe that this changes the way that we analyze data from the upcoming $10 billion James Webb space telescope. And Brett Morris and I and several people from my group and also collaborators from America at MIT, we are thinking hard about this problem now. And I would say all of this motivates new ways of thinking about clouds in the atmospheres of exoplanets. This is not an explanation about clouds, but the reason why we are interested in clouds, we seem very boring to people who are non-experts, is that clouds confuse your ability to say things like whether there's life on the planet or what the climate of a planet is. So this is why we think about that. So this is my explanation for these two new papers and I hope it was understandable.