for this evening, is on how to get to Mars in all very interesting ways, some of them might be really, really slow. Our next speaker has studied physics and has a PhD in math and is currently working as a mission planner at the German Space Operations Center. Please give a big round of applause to Sven. Thank you. Hello and welcome to Thrust is not an option, how to get to Mars really slow. My name is Sven. I'm a mission planner at the German Space Operations Center, which is a part of the DLR, the Deutsches Zentrum für Luft und Raumfahrt. And first of all, I have to apologize because I kind of cheated a little bit in the title. The accurate title would have been reducing Thrust, how to get to Mars or maybe Mercury really slow. The reason for this is that I will actually use Mercury as an example quite a few times. And also we will not be able to actually get rid of all the maneuvers that we want to do. So the goal of this talk is to give you an introduction to orbital mechanics, to see what we can do, what are the techniques that you can use to actually get to another planet, to bring a spacecraft to another planet. And also go a few more, go a bit further into some more advanced techniques. So we will start with gravity and the two-body problem. So this is the basics, the underlying physics that we need. Then we will talk about the two main techniques, maybe to get to Mars, for example, the Hohmann transfer, as well as gravity assists. The third point will be an extension of that that's called a planar circular restricted three-body problem. Sounds pretty complicated, but we will see in pictures what it is about. And then we will finally get the taste of certain ways to actually be even better, be even more efficient by looking at what's called ballistic capture and the weak stability boundary. All right, so let's start. First off, we have gravity and we need to talk about the two-body problem. So I'm standing here on the stage and I'm actually being, well, accelerated downwards, right? The Earth actually attracts me and this is the same thing that happens for any two bodies that have mass, okay? So they attract each other by gravitational force and this force will actually accelerate the objects towards each other. Notice that the force actually depends on the distance, okay? So we don't need to know any details, but in principle, the force gets stronger the closer the objects are, okay? Good. Now, we can't really analyze this whole thing in every detail. So we will make a few assumptions. One of them will be that all our bodies, so in particular the sun, Earth, will actually be points, okay? So we will just consider points because anything else is too complicated for me. Also, all our satellites will actually be just points. One of the reasons is that in principle you have to deal with the attitude of the satellites. For example, a solar panel needs to actually point towards the sun. Of course, that's complicated, so we will skip this for this talk. Third point is that none of our planets will have an atmosphere, so there won't be any friction anywhere in the space. And the fourth point is that we will mostly restrict to movement within the plane, so we only have like two dimensions during this talk and also I will kind of forget about certain planets and other masses from time to time, okay? I'm mentioning this because I do not want you to go home this evening. Start planning your own interplanetary mission, then maybe building your spacecraft tomorrow, launching in three days, and then a week later I get an email, hey, you just didn't work. I mean, what did you tell me? Okay, so if you actually want to do this at home, don't try this just now, but please consult your local flight dynamics department. They will actually supply you with the necessary details. All right, so what's the two-body problem about? So in principle, we have some body, the sun and the spacecraft that is being attracted by the sun. Now, the sun is obviously much heavier than a spacecraft, meaning that we will actually neglect the force that the spacecraft exerts on the sun. So instead, the sun will just be like, it will be at some place, it might move in some way or a planet, but we only care about a spacecraft in general. Furthermore, notice that if you specify the position and the velocity of a spacecraft at some point, then the gravitational force will actually determine the whole path of the spacecraft for all time, okay? So this path is called the orbit, and this is what we are talking about. So we want to determine orbits, we want to actually find ways how to efficiently change orbits in order to actually reach Mars, for example. There's one other thing that you may know from your day-to-day life. If you actually take an object and you put it high up and you let it fall down, then it will accelerate, okay? So one way to actually describe this is by looking at the energy, yeah? There is kinetic energy that's related to movement, to velocity, and there is a potential energy which is related to this gravitational field. And the sum of those energies is actually conserved. This means that when the spacecraft moves, for example, closer to the sun, then its potential energy will decrease and thus the kinetic energy will increase, so it will actually get faster. So you can see this, for example, here we have two bodies that rotate around their center of mass, and if you're looking careful when they actually approach each other, then they are quite a bit faster, okay? So this is important to keep in mind. All right, so how do spacecrafts actually move? So we will now actually assume that we don't use any kind of engine, no thruster, we just cruise along the gravitational field. Then there are essentially three types of orbits that we can have. One of them are hyperbolas. So this case happens if the velocity is very high, because those are not periodic solutions, they're not closed, so instead our spacecraft kind of approaches the sun or the planet in the middle, the center. From infinity, it will kind of, well, turn, it will change its direction, and then it will leave again to infinity. Another orbit that may happen is a parabola. This is kind of similar. Actually, we won't encounter parabolas during this talk, so I will skip this. And probably most common orbit that we all know are ellipses, in particular, circles, because, well, we know that the Earth is actually moving around the sun approximately in a circle, okay? So those are periodic solutions, they are closed, and in particular, they are such that if a spacecraft is on one of those orbits, and it's not doing anything, then it will forever stay on that orbit, okay? In the two-body problem. So now the problem is we actually want to change this, so we need to do something, okay? So we want to change from one circle around the sun, which corresponds to the Earth orbit, for example, to another circle around the sun, which corresponds to the Mars orbit. And in order to change this, we need to do some kind of maneuver, okay? So this is an actual picture of a spacecraft. And what the spacecraft is doing, it's emitting some kind of particles in some direction. They have a mass, M. Those particles might be gases or ions, for example. And because these gases or these emissions, they carry some mass, they actually have some momentum due to conservation of momentum, this means that the spacecraft actually has to accelerate in the opposite direction, okay? So whenever we do this, we will actually accelerate the spacecraft and change the velocity. And this change of velocity is denoted by delta V. And delta V is sort of the basic quantity that we actually want to look at all the time, okay? Because this describes how much thrust we need to actually apply in order to, well, change our orbit. Now, unfortunately, it's pretty expensive to, well, to apply a lot of delta V. This is due to the Tarkovsky rocket equation. So the fuel that you need in order to reach or to change your velocity with some delta V, this depends essentially exponentially on the target delta V. So this means we really need to take care that we use as a few delta V as possible in order to reduce the needed fuel. There's, well, one reason for that is, well, we want to maybe reduce costs because then we need to carry less fuel. However, we can also actually think the other way around if we actually use less delta, less fuel, then we can bring more stuff for payloads, for missions, for science experiments, okay? So that's why in spacecraft mission design, we actually have to take care of reducing the amount of delta V that is spent during maneuvers. So let's see, what can we actually do? So one example of a very basic maneuver is actually to, well, sort of raise the orbit. So imagine you have a spacecraft on a circular orbit around, for example, sun here. Then you might want to, well, raise the orbit in the sense that you make it more elliptic and reach higher altitudes, okay? And for this, you just accelerate in the direction that you're flying. So you apply some delta V and this will actually change the form of the ellipse, okay? So that's a very common scenario. Another one is, if you approach a planet from a very far way, then you might have a very high relative velocity, such that with respect to the planet, you're on a hyperbolic orbit, okay? So you would actually leave the planet. However, if this is actually your target planet that you want to reach, then, of course, you have to enter orbit, you have to somehow slow down. So the idea here is that when you approach the closest point to the planet, for example, then you actually slow down. So you apply delta V sort of in the opposite direction and change the orbit to something that you prefer, for example, an ellipse, because now you will actually stay close to the planet forever, okay? Well, if reality would be a two-body problem. Okay, so let's continue. Now we actually want to apply this knowledge to, well, getting, for example, to Mars. Let's start with moment transfers. Mars and Earth both, well, evolve around the sun in pretty much circular orbits, and our spacecraft starts at the Earth. So now we want to reach Mars. How do we do this? Well, we can apply what we just said. So we accelerate when we are at the Earth orbit, such that our orbit touches the Mars orbit on the other side. Okay, so this gives us some amount of delta V that we have to apply. We need to calculate this. I'm not going to do this. Then we actually fly around this orbit for half an ellipse, and once we have reached the Mars orbit, then we can actually accelerate again in order to raise the other side of the ellipse until that one reaches the Mars orbit. So with two maneuvers, two accelerations, we can actually change from one circular orbit to another one. Okay, this is the basic idea of how you actually fly to Mars. So let's look at an animation. So this is the orbit of the inside mission. That's a NASA Mars mission, which launched and landed last year. The blue circle is the Earth, and the green one is Mars, and the pink is actually the satellite of the probe. You can see that, well, it's flying in this sort of half ellipse. However, there are two, well, there's just one problem. Namely, when it actually reaches Mars, Mars needs to be there.  That's not good. I mean, this happens pretty regularly when you begin playing a Kerber space program, for example. But yeah, so we don't want to play around with this the whole time. We actually want to hit Mars. So we need to take care that Mars is at the right position when we actually launch, because it will traverse the whole green line during our transfer. This means that we can only launch such a home and transfer at very particular times. This time when you can do this transfer is called the transfer window. For Earth Mars, for example, this is possible every 26 months. So if you miss something, software's not ready, whatever, then you have to wait for another 26 months. The flight itself takes about six months. All right. There's another thing that we kind of neglected so far. Namely, when we start, when we depart from Earth, then, well, there's Earth mainly. So that's the main source of gravitational force. For example, right now, I'm standing here on the stage and I experience the Earth. I also experience Sun and Mars, but I mean, that's very weak. I can't ignore this. So at the beginning of our mission to Mars, we actually have to take care that we're close to Earth. Then during the flight, the Sun actually dominates the gravitational force, so we will only consider this. But then when we approach Mars, we actually have to take care about Mars. So we kind of forgot this during the home and transfer. So what you actually do is you patch together solutions of these transfers. So in this case, there are essentially three sources of gravitational force or Earth, Sun and Mars. So we will have three two-body problems that we need to consider. One for departing, one for the actual home and transfer, and then the third one when we actually approach Mars. So this makes this whole thing a bit more complicated, but it's also nice because actually we need less delta V than we would for the basic home and transfer. One reason for this is that when we look at Mars, so the green line is now the Mars orbit and the red one is again the spacecraft. It approaches Mars. Then we can actually look at what happens at Mars by kind of zooming into the system of Mars. So Mars is now standing still and then we see that the velocity of the spacecraft is actually very high relative to Mars. So it will be on a hyperbolic orbit. It will actually leave Mars again. You can see this on the left side, because it's leaving Mars again. So what you need to do is, in fact, you need to slow down and change your orbit into an ellipse. And this delta V that you need here for this maneuver is actually less than the delta V you would need to circularize the orbit to just fly on the same orbit as Mars. So we need to slow down. A similar argument actually at Earth shows that, well, if you actually launch into space, then you need quite some speed already to not fall down back onto Earth. So that's something like seven kilometers per second or so. This means that you already have some speed. And if you align your orbit or your launch correctly, then you already have some of the delta V that you need for the Hohman transfer. So in principle, you need quite a bit less delta V than you might naively think. All right. So that much about Hohman transfer, let's look at gravity assists. That's another major technique for interplanetary missions. The idea is that we can actually use planets to sort of getting pulled along. So this is an animation. On the lower, in the lower animation, you see kind of the picture when you look at the planet. So the planet is standing still. And we assume that the spacecraft, so the blue object, is on a hyperbolic orbit. And it's kind of making a 90 degree turn. Okay. In the upper image, you actually see the picture when you look from the sun. So the planet is actually moving. And if you look very carefully at the blue object, then you can see that it is faster. So once it has passed, the planet is actually faster. Well, we can actually look at this problem. So this is again the picture. When Mars is centered, we have some entry velocity. Then we are in this hyperbolic orbit. We have an exit velocity. Notice that the lengths are actually equal. So it's the same, it's the same speed, but just a turn direction in this example. But then we can look at the whole problem with the moving Mars. Okay. So now Mars has some velocity V Mars. So the actual velocity that we see is the sum of the entry in the Mars velocity before and afterwards exit plus Mars velocity. And if you look at those two errors, then you see immediately that, well, yeah, the lengths are different. Okay. So this is just the whole phenomenon here. So we see that by actually passing close to such a, well, planet or massive body, we can sort of gain free delta V. Okay. So of course it's not, I mean, energy is still conserved. Okay. But yeah, let's not worry about these details here. Now, the nice thing is we can use this technique to actually alter course. We can speed up. So this is the example that I'm shown here. But also we can use this to slow down. Okay. So that's a pretty common application as well. And we can use this to slow down by just changing the arrows essentially. So just approaching Mars from, in this case, Mars from a different direction essentially. So let's look at an example. And this is a baby Colombo. That's actually the reason why I kind of changed the title because baby Colombo is actually a mission to Mercury. Okay. So it was launched last year. It's a combined ESA JAXA mission. And it consists of two probes and one thrust essentially. So it's the three stages that you can see in the picture. Yeah. And that's a pretty awesome mission actually. It's really nice. But it has in particular a very cool orbit. So that's it. What can we see here? So first of all, the blue line, that's actually earth. The green one, that's Mercury. So that's where we want to go. And we have this intermediate trachease one. That's Venus. And well, the pink curve is baby Colombo's orbit. So it looks pretty complicated. It's definitely not a Hohmann transfer. And in fact, this mission uses nine gravity assists to reach Mercury. And if you look at the path, so for example, right now, it actually is very close to Mercury because the last five or six gravity assists are just around Mercury in order to slow down. Okay. And this saves a lot of Delta V compared to the standard Hohmann transfer. All right. But we want to do, we want to do even better. Okay. So let's now actually make the whole problem more complicated in order to hope for some kind of nice tricks that we can do. Okay. So now we will talk about a planar circular restricted three-body problem. All right. So in general, the three-body problem just means, hey, well, instead of two bodies, we have three. Okay. And they pairwise attract each other. And then we can solve this equation of motion. We can ask a computer. And this is one animation of what this could look like. So there are three masses and their orbits are traced. And we see immediately that we don't see anything. That's super complicated. Yeah. There is no way we can really, well, I don't know, formulate a whole solution theory for a general three-body problem that's complicated. Those are definitely not ellipses. So let's maybe go a step back and make the problem a bit easier. Okay. So we will now talk about a planar circular restricted three-body problem. There are three words. So the first one is restricted. Restricted means that in our application case, one of the bodies is actually a spacecraft. Spacecrafts are much lighter than, for example, sun and Mars, meaning that we can actually ignore the force that the spacecraft exerts on sun and Mars. Okay. So we will actually consider sun and Mars to be independent of the spacecraft. Okay. So in total, we only have like two gravitational forces now acting on a spacecraft now. So we neglect sort of this other force. Also, we will assume that the whole problem is what's called circular. So we assume that sun and Mars actually rotate in circles around their center of mass. This assumption is actually pretty okay. We will see a picture right now. So in this graph, for example, in this image, you can see that the black situation, so this might be at some point in time. And then later on, the sun and Mars actually have moved to the red positions and the spacecraft is at some other place and now, of course, field some other forces. Okay. And also, we will assume that problem is planar, meaning again that everything takes place in the plane. Okay. So let's look at a video. That's a video with a very low frame rate, something like two images per day maybe. It's actually Pluto and Charon. So the larger one is the, well, X-planet Pluto. It was taken by New Horizons in 2015 and it shows that they actually rotate around their, well, center of mass. So both actually rotate. This also happens, for example, for sun and earth or sun and Mars or sun and Jupiter or also earth and moon. However, in those other cases, the center of mass is usually contained in the larger body. So this means that in the case of sun, earth, for example, the sun will just wiggle a little bit. Okay. So you don't really see this extensive rotation. Okay. Now, this problem is still difficult. Okay. So if I put now a mass in there, then, well, yeah, you don't really know what happens. However, there's a nice trick to simplify this problem. And unfortunately, I can't do this here, but maybe all the viewers at home, they can try to do this. You can take your laptop. Please don't do this. And you can rotate your laptop at the same speed as this image actually rotates. Okay. Well, then what happens? The two masses will actually stand still from your point of view. Okay. If you do it carefully and don't break anything. So we switch to this sort of rotating point of view. Okay. Then the two masses stand still. We still have the two gravitational forces towards sun and Mars. But because we kind of look at it from a rotated or from a moving point of view, we get two new forces. Those forces you know. So centrifugal forces, of course, the one that, for example, you have when you play with, I don't know, with some children or so, and they want to be pulled in a circle very quickly, and then they start flying. And that's pretty cool. But here we actually have this force acting on the spacecraft. Okay. And also there is the Coriolis force, which is a bit less known. This depends on the velocity of the spacecraft. Okay. So if there's no velocity in particular, then there will not be any Coriolis force. So our new problem actually has four forces. Okay. But the advantages of sun and Mars actually are standing still. So we don't we don't need to worry about their movement. Okay. So now how does this look like? Well, this might be an example for an orbit. Well, that looks still pretty complicated. Now, I mean, this is something that you can't have in a two-body problem. It has these weird wiggles. I mean, they're not really corners. It actually kind of switches from sun to Mars. Yeah. So it stays close to sun for some time and then it moves somewhere else. So that's still pretty complicated. I don't know. Maybe some of you have read the book, the three-body problem. Yeah. So there, for example, the two masses might be a binary star system. Okay. And then you have a planet that's actually moving along such an orbit. Okay. That looks pretty bad. So in particular, the seasons might be somewhat messed up. Yeah. So this problem is in fact in a strong mathematical sense, chaotic. Okay. So chaotic means something like if you start with very close initial conditions and you just let the system evolve, then the solutions will look very, very different. Okay. And this really happens here, which is good. All right. So one thing we can ask is, well, is it possible that if we put a spacecraft into the system without any velocity, is it possible that all the forces actually cancel out? And turns out, yes, this is possible. And those points are called Lagrangian points. So if we have zero velocity, there's no Coriolis force. So we have only these three forces. And as you can see in this little schematics here, it's possible that all these forces actually cancel out. Now, imagine, yeah, I gave you homework. Please calculate all these possible points. Then you can do this, but we won't do this right here. Instead, we just look at the result. So those are the five Lagrangian points in this problem. Okay. So we have L4 and L5, which are at equilateral triangles with Sun and Mars. Well, Sun and Mars in this case. And we have L1 and 2 and L3 on the line through Sun and Mars. So if you put a spacecraft precisely at L1 without any velocity, then in relation to Sun and Mars, it will actually stay in the same position. Okay. That's pretty cool. However, mathematicians and physicists will immediately ask, well, okay, but what happens if I actually put a spacecraft close to a Lagrangian point? Okay. So this is related to what's called stability. And you can calculate that around L4 and L5, the spacecraft will actually stay in the vicinity. So it will essentially rotate around the Lagrangian points. So those are called stable. Whereas L1, L2, and L3 are actually unstable. This means that if you put a spacecraft there, then it will eventually escape. However, it just takes a different amount of time depending on the Lagrangian points. So, for example, if you're close to L2, this might take a few months. But if you're close to L3, this will actually take something like 100 years or so. Okay. So those points are still different. All right. Okay. So is there actually any evidence that they exist?  I could just show anything. However, we can actually look at the solar system. So this is the inner solar system here. In the middle you see, well, in the center, you see the Sun, of course. And to the lower left, there's Jupiter. Now, if you imagine an equilateral triangle of Sun and Jupiter, well, there are two of them. And then you see all these green dots there. And those are asteroids. Those are the Trojans and the Greeks. And they accumulate there because L4 and L5 are stable. Okay. So we can really see this dynamics going on in the solar system. However, there's also various other applications of Lagrangian points. So in particular, you might want to put a space telescope somewhere in space, of course, in such a way that the Sun is not blinding you. Well, there's Earth, of course. So if we can put the spacecraft behind Earth, then we might be in the shadow. And this is the Lagrangian point L2, which is the reason why this is actually being used for space telescopes, such as, for example, this one. However, it turns out that L2 is unstable. So we don't really want to put the spacecraft just there. But instead, we put it in a closed orbit close to L2. This particular example is called a halo orbit. And it's actually not contained in the planes. I'm cheating a little bit. So on the right-hand side, in the animation, you actually see the orbit from the side. So it actually leaves the plane. The blue dot is Earth. And on the left-hand side, you see the actual orbit from the top. So it's rotating around this place. Okay. So that's the James Webb Space Telescope. By the way, you can see in the animation, it's supposed to launch in 2018. That didn't work out, unfortunately. But stay tuned. Another example that has become pretty famous recently is the Chinese Keqiao Relay Satellite. This one sits at the Earth-Moon L2 Lagrangian point. And the reason for this is that the Chinese wanted to, or actually did, land the Changi-4 Moon land on the backside of the Moon. And in order to stay in contact, radio contact with the lander, they had to put a relay satellite behind the Moon, which they could still see from Earth. And they chose some similar orbit around L2. Okay. So let's now go to some other, more advanced technique. Ballistic capture. Right. Okay. So this whole business sort of started with a mission from the beginning of the 1990s. That's the Hiton mission. So that was a Japanese Moon probe, consisted of a probe which had a small orbiter inside, which was separated. And then it was supposed to actually enter orbit around Moon. Unfortunately, it missed its maneuver. So it didn't apply enough Delta-V. So it actually flew off. And the mission was sort of lost at that point because Hiton itself, so the main probe, did not have enough fuel to reach the Moon. All right. That's, of course, a problem. I mean, it's a risk you have to take. And they were probably pretty devastated. However, there were two people from JPL, NASA, who actually heard about this, the Bruno and Miller. And they were working on strange orbits, those ballistic capture orbits. And they actually found one for the Hiton probe. They sent this to the Japanese. And they actually used that orbit to get the Hiton probe to the Moon. And it actually arrived in October 1991. And then it could do some science, maybe some different experiments, but it actually arrived there. However, the transfer took quite a bit longer. So a normal Moon transfer takes like three days or so. But this one actually took a few months. All right. And the reason for this is that it looks pretty weird. So this is a picture of the orbit or a schematic picture. You can see the Earth, well, they're in the middle sort of, and the Moon a bit to the left of it. The L2 is the Lagrangian point of the Sun-Earth system. So it's pretty far out. And you can see that the orbit sort of consists of two parts. First, it leaves Earth, and it flies far beyond the Moon. So somewhere completely different towards some other Lagrangian point that's really far away. Then it does some weird things. And in the upper part of the picture, there it actually does a maneuver. So we apply some thrust, some delta B to change on a different orbit. And this orbit led the probe directly to the Moon, where it was essentially captured for free. So it just entered orbit around the Moon. And this is of course not possible in the two-body problem, but we may find a way for doing this in the three-body problem. Okay. So what do we mean by capture? Now we have to sort of think a bit more abstractly. The idea is, well, we have Sun and Mars, and we have a spacecraft that flies in this three-body problem. So the red orbit is actually the orbit of the spacecraft. Now, at any point in time, we may decide to just forget about the Sun. Okay. So instead, we consider the two-body problem of Mars and the spacecraft. Okay. Because at this point in time, the spacecraft has a certain position relative to Mars and a certain velocity. So this determines its orbit in the two-body problem. Usually, it would be very fast. So it would be on a hyperbolic orbit, which is denoted by the dashed line here, okay, or a dashed curve.  But of course, the orbit is only an approximation because in the three-body problem, the movement is actually different. But later on, it might happen that we continue on the orbit. We can do the same kind of construction by just ignoring the Sun, essentially. And then we could find that the spacecraft suddenly is in an elliptic orbit. This would mean that, well, if we forgot about the Sun, then the spacecraft would be stable. It would be captured by Mars. It would be there. It would be pretty nice. So this phenomenon, when this happens, we call this a temporary capture, okay? Temporary because it might actually leave that situation again later on, because the actual movement depends on the three-body problem, which is super complicated. So it's possible that it actually leaves again. But for that moment, at least, it's captured. And we want to find a way or describe some kind of algorithm, perhaps, how we can, well, find this situation, essentially, okay, in a reasonable way. And the notion for this is what's called, well, instability. The idea here is the following. We look at the three-body problem. We want to go to Mars. So we pick a line there. And on that line, we take a point x, which has some distance r to the Mars. And we pick a perpendicular speed, a perpendicular velocity to the line, such that this corresponds to some kind of elliptic orbit in the two-body problem, okay? So that's the dashed line. But then we actually look at the problem in the three-body problem. And we just evolve the spacecraft. And it's following the red orbit. It might follow the red orbit, okay? And it can happen that after going around Mars for one time, it hits again the line, okay? Then we can do the same construction with forgetting the sun again and just look at the two-body problem. And it's possible that this point actually still corresponds to an elliptic orbit. That's somewhat interesting, right? Because now this means that if we actually hit the point x, then we can follow the orbit. And we know that we wrap around Mars once and are still sort of captured in the corresponding two-body problem, okay? If we actually are able to wrap around Mars twice, then we would call this two-stable and, well, for more rotations than it's n-stable, okay? So that's good because such an orbit corresponds to something that's usable because we will wrap around Mars n times. However, it's also possible that you have an unstable point, meaning that we again start on something that corresponds to an ellipse around Mars. But if we actually follow the orbit in a three-body problem, it will, for example, not come back, it will not wrap around Mars, it will go to the sun or somewhere else, okay? So that's, of course, not a nice point. This one's called unstable. And then there's another thing we can do. That's actually a pretty common trick in finding orbits, et cetera. We can, instead of solving the problem in forward time, we actually go back, okay? So essentially in your programming, you just replace time by minus time, for example.  And it's possible that the point that corresponds to such an ellipse, when you go back into the past and it does not wrap around, but it actually goes to the sun, for example. We call this unstable in the past, okay? So that's just some random definition. And we can, well, we can use this. The reason for this is we can actually kind of take these concepts together and build an orbit from that. The idea being we pick a point x that is unstable. So for example, it might wrap around Mars six times. Well, some number that we like. This is the blue part here in the picture. So it wraps around Mars six times. But when we go back in time, it actually leaves Mars or at least it doesn't come back in such a way that it's again on an elliptic curve. So this is the red part, okay? Well, so we can just follow this. And then we pick a point y on that curve, okay? So this one will be pretty far away from Mars, or we can choose it. And then we sort of use a home and transfer to get from Earth to that point y, okay? So our orbit actually consists of three parts now, okay? So we have the home and transfer, but it's not actually aiming for Mars. It's actually aiming for the point y. Then there we do a maneuver because we want to switch onto this red orbit, okay? And then this one will bring us to the point x where we know, because it was constructed in this way, that the spacecraft will continue to rotate around Mars, for example, six times, okay? So in particular, at x, there's no maneuver taking place, okay? So that's a possible mission scenario. The way this is done then usually is you kind of, you calculate these points x that are suitable for doing this, okay? So they have to be stable and unstable in the past at the same time, so you have to find them. And there's a lot of numerical computations involved in that. But once we have this, you can actually build these orbits. Okay, so let's look at an actual example. So this is for Earth Mars. On the left, you see, well, the two circular orbits of Earth and Mars. And on the right, you see the same orbit, but from a point of view centered around Mars, okay? And the colors correspond to each other. So the mission starts on the left side by doing a Hohmann transfer. So that's the black line starting at Earth and then hitting the point which is called xc here. So that's the y that I had on the other slide. So this is this point y or xc here is pretty far away still from Mars. There we do a maneuver and we switch onto the red orbit, which brings us to the point x closer to Mars, after which we will actually start rotating around Mars. The point x is actually at the top of this picture, okay? And then on the right, you can see the orbit. And it's looking pretty strangely. So the red orbit is when we capture orbit, our way to actually get to Mars. And then if you look very carefully, you can count we actually rotate around Mars six times, okay? Now, during those six rotations around Mars, we could do experiments. So maybe that is enough for whatever we are trying to do, okay? However, if we want to stay there, we need to actually execute another maneuver, okay? So to actually stay around Mars. And I mean, the principle looks nice, but of course, we have to do some calculations. We have to find some ways to actually quantify how good this is. And it turns out that there are a few parameters that you can choose. So in particular, the target point x, this has a certain distance that you're aiming for around Mars. And it turns out that this procedure here, for example, is only very good if this altitude, this distance r, is actually high enough. If it's high enough, then you can save, in principle, up to 23% of the data we, which is enormous, okay? So that would be really good. However, in reality, it's not as good usually. And for certain lower distances, for example, you cannot save anything. So there are certain tradeoffs to make. However, there's another advantage. Remember this point y. We chose this along this capture orbit, along the red orbit. And the thing is, we can actually choose this freely. This means that our home untransfer doesn't need to hit Mars directly when it's there. So it doesn't need to aim for that particular point. It can actually aim for any point on that capture orbit. This means that we have many more home untransfer available that we can actually use to get there. This means that we have a far larger transfer window, okay? So we cannot just start every 26 months. But now, with this technique, we could actually launch. Well, quite often. However, there's a little problem. If you looked at the graph carefully, then you may have seen that the red orbit actually took like three quarters of the rotation of Mars. This corresponds to roughly something like 400 days, okay? So this takes a long time. So you probably don't want to use this with humans on board because they have to actually wait for a long time. But in principle, there are ways to make this shorter. So you can try to actually improve on this. But in general, it takes a long time. So let's look at a real example for this. Again, that's baby Colombo. The green dot is now Mercury. So this is kind of a zoom of the other animation. And the purple line is the orbit. And yeah, it looks strange. So the first few, well, movements around Mercury, they are actually the last gravity assets for slowing down, okay? And then it actually starts on the capture orbit. So now it actually approaches Mercury. So this is the part that's sort of difficult to find, but which you can do with this stability. And once the animation actually ends, this is when it actually reaches the point when it's temporarily captured. So in this case, this is at an altitude of 180,000 kilometers. So this is pretty high up above Mercury. But it's enough for the mission, okay? And of course, then they do some other maneuver to actually stay around Mercury. Okay. So in the last few minutes, let's have a look, let's have a brief look at how we can actually extend this. So I will be very brief here because well, we can try to actually make this more general to improve on this. This concept is then called the Interplanetary Transport Network. And it looks a bit similar to what we just saw. The idea is that in fact, this capture orbit is part of a larger, well, a set of orbits that have these kinds of properties that are wrapped around Mars and then kind of leave Mars. And they are very closely related to the dynamics of a particular Lagrangian point, in this case L1. So that was the one between the two masses. And if you investigate this Lagrangian point a bit closer, you can see, well, you can see different orbits of all kinds of behaviors. And if you understand this, then you can actually try to do the same thing on the other side of the Lagrangian point. Okay. So we just kind of switch from Mars to the Sun and we do a similar thing there. We expect to actually find similar orbits that are wrapping around the Sun and then going towards this Lagrangian point in a similar way. Well, then we already have some orbits that are, well, kind of meeting at L1. So we might be able to actually connect them somehow, for example, by maneuver. And then we only need to reach the orbit around Sun from Earth. Okay. If you find a way to do this, you can get rid of the Hohmann transfer. And this way, you reduce your Delta V even further. The problem is that this is hard to find, because these orbits, they are pretty rare. And of course, you have to connect those orbits. So you approach the Lagrangian point from L1, from two sides, but you don't really want to wait forever until it's very easy to switch. So instead, you apply some Delta V in order to, well, not wait that long. So here's a picture of how this map looked like. Well, again, very schematic. So we have Sun, we have Mars, and in between there's a Lagrangian point L1. The red orbit is sort of an extension of one of those capture orbits that we have seen. Okay. So it wraps around Mars a certain number of times. And well, in the past, for example, it actually goes to Lagrangian point. I didn't explain this, but in fact, there are many more orbits around L1, closed orbits, but they're all unstable. And these orbits that are used in this interplanetary transport network, they actually approach those orbits around L1. And we do the same thing on the other side, on the Sun. And then the idea is, okay, well, we take these orbits, we connect them. And when we are on the black orbit around L1, we actually apply some maneuver, we apply some Delta V to actually switch from one to the other. And then we have sort of a connection of how to get from Sun to Mars. So we just need to do a similar thing again from Earth to this particular blue orbit around the Sun. Okay. So that's the general procedure. But of course, it's difficult. And in the end, you have to do a lot of numerics because, yeah, as I said at the beginning, this is just a brief overview. It's not all the details. Please don't launch your own mission tomorrow. Okay. So with this, I want to thank you. And I'm open to questions. So thank you, Sven, for an interesting talk. We have a few minutes for questions. If you have any questions, line up next to the microphones, we'll start with microphone number one. Hello. So what are the problems associated? So you showed in the end this going around the Lagrange point L1. Is this also possible for other Lagrange points? So could you do this with L2? Yes, you can. Yeah. So in principle, I didn't show the whole picture, but these kind of orbits, they exist at L1, but they also exist at L2. And in principle, you can this way sort of leave this two-body problem by finding similar orbits. But of course, the details are different. So you cannot really take your knowledge or your calculations from L1 and just take them over to L2. You actually have to do the same thing again. You have to calculate everything in detail. I'll take a question from the internet. Is it possible to use these kinds of transfers in Kerbal space program? Okay. So, well, so home and transfers, of course, the gravity assist as well, but not the restricted three-body problem because the way Kerbal space program, at least the default installation, so without any mods, works is that it actually switches the gravitational force. So the thing that I described as a patched solution where we kind of switch our picture, which gravitational force we consider for our two-body problem, this is actually the way the physics is implemented in Kerbal space program. So we can't really do the interplanetary transport network there. However, I think there's a mod that allows this, but your computer might be too slow for this. I don't know. If you're leaving, please do so quietly. Small question. And a question from microphone number four. Hello. I have actually two questions. I hope that's okay. First question is, I wonder how you do that in like your practical calculations. Like you said, there's a two-body problem and there's solutions that you can calculate with a two-body problem and then there's a three-body problem. And I imagine there's an n-body problem all the time you do things. So how does it look when you do that? And a second question is you said that reducing delta V about 15% is enormous. And I wonder what effect does this have on the payload? Okay. So regarding the first question.  So you have to, well, you calculate those things in the simplified models. Okay. You kind of patch together an idea of what you want to do. But of course, in the end, you're right. There are actually many massive bodies in the solar system. And because we want to be precise, we actually have to incorporate all of them. So in the end, you have to do an actual numerical search in a much more complicated n-body problem with, I don't know, 100 bodies or so. And you have to incorporate other effects. For example, the solar radiation might actually have a little influence on your orbit. Okay. And there are many effects of this kind. So once you have a rough idea of what you want to do, you need to take your very good physics simulator for the n-body problem, which actually has all these other effects as well. And then you need to do a numerical search over this. You kind of, you know where to start with these ideas, where to look for solutions. But then you actually have to just try it and figure out some algorithm to actually, well, approach a solution that has the behavior that you want. But it's a lot of numerics. Right. And the second question, can you write me again? Sorry. Second question was reducing delta V by 15%. What is the effect on the payload? Right. So I mean, well, if you need 15% less fuel, then of course you can use 15% more weight for more mass for the payload, right? So you could put maybe another instrument on there. And another thing you could do is actually keep the fuel, but actually use it for station keeping. So for example, in the James Webb telescope example, the James Webb telescope flies around this halo orbit around L2, but that orbit itself is unstable. So the James Webb telescope will actually escape from that orbit. So they have to do a few maneuvers every year to actually stay there. And they have only a finite amount of fuels at some point. This won't be possible anymore. So reducing delta V requirements might actually have increased the mission lifetime by quite a bit. Number three. Hey, when you do such a mission, I guess you have to adjust the directory of your satellite quite often because nothing goes exactly as you calculated it, right? My question is how precise can you measure the orbit, sorry, the position and the speed of a spacecraft at, let's say, Mars? What's the resolution? Right. So for Mars, I'm not completely sure how precise it is, but for example, if you have an Earth observation mission or something that's flying around Earth, then you can have an orbit, a rather precise orbit that's good enough for taking pictures on Earth, for example, for something like two weeks or so. So you can measure the orbit well enough and calculate the future, something like two weeks in the future. So that's good enough. However, I can't really give you good numbers on what the accuracy is, but depending on the situation, it can get pretty well for Mars, I guess. Well, that's pretty far, I guess, that will be a bit less. A very short question for microphone number one, please. Thank you. Thank you for the talk. I have a small question. As you said, you roughly plan the trip using the three body and two body problems, and are there any stable points like gradient points in, for example, four body problem, and can you use them during the roughly planning stage? Yeah, I actually wondered about this very recently as well, and I don't know the answer. I'm not sure. So the three body problem is already complicated enough from a mathematical point of view. So I have never actually really looked at the four body problem. However, with those many bodies, there are at least very symmetrical solutions, so you can find. But it's a different thing than the gradient points. So unfortunately, we're almost out of time for this talk. If you have more questions, I'm sure Sven will be happy to take them after the talk, so please approach him after. And again, a big round of applause for an interesting talk. Thank you.


Hey guys, welcome back to this video. I wanted to share a couple of stories both positive and negative from my experiences slash the experiences of the people that I've worked with Through this COVID pandemic over the last year. It's crazy to think that it's been a year since like first taking on COVID patients in my hospital and I wanted to share a couple of these stories to like just kind of document Some of the things that I and those around me have experienced Obviously, this will be like HIPAA compliant. I'll change identifiers where appropriate and whatnot but This pandemic has been crazy and I think it's changed most nurses more than we realize or maybe you do realize it For me, it it's like when I'm in it I don't realize the struggle as much or I don't realize the crazy things that I'm Experiencing until I take a step back and if I tell someone about an experience or just have a moment to like reflect on What happened during my shift? It It then like hits me. I don't know. It's hard to explain. So I have let's see, sorry I have my I have them on my phone. I have I think seven Experiences That have happened to me Slash like my co-workers That I want to share that have happened during the COVID-19 pandemic. Some of these will be like Positive and some of these are are pretty like dark. So just be forewarned, but Let's start with something Positive and that was extubating our first COVID patient So when code first hit I'm in Arizona and like I said, I mean I see nurse We only had a couple COVID patients. We didn't get like full-blown hit really super hard initially And so I remember our first COVID patient This was a young patient someone that you wouldn't necessarily expect to get this sick from COVID and we We babied this COVID patient my baby I mean that like we just we did everything and anything to like we kept this patient in ice you for longer We were very hesitant about Extubating and whatnot. I want to say this patient was intubated for probably three to four ish weeks But I remember when this patient got extubated. It was like Everyone was celebrating. We were so excited. We were so happy for this patient and It was like it was a really big deal and the patient actually wrote us a note probably Several weeks after they got discharged like from the hospital and rehab and all that to kind of share their experience of being in the ICU as a patient being like one of the first COVID patients and It was it was very humbling to have like that first patient that we extubated Because there was several more that we didn't even get close to extubating which I'm gonna talk about that Sorry, like I said, all my things are on my phone Let's see. Okay. So that was a positive story. Let's do Something that's not so positive. So I Saw the lowest sets that I've ever seen in my Time as an iciners I've been iciners for almost five years, which I guess is like kind of long but not that long But I saw the lowest oxygen sats on a patient that I have ever ever seen where they are still alive and like a perfect PLEF So this was a legit sat, but the patient desat it down to three percent and this was during a tube exchange The patient was on a peep of 20 and a hundred percent and it was a very difficult tube exchange, so I Have never seen sats get that low and be like legit, but yes three percent That was a shift. I was being a charge nurse doing the charge nurse role and during that same shift I also helped assist in Intubating a patient who was pregnant and then a couple minutes later doing a bedside C section Obviously, I didn't do the C section The OBGYN was there, but we had a COVID patient that was just not doing super well. We ended up needing to intubate Baby didn't like the intubation and we ended up doing a bedside C section Which was the first for me. I know lots of babies have been delivered in ICU's Before we've had a couple I've never been there personally, but this is my first one that I was at I'm also pregnant right now and I was pregnant at the time when this happened and it was just it was like a crazy experience Yeah, I mean that's a very hard thing.  Let's see We had lots of families that got sick with COVID like several of the family members all were in you know the hospital or in the ICU And in particular we had a husband and wife who Were with us for a couple weeks not doing super well and they ended up both passing away at the same time We actually rolled their beds in together so they could both pass away and their family could Be there for that. So that was another sad story Sorry, I know these are like so sad But I know you guys probably appreciate hearing this and have experienced a lot of these things yourselves Let's see another story. We had a patient who who's Sibling was in the ICU and the sibling passed away And the other patient was still intubated and we had like a note on our like little report sheet that said like The patient wasn't aware yet that their sibling had passed away Which is sad because obviously this patient was intubated and sedated and it's like Who is going to tell them that when like that patient is also so critically ill and To hear you know that your sibling has passed away just a couple doors down like very very sad Another thing was Doctors not coding patients even if families were really insistent. So our intensivists are Amazing, but we got to the point where they basically had to say We cannot keep coding or we cannot get to that point of coding your your loved one because it's just It they they are so so so sick and that's really hard when Um I don't want to say that option is taken away But that that option is taken away from families and I feel like as nurses We know that for the most part of patients are that sick that they are coding Their outcomes are are not good. Their mortality rate is not is super high the morbidity rate super high But to like hear these conversations that the doctors are having the families basically saying like look they are so sick It's not appropriate for them to be a full code Is was kind of crazy and I know like I've heard it in little bits You know like those conversations prior to the covid pandemic But when we are in like full pandemic mode, it was like every single shift these conversations were happening Okay, I wanted to share two more stories that are both kind of happy since a lot of these have been depressing um But we had a patient whose birthday was uh happened while they were intubated in the icu and obviously a covid patient and um So for the family we actually sang the patient happy birthday and like recorded us singing it and sent that to The family obviously we had kind of formed a close relationship with the family over the past several weeks So we knew like it would they would be okay with it and it was only sent to you know the family, but it was um Just something that I'll always remember that like my co-workers and I did for this patient um I know the family really appreciated it. I don't know actually how the patient ended up doing overall But like I think it meant a lot to them that uh, we did that and then another happy story that I have is um We had a patient who had been with us for a while And it was actually like their one month Aversary of being in the icu like intubated with covid And as the pandemic was kind of dwindling down our surge was dwindling down We were closing some of the icu's that we had opened up for covid and so for this particular patient They were our like last covid patient in this unit It was actually them and one other patient But the other patient was like going to be going off the unit for a scan And so like the nurses and I were kind of like well, what should like we do? It's like this patient's one month anniversary They're like the last patient in this makeshift covid icu that we've had Like what should we do? And so um, I wasn't the nurse I was actually being the secretary at the time But the bedside nurse was like well, let's play music and give this patient like the best bath they've ever had Which is like such an icu nurse thing to do But um the patient wasn't able to tell us like what they music they liked So we just guessed based on their birth date like what type of music they would like And the nurse blasted the music Like gave this patient a really good bath bed change made them all super clean because like most of us Probably know and have experienced like during the pandemic a lot of things kind of had to go to the wayside And not saying that like patients didn't get baths But it definitely wasn't the consistency and the quality that we can when we only have one or two patients when you've got Multiple patients that are super sick a bath is not necessarily always a priority So it was kind of nice to you know put the music on Clean the patient up super super well and it was like a celebration for them But also for us of like we're closing down this unit Like this pandemic has hit us so hard But it's coming to I don't want to say coming to an end because I don't know if we're going to have another surge I'm sure maybe we might but I hope we don't have a surge That big to where we're opening up all these like makeshift icus. It was crazy. So Those are a couple stories from my time working During the COVID-19 pandemic If you guys want to share some um feel free to leave them down below Obviously make sure they're like HIPAA compliant Change genders change ages change whatever um so that way the stories are are safe, but um I hope hearing some of these experiences are Like help you feel I don't know enlightened But also that like I'm in the same boat as you guys I've been in the same boat as you guys and I'm experiencing a lot of the same things that you guys are experiencing And yeah, so I hope it's helpful and I'll see you guys my next video. Bye


Hi, I am Dr. Ashwin Raj Gopal. I am the consultant surgical oncologist at Manipal Hospital, Whitefield. This is one of the biggest myths that is prevalent right now. But as an oncologist, I can assure you that this is absolutely wrong. Research shows that doing a small biopsy does not cause any spread of cancer. In fact, biopsy is absolutely necessary. We need to find out what type of tumor that we have. There are two types of tumors. One is called as benign tumor and other one is called as malignant tumor. Based on biopsy, we can decide the type of treatment that is required. There are different kinds of biopsies. One, few are endoscopic biopsies, few which can be done with radiological imaging, and few which is called as an open biopsy. But all these biopsies are painless, which can be done on an OPD basis within a matter of about 5 to 10 minutes. There are a lot of patients who when I tell them that they need to undergo a biopsy, but they get scared and they go to some other doctor thinking that biopsy causes spread. But this is absolutely untrue.


Sorry about that little thing blinking right there. Now it's trying to string in. Not today. But now reconnection is successful. We got some jokes. Anyways, this is what we are looking at right now. Howdy, howdy, Clara, howdy. OBS just has jokes today. So here we are. It doesn't want to stay where I put it. Ornery for no reason. Even adding just that line, if I wanted to leave it, it could be a finish all on its own. I'm going to leave this rough edge on the outside and then try to blend the inside bit. Also, I'm seeing that I've made it put a little bit too much because it's already eating into my yellow and also green. But geodes aren't perfect, right? Right. So in that little squirt bottle I had is just 91% isopropyl alcohol. I keep it in different size little squeeze bottles for ease of use. I don't want anybody getting confused because I'm using a whole bunch of different little bottles of liquid. See, that part looks geody, right? It looks geody. Yeah. Now how you do this and keep it looking awesome all the time. When Clara comes up here, she's going to do an alcoholic piece of me. I don't know what we're going to do. Do I ever? I think there's two. You guys haven't seen Clara's alcoholic videos. Make sure you check them out, especially if you like drag any things. She is Mother Dragon. I keep having to remind myself not to mess with it too much. Let the inks kind of do what they're going to do because inks on all on their own have some awesome just effects just because they're awesome. I mentioned they're awesome. Let it for a minute. I'm going to move on to the green. I would do the yellow, but I feel like if I add too much to it, it's going to disappear because it's already kind of not there. What's that, Joe? What can I be of assistance? You know, I give you practical advice from my experience. I don't do that many molds. However, the FAM is very experienced in all things resin and we're always happy to help. So like Clara said, bring it. I like this look so much. I have to stop myself from letting it kind of take over to where it doesn't make sense anymore. Oh, currently that's what I'm trying to avoid. This is Monday part two. I feel like it is. Pretty sure. I'm adding more ink there because some of the red transferred my paintbrush. What is the best resin and can I mix mica powders in the resin? It's a large tic-tac-toe mold from Julie Butts in Australia. She's pouring your heart out. I love Julie. She's great. Super talented. Get a casting epoxy when you're doing molds. I don't care honestly how thick it is. Your result will be so much better if you use a resin that's made for what you're trying to do. And so that's why I always push people to get a casting epoxy. For example, the B that we poured this one, it is pretty awesome. Not just because Jeff made it, but in terms of like clarity. We use any heat because I don't want to melt the mold. And it did just fine because the resin is for mold, so it doesn't hold onto bubbles like art resins do. Less than a quarter inch. It was less than a quarter inch. You could definitely get by with not using a casting epoxy because it's very close to the thickness that like art resins are poured, but still. The other thing I'm going to have issues with is you're going to almost always end up with a flimsy piece out of them just because of how shallow you have to pour. I see it in groups all the time. People making comments that they don't know why their mold is still flimsy after x many days. And it's almost always because they use the wrong epoxy or they had to pour it really thin because the mold itself is super thin. Just another unfortunate thing about casting resin. Excuse me. Jeff just had an epiphany in use. A little knife over there. Great tips, Clara, as per use. So after I get these colors done where I want them, I'm going to add a few to create depth within the colors, the different colors. At least in theory that's what I have in my head that I'm going to do. Y'all know that inks are going to ink and resin's going to resin. I still don't understand why I don't have a shirt that says resin's going to resin. That's not this video. Excuse me. It's a heavy mold. She made the channels deeper than the channel. She made a quality mold for you. She's awesome. Clara said, hey girl, I don't know why I put more blue there. It's already getting kind of saturated. Do you guys think I should add glass to this piece or nah. Glass is a very Erica thing to do. That is very a point. That'd be true to your brand people. We're good with the blue. And we're going to go on to purple. Which one did I use? Are you going to use purple? Yes, glass. I was going to find large molds for your market girl. That would be an expressive mold. The reason why molds are so expensive is because cell phones aren't cheap. That looks so fun. I wish I could leave that, but it's going to end up growing where I want it to be. I have to brush that out just a little bit. Speaking of on brand, I'm thinking about adding some splatters as well. It's weird. I want to be on brand, but not be too far away from typical geode, but I don't want to be basic geode. Strong, really real. I'm just blending the new wave of color in while pushing some of it backwards to create the darker lines. I'm trying to remember to talk my way through some of this because sometimes I just do art as if we're just all hanging out, art-ing together instead of instructing people. I'll do better. I swear. Hi, Bea. I'm making an apology. Do you think I should add?  You're going to either have to do a couple layers or use a casting epoxy. Are you going to be talking about doing a cross, not rainbow colors, but more realistic looking? Cross, not rainbow colors, but more realistic? So are you asking if I thought about painting a cross that has no rainbow colors, but just painting a realistic cross? I haven't tried the crystal resin you're mentioning. They have good reviews, just like you see in many pictures. I guess I would need to see a reference. I don't want to get it wrong. This purple is looking mighty similar to the purple I have in front of it. So I'm either going to have to use, I'll probably just use more ink since it's the end to keep it darker unless alcohol. Just google it, okay. But if I google a realistic cross, I want to just bring up like wooden crosses. I just want to make sure that I'm looking at the same thing that you have in your head. And a lot of times that can't happen unless I'm directed to a specific image. You know what I mean? I would just hate to look at something because you asked me to do something and then have been studying the wrong reference, which is something that I would do. I'm just trying to give this outside purple some motion without making it look exactly like the purple right in front of it. Like stagnant color lines. Mist boring. Wooden cross with detail. Okay. Yeah, standpoint art kind of is not cheap. It used to be 55 per half gallon, but raw materials went up. So it is now $60 per half. It is a good price. Honestly, you get what you pay for with resin, and I know that could come off not as genuine from me since we're very close with stone coat. That being said, if you ask me about a specific resin that I have used, I'll give you an honest opinion on that because we're review channels, well, and I'm not going to steer you guys in the wrong direction. I'm putting up these inks before I make a mess, and then we're going to add to this geode. Well, thank you, Elena. I try to come off as genuine as I can possibly be while being fair and unbiased. I know there's probably definitely people out there that look at our channel and are like, they're not going to get an honest review because they're friends of stone coat. Which people are entitled to their opinion, no matter how wrong or misguided it is. I don't get upset. Now we have this. Do you want to do glass or shall I just do glitter glass? While we're thinking about that, while we're milling that around, I'm going to spray seal this. For this, I'm using Eevee Archival in Matte, if they met. Lisa, how are you doing? So, I have with some heat to dry it up. I'm filming from my side right now, but I will definitely check it out as soon as I get off till I see it. I'm glad your knee shirt really went well. I'm really digging how this looks. It's a little bit rustic for my ultra refined case, which is not very ultra refined, but you know what I mean. Glitter glass. Definitely going to use glitter glass either way. I'm just trying to decide. I want to use fire glass as well. Now, I did just seal it in with the Eevee Archival spray, which would bless you. Which makes your ink smooth less if other alcohol is on them. For example, if I put a couple drops down and leave them, it's going to reactivate. That's the word. It's going to reactivate the colors. However, what I'm sealing it in for is not to protect it from more alcohol. It's to protect it from the resin because sometimes resin can fade your inks or change the color of them completely. I like adding a little splatter here and there to make it not as perfect. You know what I mean? That's the Marabu ink.  It works. You still want any water in there. No. Whether I should use fire glass and glitter glass or just glitter glass. I still want it to hide. Actually, I'm not going to use this fire glass. I'm going to use some other fire glue. Actually, I wonder what would happen. Okay, I'm just going to do a quick test real quick. Okay, it'll take just a second. Just hear me out. So funny when I approached Jeff and I opened the phrase like our conversation with, okay, just, okay, hear me out. I'm just going to drop this alcohol over the glass, even though I've already sealed it. Just dropping a little bit all up in there. I just want to see if it'll kind of take on the texture of the glass as it dries. But I'm not just dumping a whole bunch of alcohol on there because I don't want it to like kind of blob out. Now that I have alcohol down, I'm just going to help it evaporate. We did James Clara. Okay, don't do it on high. Don't do it on high. Don't do it on high. I'm hoping and foreseeing that this kind of works kind of like how you do the alcohol bubbles, not quite as rainbow-y. So yeah, Megan, for sure that, but I'm not trying to color my glass. I just want to have the glass kind of texture or print of the glass. Now let's pull one up a little bit. It might be a thing. Okay, now these are going to be red, so I'm going to have to put them somewhere. I don't want to just scoop them all up because like smear any not dried alcohol that's down. Y'all, if this works, it looks like it will. Then what? I can't put, I don't know, put glass and hide it if it looks good. Maybe we'll just do fire glass or I mean glitter glass. Yes, I want to look glitter glass. It looks like right here, not right here right there, but like the same process but with glitter glass. Right now I'm going to have to get out some like actual crystals, prop ink on them, see if it'll do some coolness. Megan, I sure can let you know about the picture you sent me, but like I mentioned a little bit ago, I'm filming from my phone, so I can't check it right now. I don't think that one was going to let go of it then. So I don't know that this so much looks like glass, but I'm digging the texture, but I need your guys' opinion. I need to check in on Ann Marie. I haven't logged on to her channel in a little bit. Who? Eddie G? Yeah, I have that in her cart. I didn't read enough reviews on it. So what do you guys think? Absolutely, Arlene. Looks better with the crystals? Yeah. We'll stay a little, oh it's dry. It'll stay like that. Ann Marie is still classy and awesome, sweet. I think I liked it better with the glass on as well. So let's mix some resin, seal that in, but I'm going to investigate this further for sure. Do you want to try something before we move on? Let me try this. I just want to see what the glitter glass looks like with pink. Sorry today. Now I'm just going to take my alcohol ink and just drop it over the glass that I just put down, and hopefully I'll get a fun texture out of it. Hopefully. We'll find out together, yeah? Not high heat because I don't want to project all of my glass everywhere, which I'm doing anyways. Hey, David, thank you so much. Man, it's all that's doing in the early hours. Hey, Evelyn, I know YouTube does indeed forget to send out notifications sometimes. This looks like it's going to end up with a fun texture. Hey, Pamela, how are you? Actually, y'all, I may not resin this. So sorry about all that movement. That interesting texture to it and the glass kind of stuck. So now I'm wondering if I want to continue this and put like with the glass and then drop alcohol on it. Just jump to resin. What do you guys think? I'm going to put at least some more glitter glass here either way. Texture of glass all of it, but then I'll be wasting the glass because the resin's going to cover it up. In the meantime, I'm going to go ahead and mix it. Okay. I wanted to see the texture. So now I'm wondering, not wondering, should I continue this around? This is a 30 two. I'm trying to figure out if I want to resin it or I feel like probably because that matte spray really glass or resin and glass. Resin and then do your, you know how they have the lines? Maybe do like lines, highlighter, but a little brighter. You know how they do the lines in some of the geodes? Resin and then lines or lines in that resin? Do you need resin on anything? This is a trick for getting every last drop out. Just be careful doing this. Anytime you heat resin, it's going to want to set up more quickly than you probably want it to. It's so crazy. It feels like it's going to get there without me having to finagle it. And then it gets so close. And then J.K. Thanks for playing, J.K. Just me. Cool. Why didn't you chat today? Either the food is really good or so glad I'm not alone. Love me some popcorn. I think I'm going to mix a little bit of abalone into this, all of it, and then that'll be that little bit of extra sparkle for my everything. Big slady swimming, yellow and green peppers. Is that a whole sentence? Word. Denise, the answer is Clara. Clara, let that dog bell. I'm not doing the other part because I sound silly. What if we like, okay, here's my turn of thought. I was going to say I couldn't be a baker because they have to like, mix all the things all the time. And then I was like, J.K., they have stand mixer and I was like, wait. Stand mixer, this should be. You're welcome, JJ. I hope it's ready for the rest of the day in the most sweet way I could ever say that. I hope you're saying yes to the fact that it's been three minutes since I started mixing, but the dog's out. I'm not going to do the other part. Still not. You're only doing one? You sounded like the song makes it a little bit of tool time, Taylor. 32 inches, I believe. I'm going to start giving discount codes for the website, like at the end of my videos, and then have it only be live for like an hour. We'll have to watch till the end. I started this video with like 50 people and 20 have all gone to let the dogs out, apparently. Evelyn says, I hear you, Jeff. Thank you, Clara. The other time she tells me that I have mixed long enough, I feel like I got early parole. I don't know what that feels like in real life, but I feel like it would be that exciting. Okay. So I know that we have like the trowel and all that to move your resin around. And I love our notch trowel. You don't know what that is. It looks like, wait, it looks like this. You can move your resin about and have it lay evenly across your everything. I just really like the feeling of my hands in resin. Okay. So I'm probably the only one, but I'm going to ask anyways. Am I the only one? I'm probably not. Okay, I have a question. In the very beginning, when you open it, it doesn't have you an option that you can enter the size and all that? Well, it says, I guess it depends on how high def you want it. I don't know if you guys heard Jeff's question, but any of you guys proficient with procreate? It's not want not. I'm going to have to go put some of this resin on the piece we did yesterday because I didn't have the clear, put over my glass. So that's going to have to happen. But I'll do that in a minute later. Denise says I procreated once and that was enough. But you're experience is handed. I don't know. I can't even, I don't even know how to words it so that I won't get hit with all of that. That's what she said. Helen, will you come over here so you can read it? Because I'm going to go put this on my alugia. I made the paste. I just want to know. Okay. So, Claire, now I just want, whoa. Now I just want to make it to where when I, when I push, like when I'm drawing, I don't want it to be really thin and then I could push it to be thick. I just want it to be a solid line. I just want to know somebody knew something in there.  Technical pen. All right, I'll try that. I think that's what I use, but I just got to get used to, you know, the pressure thing. All right, I appreciate it, JJ. What do you think about my this? It looks like you're going to need a little bit of playing notes you have as I need more resin. I forgot to say it's beautiful and amazing. The assistance. It shows. No flicking your pen. I'm going to assume that's PG though. So I'm just going to use the red fireglass that we pulled off of this piece earlier because it was not want not. This would look the exact same whether I dropped it in alcohol ink earlier or not. I feel like I should do another round glass. What do you guys think about doing it between the green and the yellow? I guess I could just make it a thinner line right there or should I do that? What do you guys think? This is a group project. Hi, Lisa. What's up, Sam? Where else should I add the glass? Or should I? Should I just not to hang out with resin? She's going to tell me. All right, figure it out in the field. Thanks, Joe. I agree. The colors do liven up with the addition of resin for sure. It gets thinner. Come on. So essentially, since this is not a mirrored fireglass, the glass is kind of going to disappear a little bit in the liquid of the resin. However, bring it to life a little bit with bubble beads and glitter glass. The thickness of this line of glass. Nature is not perfect. Actually, I think I like that. Not going all the way around. Look like Australia. That's what I was going for. I missed a little bit. Does the bubble bead disappear under resin? It won't because it's got a slight color to it. It's got like a bit of an opal look. They're clear, but kind of not also. I think actually, I'm going to add some glass just here. Just a little bit. Just a little touch. A little sum sum. Not to cover up that texture, but just because it just looks like something needed to be right here. I actually didn't mind that texture, and I would have done it all the way around, but I don't want to waste my glitter glass. It looks like I'm covering up things that I detailed in the ink, but it's just because I see extra detail in those areas. And until I get the look right, I'm just going to keep trying to add in those areas. JJ and Clara. Have you guys ever tried to make an NFT or anyone else in the room and in the resin that I was using for my beads? You want to get resin in your bubble beads? It looks very subtle. That's one of those things once you're looking at it, you're like, ah. I'm going to use these bubble beads to trail off further. Just very small shards of glass. I'm going to just use it all. Clara said, what is an NFT? I'm going to take it as a no. An NFT is basically a crypto piece of artwork. It's digital. I don't know a whole lot about it just right now, but it's going to be a big thing in the crypto world.  It stays the artist forever. When I first heard that description, I was like, it's pyramid scheme, but not really. So with every sale of like, if Jeff bought a piece from me and then he resold it, I would get a percentage of the resale value. It always stays like the artist gets a commission every time it's sold, which I think is pretty cool. Down so you can see what the beads on the resin looks like. Sorry, sorry. So those are the bubble beads. You can kind of tell the iridescence, but since the light on my camera phone is kind of extra, it's hard to see the iridescence, but you can really see it in person. But I like the way it kind of feeds off of the glass, so it's not a hard stop. Finger was gross looking. So it's not a hard stop of where the glass is and then just nothing. So I kind of trailed it off. I think it looks very whimsical as far as a rainbow geode goes.  Bubble beads. Bubble beads is what I meant today. Glitter glass, it's very fine tea tiny shards of glass. If you purchase this from me or anybody else, be advised it is glass and it will cut you if you don't respect it. This you won't be able to really see until the light hits it. So it's very subtle sparkle, which I'm a fan of in your face sparkle, subtle sparkle, sparkle, sparkle, but sometimes you need a subtle sparkle and this is just the product for it. I have the bubble beads, the fire glass and the glitter glass on my website, but unfortunately I don't sell alcohol in luckily there's a million places you can actually get alcohol. Like in the description box below this video on our Amazon link. Shameless plug. Buy this stuff by the kilo and I sell out of it all the time. Bee can I borrow your face? I like how it like blended this these two colors together this seam of texture. It's very cotton candy. Joe the working time for art coat is two hours. I love this piece. I love the progress that we made with it during this live feed. Let me know what you guys think. I agree. It's very whimsical. So yeah, I'm going to take this to the desk free zone and Evelyn who told you I was engaged? Yes, May 1st 2022. Joe if you, I don't know if you've ordered anything from my website, but next time you do just drop a note and I'll send you a like an eight ounce sample to try. So this Saturday I'm going to have a fire sale for wedding funds. We have a lot of art we need to sell priced to move and would love to have all you guys there. Tell all the people you like. Yeah, I know it's breaking away from the edge. Some of the purple. So I'm just I'm gonna have to baby it. Tell all the people that you like and then some people that you don't like about our fire sale help support the cause because I need Joe and you kind of want another. You never know what someone's going through. Boy cat's like, so what are we doing? Anyways, hope you all have an amazing day. We love you guys so so much. We'll see you guys tomorrow. 6 p.m. Central till next time. Bye. I said bye. We're the ex that can't reach it. That's sticky fingers.


Hello friends, I am Sanjay Gupta. I welcome you on Sanjay Gupta Tech School. In this video, I am going to explain how you can calculate sum of diagonal of 2D array. So I will be explaining two methods. First will be using two loops or using nested loops and second method will be using only one loop. So with these two ways, you will see how we can calculate sum of diagonal of 2D array or matrix. So before starting the explanation, just one update. If you go to detail or description of this video, you will find links of various playlists related to C language. So you can follow them. Now I am going to explain this first way that is using two loops or using nested loops. So let's say we have a 2D array whose dimension is 3 by 3. So remember one thing, if you are going to calculate sum of diagonal, so the dimension of your matrix will be same. Low and column size will be same. Then you can calculate diagonal sum. And these are other variables that I am going to use. Then the sprint net will display the masses and the 3 by 3 matrix on output screen. So these two loops will be reading values from user. So this is the reading part of 2D array. So after this reading part, now I am going to explain the second part that is to calculate sum of diagonal using two or nested loops. So I am starting one more part of loop. This is first loop that is for rows and this is second loop that is for and inside this I am going to write the logic. So let me first draw a matrix. So let's say this is 3 by 3 dimension matrix. So these are the indexes 0, 1, 2 row index and 0, 1, 2 column index and let's say these are the values 1, 2, 3, 4, 5, 6, 7, 8, 9. So we need to calculate some of these values 1, 5 and 9. Now if you notice that 1 is available at 0, 0 index. So 1 is available at 0, 0 index. Then 5 is available at 1, 1 index and 9 is available at 2, 2 index. So if we see all the elements which are on diagonal position having same row and column indexes. So inside this loop, I am going to apply a market condition that is if Iw equals to j. So I is representing row index and j is representing column index. So if both are equal, then I can apply as equals to s plus a of ij. So this way all the elements which are available on diagonal position will be added. And after completion of this loop, after completion of this loop, I can print the term on output screen with the help of this printf. So this way this is the first method which is using two or nested loop that I explained you. So in this method we compare both indexes i and j. i is representing row index, j is representing column index. If both are same, then only these elements will be added otherwise not. Now I am going to implement this method that is using one loop. So I am going to convert this implementation to one loop only. And you will see we will be able to have addition of diagonal. So I am removing these statements. So here you can see we have only one loop. We have only one loop and here I am writing these. So this is only one loop and with this one loop I am able to add the locations or the values which are available on diagonal positions. So now iterate this loop with me. i is 0, i is less than 3 that is true. Now check this a i i on both index and hk i. So i is 0, so it will become a 0 0. So a 0 0 index will be added to s. Then i will be implemented to 1. So 1 is less than 3, it is true. Then indexes will be a 1 1. So we require 0 0 1 1 2 2. So we are putting same values on both row and column index. So if we put same values on both the positions. So automatically we will fetch diagonal positions of 2D array. So those will be added to s. After completion of this loop, some will be printed on our code screen with the help of s. So this way I hope you understood how we can calculate some of diagonal elements using simple loop only. So this is very important. You should know how we can process diagonal using one loop only because row index and column indexes are same. So don't use two loops to calculate some of diagonal. You can use simple loop to do that operation. So I hope you understood both the ways like how we can calculate some of diagonal using master.tool loops or how we can calculate some of diagonal using one loop only. So I hope you understood whatever I explained in this video. If you want to watch more programming related videos, you can open my channel, go to playlist or you can go to detail or description of this video where you will be having links of various playlist related to C language. So do follow them. Thank you for watching this video.


just had a baby boy. This was our third child so we actually consulted Dr. Lina Sridhar and we had a great experience throughout. We could make out that she cares for my wife and our to-be-baby. She always gave us complete information, good information and she also kept us focused on this because you know it's easy as parents of the third child to you know think that we know everything and everything's under control but she always kept us informed and kept telling us what could go right and what could go wrong and where we should take care and this is where I think her guidance was excellent and I could always find out that she cares for us so that's about Dr. Lina and even now during the operation I think she's taking great care of my wife so we are now happy with baby boy and Dr. Lina family. Thank you Dr. Lina. As far as Manipal is concerned Manipal is a name that we trusted right back from Bangalore because I'm from Bangalore so I've known Manipal right from my childhood and then I've been there one of my cousins used to teach in Manipal as well so we know Manipal well this is the name we trust and that's why we came here knowing that it's in Duarka when we moved to Delhi so that's why we came to Manipal and I must say that I'm not disappointed at all. Manipal's kept it standard. Thank you.