 It's about time to start. There are no objections, and I suppose I should start now. OK, so I'll start. It's a pleasure to be here. I realize that it's difficult to follow Delia. Delia did a great job last week. On the other hand, I realize that we were actually supposed to inspire each other and inspire others. So thinking about how Delia had such an immersive experience for us, maybe I can do something similar. This will be a series of talks. For the first talk, there won't be any real immersion, unfortunately. Maybe in the second one, I already have an idea of something I want to do there. This may only be a series of two talks, but who knows? Whenever I start putting stuff together, I realize there's so much material that turns into a series. So this is also inspired by Rob Knopp. He gave a talk about a year ago, I think it was, on epicycles, which is a way of interpreting orbits in a geocentric universe, the universe of Ptolemy. So I thought, well, real orbits are interesting, too. So maybe we should talk about real orbits. So let's talk about this. For those of you who don't know me, I'm Dr. William Wall. I'm a researcher at this Institute in Tornacín, Puebla, Mexico. I'm an observational astronomer specializing in molecular clouds and nearby galaxies. Here is a picture of the P, far from our institute. So this talk is going to be a little bit different. I'm going to make it a little bit more technical. I think the audience can handle it. And I think that, even if I put in a bit of math here, many of you will appreciate it as your fellow science educators and scientists. And for those of you who have trouble with math, I believe I can make it reasonably interesting, or at least less frightening. If you have trouble with math, the usual reason, big part of that is it hasn't been taught to you in a way that works for you, not that I will necessarily do any better. But I think I might be able to make it a little bit less probable with it. And there won't be that much explicit math. Anyway, so here we see some elliptical orbit and a circular orbit, which is basically an elliptical orbit with an eccentricity of zero. Like, well, perigee and apogee, the near points and the far points in the orbit. So these are things that we're not going to go over in the first talk. The first talk is mostly about the physics that underlives orbits. And the next one will be more about the orbits with a little bit less physics. Let's look at the organization here. So we'll start off with some definitions. And as I'd like to do some motivation, why are orbits interesting? There are a number of reasons. We'll only be talking about the basic physics, as I said today. The second part might be another two or three parts, depending on how much is put in here. I think it's going to be at least three talks, but we'll see how that goes. And I'm afraid that the appropriate peril for ground crew is beyond the scope of this talk. So I did use the internet extensively for my talk, as I did my talk on the moon. There's more original material here, original figures. This is from Wikipedia. I would you define an orbit. It says here, an orbit is the gravitationally curved trajectory of an object, such as a planet around a star or a natural satellite around a planet. And it normally refers to repeating trajectory, but could also refer to a non-repeating trajectory. And this is a definition which I like. A lot of information in it. But I also prefer something a little simpler. So I wrote my own definition. Bodies are an orbit when they fall around each other due to their mutual gravity. Doesn't have quite as much information, but it's simpler. Here you see skydivers. They say they're in free fall. It's not really free fall because there's air resistance. But you can imagine that these bodies are far out in space, far from other celestial bodies. They would actually orbit each other due to their own gravity. And I worked out, based on their distances, they'd have orbital periods. It would be some billion years. It's kind of interesting. We don't think of our bodies having gravity, but OK, let's check my fan mail here. OK, so some people. No, yes, that's true. If you're talking about the Earth nearby, then yes, it will hit the ground. But I'm saying far out in space, whether far from any celestial bodies. Yeah, some people have some unfortunate shaz that you broke up with calculus, because I will be talking a bit about calculus here. But you don't really need to roll calculus to follow what I'm talking about. OK, so why are orbits interesting? There are a number of reasons why orbits are interesting to understand days, which is not just about the rotation of our planets, but also about their orbits, because that affects the difference. Seasons, of course, years, tides. We have to understand climate. We have to understand the Earth. Other heavenly motions, such as eclipses, which involve shadows of one body on another. Occultations and transits are one-to-one body blocks another. And of course, risings and settings of objects. Sun, that does depend on this orbit to some extent. And of course, artificial satellites. You can't put something into orbit, into the right orbit, if you don't know anything about orbits. And artificial satellites have reshaped our worldwide civilization, because of communications around the world, which we're able to do right now over the internet. So science circle, so communication satellite, and of course, global positioning system, which we rely on almost every day, or can rely on. It says, reshaped our civilization, revolutionized our civilization. It wouldn't be possible if we didn't understand orbits. And of course, we can do some real kick-ass science here. We can detect gravitational waves, for example. We can detect gravitational waves, a way of detecting probing the universe without electromagnetic radiation. Excellent planets. Keep in mind that these motivations, they could make interesting science circle talks in the future. And of course, space travel. If you want to explore the solar system and other solar systems, we have to understand orbits, exploit, for example, the moon, or asteroids, or have space habitats, space colonies. This is a subject which I found fascinating when I was a teenager, way back in the 70s. There was a physicist, Gerard O'Neill, who was looking in detail into space colonies. Huge rotating cylinders in space, and they're so huge. They're kilometers in size. You can have landscapes inside. You can have lakes, weather inside. This is a thoroughly fascinating subject. Now people living in space colonies. And where would you put them? How would you have them orbit? Truth is you're supposed to put them in the Lagrangian points, stable Lagrangian points. Moons orbit around the Earth, and we're not going to talk about Lagrangian points this time. But you have to know something about orbits to do the well, let's see, let's see, when I see those sized math, so space station from just, let's be more monoliths, okay. Things to know there. Yeah, I mean, Vicks, right, we really need to know calculus to really understand in detail the math of orbits. I'm going to be talking a little bit about calculus indirectly. I'm not going to give you a little calculus lesson here, but a bit about various, and another reason, we protect our Earth from collisions, those poor dinosaurs. If only they had understood celestial orbits, maybe they'd still be around. Now, as I said, I'm just going to talk about some basic physics, because there are a number of misconceptions out there, which I'm going to try to dispel, at least a couple of them. And the basic physics, if you understand the basic physics, then you can understand orbits a lot better. There will not be a quiz at the end. If you can't follow all of this, as Shantel has mentioned, I've given her a PDF of this talk, and if you say that Bill sent you, she'll give you a discount. Start with Newton's laws of motion. Actually, Kepler's law is the planetary motion, preceded historically Newton's laws of motion, but I want to start with the physics, and all relate them to Kepler's laws of motion. Pony and gravity, we're not going to talk about relativity this time around. Weight versus mass, those are two separate concepts that we should understand what they are. Velocity and acceleration, especially centripetal acceleration, that's important. And that leads to a discussion of fictitious versus true forces. Centrifugal force, for example, is fictitious, and we'll talk about that. Velocity in a circular orbit, and some basic concepts of work and energy, because with work, I can introduce concepts of potential energy, and from that the potential and the kinetic energy, and then from the total energy, which we can get the escape velocity from, and understand orbit of trajectories a little better. If you have different total energy, you have different trajectories, different orbital trajectories, basically conic sections that we'll mention briefly this time around. Newton's laws of motion. Okay, it's amazing how Newton came up with these laws of motion on the surface of the Earth because there was a little dissipation and friction on the surface of the Earth, so it's not clear how he was able to do this. First law, a body in motion stays in motion, and a body at rest stays at rest, unless acted upon by an external unbalanced force. That's just saying a body resists acceleration. That resistance of acceleration is known as inertia, but inertia, the amount of inertia a body has is basically its mass. And if you apply such a force to a body, it results in an acceleration A, this is a vector quantity, magnitude and direction, that is proportional to the force, but inversely proportional to mass. This is also called the inertial mass of the body. And you can put those proportionalities together into an equation, F equals MA. Normally for a proportionality, you need a constant proportionality here when you're converting it to an equation. Here it's just chosen to be one. So you have the mass in kilograms, acceleration in meters per second squared, and this gives the force in Newton's, and a Newton is equal to a kilogram meter per second squared. And that's related to the weight. The weight is a force, and the mass is the amount of mass in the body, that's the measure of its inertia, is the mass times the acceleration due to gravity, gives you the weight of the body. Now that's, this of course mirrors this equation up here. This on the surface of the earth is 9.8 meters per second squared, which means that a body that's falling without air resistance will fall 9.8 meters per second faster than the previous second. So each second is going 9.8 meters per second faster. And of course, Newton's third law is probably the most famous for every action, there's an equal and opposite reaction, which is basically saying that momentum is conserved. So you define this quantity mass times the velocity, that's a vector quantity. It's conserved in interactions between bodies. That's why this was defined. So why would you multiply this by this? What does it give you? It gives you, take for example, this rocket on the launch pad. It's total momentum plus its fuel is zero before the launch. During the launch, the total momentum of the rocket plus its fuel is still zero. And after the launch or during this is launching in space, total momentum is still zero. Rocket has total momentum upwards as a momentum upwards, but the momentum of the fuel or the exhaust gases downwards, so the two momentum cancel each other. So they still get zero momentum total. And angular momentum, I'm not gonna talk much about in this talk, but it's a useful quantity when you're considering orbits, one body orbiting another orbit, orbiting another body. So it's basically the linear momentum times the radius or the distance from the very center of the orbit, gravitational center of the orbit. And it's also a vector quantity, but the vector points perpendicular to the orbit. Let's see. So one question, can we change references instead of the Earth and calculate its orbits? Take the sun as a reference, does it work to calculate Earth orbits? Yeah, I suppose you could do that, but you're talking about very tiny perturbations on the distance from the sun to the Earth. So I'm not sure that's particularly practical. Yeah, that's true. Rick's right. You can use whatever reference you want. Some references are just simply easier to calculate with. Okay, so. And this is sometimes called Newton's fourth law. It's his law of gravitational attraction. The gravitational attraction between two bodies, and he got this from Kepler's laws of orbital motion. Gravitational attraction between two bodies is proportional to the product of their masses and inverse the proportional to the square of the distance between them. So here I have that expression. You put those proportionalities together into an equation, but you need a constant proportionality here. The reason you need that is because you want this force to be on the same scale as the force in law number two, and you want the inertial mass to be the same as the gravitational mass. This is gravitational mass here. This has been tested in the laboratory. You don't have, for example, gravitational mass is the 1.1 power of the inertial mass or something like that. It's always to the one power. They're proportional to each other, and you can make them equal to each other by choosing the G appropriately. This R hat here, you can ignore that numerically because it's just a one, it's just a unit vector showing you the direction along which the force points, and the minus sign means that minimize the length of the vector trying to get all our points together. So one way to think of an equation, for those of you who are not familiar with working with equations, is you can think of this as sort of like a recipe with different ingredients that you mix in, and you get a scrumptious dish that comes out of your fine dish. The great thing about the equations is though you can interchange your final dish with an ingredient, and the ingredient can become a mixture. First, I just follow certain rules, making sure that the two sides balance because this is equal to learn balance here. Another thing to keep in mind is if you're looking at a formula and you wanna have some kind of an equation or a formula, and you wanna have some kind of interpretation as to what it means, and one thing to look for is when you see these exponents, this is greater than one, this is two. So that means that this ingredient in your recipe is particularly potent, and that's important. If you decrease this radius by a factor of 10, the force goes out by a factor of 100. And again, the force is a vector, so as I said, has a direction as well as a magnitude. This potency that I'm talking about, especially when you're talking about kinetic energy and so on has life and death consequences. So we will get to that later. Constant, oh, okay. Vicks asked an interesting question. This is a question that astronomers have looked at. Is there a change in this gravitational constant over the history of the universe? Because you can look at very high redshift objects, objects that are very far away, and so you're looking back in time. Is there evidence that this G is changing over time? And so far, I would say, yeah, there is no evidence and nothing clear, no clear evidence for this. However, can you interpret the current cosmological models in a different way as if this G were changing? I'm not sure that that's been totally ruled out. Current cosmological models do not require a varying G. That's what I was saying. So I would say for now, we'll say no, but it's not totally ruled out. So let's have fun with these different recipes. Now we're gonna combine recipe number two with recipe number four. The weight must equal the force of gravity on the surface of the earth. So what we're gonna do is we're gonna put, let's see, our astronomer on the surface of the earth. And yeah, he's being a little bit distorted by the gravity here, but you know, as long as his green shirt keeps glowing, he's a happy camber. So the weight, as I said before, is the mass of the astronomer times acceleration due to gravity on the surface. This is gravitational constant as I said before, so it's distinct from this G. And it must equal the force of gravity on the surface of the earth, which is sort of radius of the earth. So you're sort of mixing two recipes together, but you see something a little bit stranger. You see that, you know, your output, your final dish has the same thing as your, as an ingredient here. So your ingredients coming out in the same way that'll weigh in. And what this means is that that particular ingredient, I know this is gonna shock all of you, but it means that the astronomer is irrelevant. And yes, that's extremely shocking. How can the astronomer possibly be irrelevant? Well, in this particular case, he is, and what it's saying is whatever the mass is on the surface of the earth, it doesn't really matter what the mass is, that mass is gonna accelerate towards the surface of the earth, regardless of what that mass is. As in the mass of the earth, but not the mass of the body. So pre-falling bodies accelerate at the same rate. So a feather will fall, pre-fall like a stone if there's no air. This has been shown. Pre-falling means accelerating, fairly to the gravity of no additional forces present. So that equation, let me go back to that equation first of all. This equation is interesting because you can calculate stuff with it. This has been measured in the laboratory. This has been measured in the laboratory. This, measuring this is really difficult in the laboratory. You're really sensitive to measurements. But if you had the race of the earth, you could actually work up the mass of the earth. How many kilograms is the earth? So all you have to do is the race of the earth. And yeah, you can look these things up on Google Earth, which of course, or on Google I should say, and I get some help from Google Earth. And sometimes it's fun to calculate these things. So let's see here, this is Toronto. This is the Tower, this is Lake Ontario. Where am I? I'm in Niagara on the lake. I was visiting my parents back in 2009. And I have my camera has a good level of optical zoom, 18 times, which I suppose was impressive back in 2009. And I said, oh, this is a good view of Toronto across the lake. So I took a shot, I thought that was kind of pretty. And then I realized, wait, the lake here, the surface of the lake, it looks like the lake is flooding the streets of Toronto. I'm seeing the curvature of the earth here. Which means that I can work out the radius of the earth from this picture. The problem is, I didn't remember where I was standing because it's a uneven surface. How high is the camera above the surface of the lake? I didn't know what that was. Anyway, I was able to calculate, it's between 59, 30 and 7,060 kilometers, correct numbers in between. So, I mean, I'm gonna repeat this experiment sometime and I'll go back and I have a better camera now. Well, the point is, if it's a third day, I might try this experiment and see how close I can get. This is kind of gonna be the state of Toronto. So I can take these numbers and I can work, you can work these things out yourself. You might not be familiar with this notation. I'm sure many of you are, but if you're not, 10 to the minus 11 here means that that decimal point has moved over 11 places to the left. The six means moved six places to the right. So you can do this calculation yourself if you want, work out the mass of the earth. Just make sure you're using the proper units. And the mass of the earth is about six times 10 24 kilograms, I mean six with 24 zeros behind it, or six with 21 zeros for if you're talking about tons, metric tons. So that's the mass of the earth, six, six billion tons in American units or American numbers. The Brits use American now, but it's 6,000 trillion in British. And that number is useful. We'll use that later, because I've had people ask, what's the weight of the earth? Well, the weight, no, that's not a good question to ask. Earth is basically weightless and free fall around the sun. So what is zero weight? But that's not what they're asking. Yeah, okay, yes. I'm gonna use my vacation photo to do science. Well, you know, some people for vacations, they do like to do science. Crazy as that sounds. So that's not a good question to ask. The question is asked mass, and you have to know the difference between mass and weight. That's easy enough to illustrate here. Imagine we have, okay, we have our astronomer again, moon here, and the acceleration due to gravity on the surface of the earth, 9.8 on the moon, it's about one-sixth of that. Imagine you have a 60 kilogram person, which is sort of an average weight or an average mass, I should say, of an adult. I would guess more or less 60 kilograms on the earth and 60 kilograms on the moons, because mass doesn't change. That 60 kilogram person on the earth weighs 588 newtons and 97.5 newtons on the moon. And you can use British imperial units, weight, I mean, pounds actually do measure weight, and the slugs is basically the unit of mass in the British imperial unit. The conversion from pounds to kilograms actually depends on the gravitational field you're in because you're comparing mass with force. But the point of all this is that mass is that property of matter which manifests itself as inertia or it's the amount of matter in a body. It does not depend on the gravitational field. If you're in free fall with a couple of masses and you want to know which one is the heavier, or should I say larger mass, you just push on it and see which one resists your push more than the other. Whereas weight is the force with which gravity pulls down on a body, or how hard it presses down on a low surface. Okay, the Brits might use sexillions, but I think it's defined a little differently. That would be 10 to the 36 original British weight. Now, let's talk about velocity and acceleration, and of course, acceleration is the rate of change of velocity. These are both vector quantities, but keep in mind that these vector quantities can point in different directions, and that's an important consideration. If you can understand that, then you can understand orbits, because orbits will very much work that way. Only if the acceleration has a non-zero projection onto the velocity vector, will it change the magnitude of the velocity vector, as magnitude is called speed, and the direction is called direction. So let's look at some representative cases. Case one and case two are almost the same thing, really. Let's see, finding mass in terms of inertia. Now, well, okay, it seems circular to me. Well, I'm not sure there is a difference between mass and inertia, there are not. Okay, so if you have a velocity vector and the acceleration vector, they're aligned, then the speed increases, but the direction is constant. If they're anti-parallel, the speed decreases, and again, the direction is constant. Well, let's consider this case three, and you can imagine that all possible cases are a superposition of those three cases. Acceleration is perpendicular to velocity. In this case, the speed is constant, and the direction change at a constant angular velocity, omega, which is measured in radians per second, and the radian is about 60 degrees. It's basically the angular size of an arc as seen from the center of the circle, an arc that's as long as the radius of the circle. So it's a little bit less than 60 degrees. That's 180 over pi, which is 57.295. And you can understand why I want to consider this particular case, because if you want to think about circular orbits, well, this is what's happening in a circular orbit. So again, we'll examine the case. Okay, this, we'll take a projectile, which is not usually affected strongly by air resistance. A good old bowling ball here. Throw it upwards, and of course it'll come down again. And all through its path, again, assuming air resistance is negligible, the acceleration is constant. Velocity is going in the opposite sense here, so this is case two, plus it's going in the same direction here. Even at the top, when the bowling ball is not moving, acceleration is non-zero. It's still at this value of one G. Keep in mind, we also use this G here. We use this to mean a measure of acceleration. If something is accelerating at 9.8 meters per second squared, for whatever reason, even if gravity, all about a one G acceleration, measuring acceleration. Now, if we repeat this experiment, imagine you shrunk the earth down, you remove the air, then this ball is going to fall, and shrink it down, or shrink it down to a point, and the ball is basically going to be in an orbit, but it's a radial orbit with zero angular momentum. Such radial orbits more or less exist in nature, but it's kind of an interesting experiment. When you throw something, you're actually putting it in orbit, at least temporarily, and figure out how far something moves when it's accelerating. Okay, yeah, so tagline is measuring, it's not massive measure of the inertia of an object to a force intrinsic quality of matter confined to any local limber. I read Artemis once, okay. So you're talking about, tagline is talking about, I think it was Mach, who's a philosopher and a physicist, and he said that, how does a body know it's accelerating? The reason a body knows it's accelerating apparently is because it's accelerating with respect to the rest of the universe. And so the universe is pulling on it, and that's what causes inertia is. I don't know if that's fallacious or not. Yes, that's right. I wouldn't catch a bowling ball after you throw it up. That's a good point, Vic. Okay, so let's work out the distance. If you have velocity, velocity stays constant, here's your velocity as a function of time. It's this flat line, okay. So the distance traveled, of course, is v times t, which is just basically the area in the rectangle here. Velocity times time. Now suppose instead you have linear acceleration, that means constant slope, slope A is the acceleration. How far is the body moving? Well, again, it's the area under the curve. And calculating the area under that curve is very easy. It's gonna be the same as the rectangle, but multiplying by a half. So the height of the rectangle is A times t. This is the velocity at any given moment is acceleration, it's time. And of course, you have to multiply by the base of the triangle, which is t itself. And you get one half A t squared. So this is something to keep in mind. Here we have another recipe, and this particular ingredient is potent. What it means is if you have a falling object, the acceleration is g downwards. If there's no air resistance, it will cover distance very quickly indeed. It also means something else. I mean, this is gonna be related to kinetic energy, which we'll talk about later, but it means the kinetic energy is gonna go like velocity squared. And that means that rapidly moving objects, if very rapidly, that leads to very high kinetic energies and life or death consequences, which we'll talk about later. It also means something else if you have this t squared. It means when you have a projectile, if you throw a projectile, you're gonna have, if it's gonna follow a path, if you throw it roughly horizontally, you're gonna have a path that's parabolic, which is what this is. Constant downward acceleration, g. You throw this bowling ball. Well, I think that would be a real effort, but this is basically case two, a mix of case two and case three. It's not a pure case two or pure case three. Pure case three here. Acceleration is perpendicular to the velocity at this point, and you have a mix of cases one and three. And this shape is called parabola. And yeah, Vick's talking about 70,000 miles per hour to get into orbital velocity. We're not gonna talk about orbital velocity just yet. We're gonna come up on that soon though. So that parabolic path, now I put it over here. I put it over here and imagine again, you shrunk in the earth. The earth is now very, very tiny. So this parabola now is actually the end or one end of a very eccentric ellipse. So again, when you throw a body, some projectile roughly horizontally, this parabolic path can be thought of as part of the elliptical orbit. So when you throw something, you're just kind of putting it into orbit, at least temporarily, which I think is kind of cool, or at least mildly interesting. Secondary equation of conic sessions. Yes, well, I'm not gonna talk specifically about conic sections yet, tagline. I'm gonna talk about those in part two. Talk about them a little bit. And then there's this circular path that I was talking about before. And of course, this is important for a circular orbit. You have a body that's moving at velocity V. And yet it's accelerating towards the center. This is important. This is extremely important for a circular orbit. Because what this is saying is, why doesn't an orbiting object fall to the ground at our feet? Well, the truth is, it is falling towards the Earth in the sense that it's accelerating towards the Earth. But it has tangential velocity, which keeps it from reaching the ground. The tangential velocity keeps it at a constant distance above the surface of the Earth. So yet it is falling, but it's falling around the Earth. This is a concept known as Newton's Canon. And I'm gonna talk about that more in the second part. So this circle has a radius R. And the object is moving with an angular velocity, omega, so many radians per second, with respect to the center of the circle. Omega is simply this V over R, some radians per second. So we can calculate these centripetal accelerations. It's fairly easy, it's fairly simple math. So you have a velocity vector V at time T. And then just a slight time after, the infinitesimal time after. And it's gonna be the same length, but pointing in a slightly different direction. An angle D theta, which my arrows didn't come up quite right here, but tiny angle D theta, which we can measure in radians. You can think of this as holding your thumb, let's say this is your thumb. This is the difference vector, which we're trying to find. This difference vector changes is corresponds to this time delta T, or DT. This D theta is a mega DT, and radians per second times seconds is radians. So this divided by this, gives the acceleration divided by the DT. Getting this, the length of this is very easy. It's like holding your thumb up at arm's length. If I hold my thumb up at arm's length, it's 58.5 centimeters away, give or take a centimeter. And I can figure out, if I can't put a ruler up to my thumb, I can work out the width of my thumb by its angular size. If I know it's angular size, I know that my thumb is actually two degrees in width, and I need that in radians. So that's one 30th of a radian. So one 30th of, say, 60 centimeters is two centimeters. So I know my thumb is two centimeters wide. So that's what we're doing here. We'll say the arm is V long. Your arm is V long, and your thumb is D theta wide. So you can just multiply those two. That's what's happening down here. Multiplying your arm length by your thumb's angular size, and you get your DV. And of course, D theta, as I said before, is omega DT, and it goes to V over R, so you can put them all together. And if you don't follow all this, again, you can get the PDF from Chantal. And you end up with V squared over R, or you can put it in terms of omega squared. But again, okay, we see the square here, the two here. That's important. So this velocity is an appotent ingredient in this recipe. So, acceleration has a strong dependence on the body. So let's have some examples. I mean, one of the reasons I like playing with these formulas is you can actually have fun with numbers. And I have fun with numbers. So let's see what happens. When we look at, say, this fighter jet in a high G turn, 140 kilometers per hour. But suppose the radius of that turn is 800 meters. That's almost a kilometer. And what is the centripetal acceleration? It's 5.4 Gs. If you were sitting in there, you'd feel five times heavier than normal. You'd also feel gravity pulling downwards, but you'd also feel five times your weight being pressed into that seat. Because the jet is forcing you into a circular path. Forcing you into some triple acceleration. Here's another example. I'm driving home on Highway 1, which is, I'm supposed to be the trans-Canada Highway. It goes through Vancouver, Metro Vancouver. And I am heading back to Horseshoe Bay, which is in West Vancouver. And way up here is the ferry terminal, which has ferries that go to Vancouver Island, among other places. But I wanna actually get it over to the apartment over here. So I have to go on a very specific lane here, which is kind of narrow. And if it's narrow, you don't wanna have curves with varying radius of curvature, at least not quickly, driving something like that. So I can use Google Earth to work out, you know, what's the radius of this circular path? It's 290 meters. I'm going probably a little faster than I should be going. Watch per hour, radius 290 meters. So my centripetal acceleration is 1.3 meters per second squared, or 0.13 Gs roughly. Yeah, you would feel that. I mean, that corresponds to actually leaning over towards the center of the curve by eight degrees to feel vertical. And as I said, there's a strong dependence on the speed. And I'm gonna talk about that in a moment. Now, so imagine instead, we're standing on the surface of the earth. We're all on the surface of the earth, of course, at different latitudes. I chose to be on the equator in this particular diagram. Angler speed. You can work out some triple acceleration. It's, you know, 3 milli G. All of us are at different latitudes, of course. So you're gonna have slightly less because you're at different distances from the rotational axis of the earth. No sign of your latitude. But you're gonna have a few milli Gs of acceleration, which means you're not gonna feel it. But as I said before, velocity is important. It has a strong effect on your centripetal acceleration. So I'm going 70 kilometers an hour. I probably shouldn't. I should probably go 50 kilometers an hour. What do you see what happens here? This drops by a factor of two. By a modest change in velocity, you see that the centripetal acceleration drops by a factor of two. This is important if the surface is slippery. You don't have enough friction to supply the necessary centripetal force to keep you on a circular path. This is also important. Speed is also important for kinetic energy, which I'll talk about later. And this leads to discussion of fictitious and true forces. Yes, that's a good point, Vic. Very often highways are banked, but I'm not sure that it's necessarily a good idea to bank the highway in this particular part because it depends how slippery it is. Banking becomes dangerous if the highway becomes slippery. Okay, so let's talk about fictitious forces and true forces. This is an experiment you can do, and I recommend you test it as a car passenger, not as a car driver. Do not do this if you're driving. Oh, here I am in this lane. I'm driving along, and I have, say, a passenger who's looking inside the car. You just look inside the car. You don't look outside. And then you'll perceive this mysterious force pushing you to the outside. But if instead you don't look just inside the car, if you look inside the car, you don't see anything other than what's inside the car. So you perceive some mysterious force pushing you up. But on the other hand, this is something I have tried when I was not a driver. They cut a landmark as you're going into the curb. And what you'll notice is, if you keep staring at that landmark, you'll notice that your body's just trying to go on a straight line. It's being forced into a circular path. It's forced into centripetal acceleration. And this so-called centripetal force was nothing more than the inertia of your body, trying to maintain a linear path. So if you're in your surroundings, like your vehicle, or experience acceleration, you're in a non-inertial frame of reference. And you'll perceive certain forces, the inertia of you and your surrounding, will be perceived as forces, like centrifugal force or Coriolis forces, which are important in, for example, hurricanes. And we won't be talking about Coriolis forces today, that are apparent in rotating frames of reference, but are fictional. True forces, in contrast, a person in an inertial frame of reference, moving at a constant velocity, will see the non-inertial frame as accelerating due to the outside true force, a centripetal force. So let's look at one of the true forces in a circular orbit. The centripetal and gravitational forces are relevant. But the gravitational force, of course, is the centripetal force, so there's really only one force. So let's equate them. Have another little recipe here. Orbital velocity, we're talking about circular orbit. So this is simple. The mass times the centripetal acceleration gives you the centripetal force. This is the gravitational force. And you can simplify that expression very easily. And you get the orbital velocity. It's a simple expression here for circular orbits. And of course, now that we have this formula, we can have fun with numbers. I supplied the numbers here so you can do these calculations yourself. And if you're orbiting, say, the ISS or orbiting the Earth, I picked to say 200 kilometers up. I think the ISS is maybe a little higher than that, like 400 kilometers, but I won't change the answer much. But it's orbits at about 7.8 kilometers per second or about 17,400 miles per hour. And we can look at the Earth orbiting the Sun. I've, again, given you the numbers and you need to do this calculation. The Earth goes around the Sun at about 30 kilometers per second or about 67,000 miles per hour. Yes, Vic, that brings up a good point. You're talking about trajectories or artillery shells. Yeah, from what I understand, my grandfather used to tell me this about how the Navy, when the Navy went into the Southern Hemisphere for the first time, the shells, long distance shells, would keep missing the targets because they weren't properly correcting for the Coriolis force changing direction in the Southern Hemisphere. So yeah, that is an interesting subject in itself. Okay, we can now talk about escape. Well, not quite yet. We're gonna go to talk about the escape velocity. We have orbital velocity. We know the expression for that. To get to escape velocity, we have to go through certain concepts, like work and energy. So work is the product of the force and the displacement it produces. So imagine the simplest examples. You have a person here going up the stairs. The force across is their weight. They're overcoming their own weight. 60 kilo ampersand, 508 newtons going up. One story is three meters high. So the work done is 1,764 joules, about two kilojoules. If done in seven seconds, a leisurely pace, that's 252 watts or about a third of a horsepower. Not entirely relevant. This is an aside here. So a little bit less than two kilojoules. Then we can talk about potential energy. This is the energy of position within a field and it's equivalent to the work necessary to move a mass to that position within the field. Or you can say it's, you can also talk about electric charge within an electric field, it's the same idea. So that person would have gained 1,764 joules of gravitational potential energy, which could be converted to potential energy if you then jumped for a witch horse. Then we talk about the potential which is potential energy per unit mass for a gravitational field, joules per kilogram or electric field, it would be joules per coulomb for a charge, which is volts. The potential, so it's joules per kilogram because as we found out, this doesn't change for the mass within that field and I have the moon going around the potential well of the earth here. We'll talk about that in a little while. The field is the force per unit mass for a gravitational field, which would be the same as acceleration. So the question is, why do we have potential and field? What's the advantage to have in both of these quantities? And we'll talk about that in a moment. The field is the negative gradient of the potential. In other words, it's the downhill direction and slope. This is an intuitive way to represent the field. The potential, now I live in, when I'm in Canada, I live in British Columbia, it's a beautiful landscape, lots of hills and valleys and mountains and here in Mexico, as a matter of fact, there's this volcano in the distance. So there's an interesting landscape here. I've been to Europe and I've seen the Alps. So you can picture that landscape and you can picture if you had a marble rolling around in it, you know which way the marble would accelerate because you know the downhill direction. So you start with a beautiful landscape, which is basically the potential and you can attach to that potential, you can attach a bunch of arrows, which would be the field and that field tells you the acceleration. So this is a simple way of representing the field, that's why this is defined. So let's talk about the potential of the entire solar system. I wanted to find a three-dimensional potential where it showed the potential surface, the surface in two dimensions and the potential is the third dimension, I only found a slice through that potential. This is a bit contrived because you don't have the moons and the planets all lined up in a single slice in the solar system. So first of all, you'll notice that there's this large-scale variation which goes across like this. This is the solar potential, this is the potential well of the sun and impressed within that large valley of the sun are these tiny valleys representing the potentials of the planets and their moons. This is a tiny potential too, believe it or not. And you can imagine if you had a body, orbiting body, it's almost like having a marble rolling around in these potentials, now how they would accelerate. And if they're not in a velocity, they would actually, the velocity would actually fall into the well as it was going to begin with. There's no initial velocity that's at right angles to the direction to the side Jupiter. It would fall in. And this is not a perfect analogy. Things don't need to roll around in space. There's no dissipation in space. Don't need to roll to move, but it's at least gives you an idea of why the potential is defined. As I said, I wanted to have a potential surface here and maybe I'll create something like that sometimes just for the fun of it. But I did actually for the Earth, this is the Earth's gravitational potential. Well, I did create this myself using Microsoft Math and Irfan View to enhance the colors and Libre Office to make the diagram, label the diagram, put it here. So this is the Earth's potential well and I don't have the Moon's potential. But it does give you an idea of what's happening with the Moon. Is it falling towards the Earth? In a sense, it is. It's accelerating downwards. These blue arrows represent the accelerations within this field. As it gets steeper, the arrows get longer. But it's always downhill. Those are the accelerations. But the velocity can be different. So the Moon is accelerating towards the Earth, but maintaining more or less constant distance from the Earth. As it accelerates towards the Earth, it's still maintaining a roughly constant distance. How important is the Moon or maintaining the Earth's orbit and velocity? The Moon is not really important in maintaining the Earth's orbit. It doesn't really perturb the Earth's orbit around the Moon very strongly because the Moon is so much lighter. The Moon, however, does have an important effect on the Earth's rotation. It stabilizes the Earth's rotation, but that's another thing to consider. Is it fair to say that Jupiter has a stronger gravitational field than the Earth? It has a bigger field. It depends how you define big because the field, of course, extends forever. You can extend it to say when the force is less or the acceleration is less than the quantity smaller than the orbiters. Okay, so here's this gravitational force. Now, as I said before, we want to go from force. We can think of terms of work. If we move an object, change this R here, then we take the force and we do it in tiny steps. The force at each step and the stepping at R. We move it from infinity down to R. So we're adding these things up in tiny steps and that's integral calculus for those of you know calculus. And it's actually equivalent to multiplying by R in this particular case. And this is what you get for the potential energy of a mass M in the Earth's gravitational field. It goes like one over R here. So it's almost like force times this distance R. Now, keep in mind that this force is, this potential is zero at infinity and it's negative on the surface of the Earth. And that's important because what that means is that we're bound to the surface of the Earth. But we can talk about the potential in terms of the potential energy is the mass, is this potential energy divided by the mass. And so we get this expression here for the potential. Zero point, as I said before, occurs R equals infinity. That's mass has escaped the potential. Well, if it has zero potential, the bound object has negative potential. So exactly how strongly are we bound to the surface of the Earth? Again, we can work out some numbers. It's 62.5 minus 62.5 megajoules per kilogram. And that's a lot. So it means if you consider a normal sized adult, you need a few gigajoules of energy. You have to add, bring this up to zero. To bring this up to zero, you have to add energy to it so that you can escape the Earth. A amount of energy you have to add is equivalent to two million stories upstairs where space travel is expensive, meaning it's difficult to do. And this is an illustration of that. Finally, we get to the concept of kinetic energy, which is the energy of a body due to its motion. And again, we use work to work it out. We use work to work it out. Always takes work to work it out. Here's an example of kinetic energy. So interchange between kinetic energy and potential energy when you're on swings. This is my family undergoing a second childhood. This is my daughter, my sister-in-law, my mom, and my wife. So family on swings, when you're at the height of your swing, you have maximum potential energy and zero kinetic energy. But at the bottom of the swing, when you're at the bottom of your swing, you have maximum kinetic energy and minimum potential energy. And the reason I show this is because elliptical orbits are like this. You have this interchange between kinetic and potential energy further away and then closer again, orbiting. So what is the work required for the mass to accelerate a mass to velocity v? For us is mA, as usual. The distance, we've worked out the distance before. When we multiply these things together, assuming that they're in the same direction, mA squared, t squared, but of course, At is the instantaneous velocity, so we get one half mV squared. And this is a very important consideration as I said before. Velocity squared, kinetic energy has a very strong dependence on velocity and this has important implications. You can have too much kinetic energy sometimes, which is then converted into the work of fracture materials like the steel in the car, the bones in your body. Imagine, for example, you're driving around in a parking lot going at 10 kilometers an hour, which is double walking speed. And you hit something, something solid. That would be a bit shocking to damage the car, but no big deal. Imagine instead, you're not moving at 10 kilometers per hour. You're moving at highway speed 10 times faster, 100 kilometers an hour. You hit something solid at that speed. The car is totaled, chances are you won't survive even with good airbags. The damage that you will suffer is not 10 times worse, it is 100 times worse. Drive safely and slowly because kinetic energy depends very strongly on velocity and the damage will be proportional to the kinetic energy. Yes, I'm preaching, but I'm preaching mostly to myself because I have a heavy foot when driving. Something to keep in mind, not my heavy foot, but be safe when driving. So this, of course, leads to the concept of total energy, which I mean the kinetic plus potential using the word total is probably not the best word for it because you can think of other forms of energy. But you basically add the kinetic energy to the potential energy. And this total energy is important because it can be negative, zero, or positive sometimes. That means the orbit is bound, marginally bound or unbound. And this leads to different conic sections here. The term is the orbital trajectory and we'll come back to that in a moment. But let's look at the case of zero because when the energy is zero, total energy is zero, you have just enough energy to escape. So you set that to zero, so it's kind of strange. You mix the ingredients together in your recipe and nothing comes out. That's kind of freaky, actually. But you can rearrange it. And then you have, well, this is deja vu. It looks like you've calculated this before. I wonder if deja vu is recursive. Can you have that strange feeling that you've had deja vu before? Something worth thinking about. But it's not quite the same thing. It's actually root two times the orbital velocity. So you only need to increase your velocity by a 41% or a factor of 1.41 to escape the gravity of mass M. But if you talk in terms of kinetic energy, of course, you square everything. This becomes a two, which means that the energy you need to escape when you're in a stable orbit, you already have half the kinetic energy you need to escape. And that doesn't matter what the distance is. You always have half the kinetic energy you need to escape if you're in a stable orbit, which is one reason why it's better to build spaceships in orbit because you already have half the energy you need to escape. It's also, I think, plays a role in having multi-stage rockets. There's an advantage there, too, I believe. But okay, and of course, we've got fun with numbers. What is the escape velocity for the Earth? It's 11 kilometers per second or close to 25,000 miles per hour. And for escaping the solar system at the distance of the Earth, 42 kilometers per second or like 94,000 miles per hour. It's actually easier to escape the solar system and fire something into the sun not because of energy considerations, but because of momentum. And so I have a table here to summarize everything. Those three cases, the total energy, less than zero, equal to zero, greater than zero. It's less than zero. It's a bound orbit, elliptical. The eccentricity is zero up to nearly one. So many examples of this, Earth orbiting the sun, moon orbiting the Earth, short period comets, marginally bound. This is a knife edge. You're never gonna have something that's exactly equals zero. This corresponds to an eccentricity of one or a parabola. Like long period comets, they come in from the Oort cloud. They seem to have, when they reach perihelion, that means closest point to the sun. They seem to look like they have equals one. So astronomers spend a long time observing some comets to see if they can work out whether this is slightly more than one or less than one. And of course, there's hyperbolic orbit when E is greater than zero. That means this is something that's come from outside the solar system. Oh, a mua mua. Say that, oh, a mua mua. I have to practice. Anyway, eccentricity is slightly greater than one. Significantly greater than one in terms of uncertainties. And then there's Borisov. Exetricity here is 3.34, much greater than one. So that's almost like a straight line. Quite impressive. The thing about orbit is that they are predictable. Yes and no. It depends. In the long term, yes, a circle of zero, eccentricity, that's right. Orbits are predictable. If you're talking about just a two, a restricted two-body problem, then you have a restricted two-body problem means one body is much smaller than the other, much less massive. The problem is you just have a solar system with many bodies and they perturb each other. So it's not quite as predictable as you might think. And we're not gonna talk about that this time. So this is a review of everything we've done. I'm sorry, I seem to have gone a little bit long here. Talked on all these subjects here. And part two is not ready yet. Gonna be ready after some time. I mean, I'm gonna talk about Newton's cannon, of course. My son, Dylan, who had a beef with me and he wanted to shoot me with a cannon. Instead, what he did is he stuffed me in the cannon. So here I am, stuffed in the cannon. And then he tried to fire me around the earth, but now it didn't quite work. So, okay, okay. Mosul velocity is still too low. Okay, so now I'm falling around the earth. That's good. That's how Newton's cannon works, which you'd call this Dylan's cannon. And then the capitalist laws of planetary motion, which we'll talk about next time. And these are the topics that I have to consider yet. I'm gonna try to have a 3D model. Moving model, ski effect. What real orbits, as Lagrangian point, seasons, procession, eventually Michael, the white and rich cycles, occultations. And one of the other references where I got my material, you can check out the PDF on tell. That's pretty much everything. Thank you, thank you. Need an AO where I can take a battle. Thanks, Shado. Thanks, Dave. You, Shaz. Thanks to you. Questions, comments? Thanks, I got that. Can you drive a golf ball into orbit around the moon? You can potentially fire a golf ball into space, reach a skate velocity, and it will reach the moon. The problem is how do you get it into orbit? You have to dissipate the velocity or it will just keep going beyond the earth. So, escape the earth, escape the earth and the system. To make it go into orbit around the moon, you have to change its velocity somehow. If you had, let's say, another golf ball, now you need something to, you need like a third body involved, or you need some kind of atmospheric braking, but there's no atmosphere on the moon. Oh, from the moon surface. Oh, that's an interesting question. From the moon surface, you can reach the skate velocity. The skate velocity is much lower on the moon. Don't remember exactly what the number is. Something like factor five lower, I believe. Yeah, and you can make it escape the moon, but not escape the earth, so you could actually knock it off the moon into earth orbit. Yeah, I think you could possibly do that. Yeah, whereas temperature, total energy. Yeah, like I said, total energy wasn't the best thing to call it because there are other types of energy that which I've not considered. I'm only considering kinetic energy plus the gravitational potential energy, the temperature of the body. Five times, yeah, so that is roughly five times. Temperature can matter in a sense that if you have a rocket which produces very hot fuels, if it needs to be hot in order to expand in order to give you thrust, there are other ways of thrusting which, like ion drives, they're more efficient. Your fuel doesn't need to get hot, so all the energy goes into the thrusting. But the problem is this, even though you have what's called a high specific impulse, you don't get a very high thrust. Yes, water hazards on the moon, yes, not too many of those. I think they have found evidence for ice and crater on the moon, I think they did find evidence for that. It's not exactly liquid water. Yeah, five times less. Two kilometers per second is nothing to sneeze at. Like three, eight kilometers per second, yeah. But easier to achieve than escaping from the Earth. Yeah, that's why they have this idea for the Lunar Gateway Space Station. It'll be in orbit around the moon because it's easier to escape the moon than it is to escape the Earth. I think it's closer to one-fifth. 2.38 divided by 11. Yeah, like 2.4 times five is 12, so it's definitely less than six different, it's a factor of six different. Less than a factor of six, less than a factor of five. If you really are interested in understanding a bit more detail, I recommend getting the PDF from Shantel. Yeah, a rifle bullet, 1.5 kilometers per second, yeah. Well, the escape velocity from a black hole. Yeah, you're quite welcome to take one. Escape velocity from a black hole, whenever you say, talk about the escape velocity from something, you have to talk about not just the mass of the body, but where you're starting from. You can escape from a black hole if you're well outside the event horizon. So it's not necessarily more than the speed of light. It depends where you start. Yeah, I mean, Shaz, that's a good point. If you have questions, if any of you have questions about the PDF, you can contact me, you can, I am me, and if I'm not online, of course, I can go to my email. So you're quite welcome, Mike, thank you. Thank you, most of the source. Well, that's an excellent question tagline. It's not a space junk. Yeah, that is a severe problem. Is it ever gonna be so severe that we can never launch anything safely? I'd like to think that's not gonna happen. I think that they would, I think there are plans to actually remove the space jump. I hope those plans succeed. It is dangerous because you're having stuff that's speeding like a speeding bullet, and that can cause a lot of damage. Talk about kinetic energy. That those pieces can have a lot of kinetic energy and cause severe damage. Yeah, well, that's another problem, plastic in the oceans. Yeah, I think you're right, Shadow. I mean, that could be very lucrative. If you find someone who's willing to pay you, I mean, space agencies might be willing to pay you to remove space jump. So that could be great for certain businesses, as a matter of fact. USA commissions, okay. Probably be there tomorrow. Thank you, Zylin, Zillin. Yeah, it's actually, in a way, it's worse than what you're suggesting, Shiloh. Not necessarily hitting a piece of hitting an entire satellite, but if you hit something that's really tiny, it can penetrate like a high-speed bullet. So in a way, that's worse. Thank you, Delia. Yeah, if you build something in orbit, that's true. If you build something in orbit, you could create more space junk, and you don't want that to happen. So I think the astronauts now, when they go into space, they're very careful about what they let out of their spacecraft. They try not to lose things. Yes, well, I think numbers are fun. Numbers represent the secrets of the universe. Okay, yeah, this is an idea that I mentioned before, in McKenzie. You can't use the sun as an incinerator that easily. It is very difficult to launch something into the sun because the Earth has a lot of momentum, and when you launch something from the Earth, it'll have a lot of momentum, and you have to basically fire it backwards from the Earth, and you need gravitational assist in order to actually even reach, for example, the planet Venus, Mercury, and the gravitational assist to slow down the spacecraft using other planets. To get it into the sun would be a major effort, help healing surfaces. Yeah, that's a good idea. I think those actually exist, some of the simple problems that they are. Thanks for that, Fred. Satellite collision, yeah. Ticketer satellite collision, yeah. Yes, yeah, he has these satellites in orbit. It's supposed to be cell phone satellites. I forget what they're called, but astronomers aren't too happy with those because they can brighten the sky too much. That's one of the reflections. Thank you, Thomas. Internet, okay, internet satellites. Okay, interesting question. If you shoot a handgun in the sun, yeah. Traveling in a straight line in space, especially in the gravitational field is not that straightforward. In order to actually, yeah, you'd have to shoot in a direction against the Earth's orbit in order to reach the sun. If the bullet were extra fast, if you could somehow shoot a bullet that's close to the speed of light, for example, you could shoot in pretty much a straight line. It wouldn't be much deflection at such a high speed. But since we can't fire things at such a high speed, we need to use the Newtonian mechanics to figure out how to, we have to pick particular trajectories, parabolic, use something called the Hohmann transfer orbit. Yeah, I mean, what you're suggesting, tagline, is to use celestial body as some kind of gravitational assist. And you can do that. You can make the object, for example, Jupiter. It can help you fire something out of the solar system or fire something into the solar system. It depends on where your spacecraft goes behind Jupiter as it goes around the sun or goes in front of Jupiter as it goes around the sun. So you can slow down your spacecraft or speed it up using Jupiter, for example. Yes, that's exactly right, Dick. Avoid your gravitational assist from Jupiter. Very often they use Jupiter to help. And if you go behind Jupiter and have it swing around about 90 degrees or so, then it will pick up Jupiter's angular momentum. So Jupiter actually slows down a bit as the spacecraft speeds up or as it slows down a negligible amount. Yes, that's right, tagline. If you slow an object down in orbit, it'll eventually, it'll start moving inwards and it'll speed up again. But it'll have less angular momentum because it's closer to the sun and angular momentum goes like the distance that you are between the distance on the radius of the orbit. So it'll move inwards slightly. Yeah, I explained that, Fred, in my talk on the moon. Maybe you weren't there, so I can explain it again. It's not really that difficult to understand if I had the diagrams with me. Basically what happens is the moon has, tidal forces are basically differential gravitational force. So the moon is pulling on the near side of the earth. Harder than it's pulling on the center of the earth and that's pulling harder on the center of the earth than the far side of the earth. So that stretches the earth into a bulge. And that bulge, because the earth is rotating, moves ahead of the moon. And so the bulge has a gravitational force on the moon and it's flinging the moon outwards. As the moon flings outwards, it slows down the earth's rotation. There's a trade-off in potential and kinetic energies, orbits are not usually circle. Yes, that's why I had that swing analogy where I had my family on the swings. It's sort of like being on a swing where you, when you're at the apacus point where you're farthest from the body you're orbiting, you have maximum potential energy but minimum kinetic energy. And then when you reach the periapsis point then you see how the moon creates lags. What do you mean, Barry? What kind of lags? Okay, well, yes. Oh, I see. Okay, I see what you mean. I wasn't thinking of lagging that in those terms. So I have a question for people here. Were you able to follow the math? Okay, I mean, we're sure many of you are familiar with the math and it wasn't a problem. Anyone have any troubles following the math? Okay, sorry about that. I mean, if you did have trouble with the math, as I said before, I got the PDF from Chantel and study it and you can ask me questions or do your own research. One of the things that I find sometimes when I'm looking at slides like this is I need a moment to look them over before I'm actually listening to the lectures. So I kind of found out about this maybe an hour ago. Math has never been my strong suit but if I spend a bit of time with it, I can usually get it. Like I said before, most of this was quite familiar. I've taken, you know, physics. I've taken some calculus and so forth. But just to kind of wrap my head around it, I need to spend a few moments with it beforehand. Yeah, that makes sense. I would say that the concepts I've presented here are basically quite simple but the problem isn't that a given concept is difficult to understand. It's that if you could introduce a number of concepts in a talk, it starts to become a bit confusing because you can separate or you have the next one and it is helpful to go over the talk. Thank you, Todd, Todd. Thanks again, Mike. Okay, Shaus, take care. Thank you, Tiglan. Thank you, most of the source. You're seeing me, most of the source. I should have put that up on the internet. Most of the source. Ah, I'm interested in what most of the source looks like. Yes, it did work reasonably well. Thanks, Tiglan. I had a lot of trouble working with it on Thursday to get it set up, but this time I didn't need your help. I appreciate your help the first time around. But I did eventually figure it out and work it out myself. I wonder if there's another way of getting it to Shiloh. Yeah, that's a good idea, Tiglan, because I was thinking I could maybe include the videos as well, but the videos on some kind of Google Drive. Yeah, that's right. I mean, I'm thinking that I could create my own video and share it that way. That might be a good idea to try. Thank you, Shiloh. Bye, Mike. Thanks, Shiloh. Thank you, Braga. Yeah, thanks, Vic. Your comments and answers were useful, as usual. Thanks, Cass. Take care. Bye, Delia. Okay, it's almost lunchtime for me. Thanks, Tiglan.