Kepler's First Law: Planets Revolve around the Sun in an Ellipse

Transcript

Kepler's Laws of Planetary Motion. So we now describe one of the great accomplishments of calculus by showing how the material of this chapter can be used to prove Kepler's laws of planetary motion. So after 20 years of studying the astronomical observations of the Danish astronomer, I think it's pronounced Tycho Bray, the German mathematician and astronomer, Johannes Kepler, born 1571, died 1630. formulated the following three laws. So again, so he formulated these just from the data from this guy here, this astronomer. So he basically summed up his findings as these three laws, so Kepler's laws. So law one, a planet revolves around the sun in an elliptical orbit with the sun at one focus. And again, just quickly recap on the focus. An ellipse is just basically defined. You have two points like this. The distance from any, yeah, basically the distance from this focus one. It's focus too, like basically have a point here. The sum of these distances is going to be constant, and I'll recap on that soon. So, and then the sun's going to be at one of those focus. So, law, too, the line joining the sun to a planet sweeps out equal areas in equal times. All right, and yeah, that one is basically, if you have a, again, if you have the sun here, say the planet's orbiting like here, so, and it's moving this way. So after this amount of time, this area, is going to be well the same as if it was moving across in the same time. So these areas are equal. So this is equal to the line. All right, that's pretty much how it is. So basically, the line joining the Suntoe a planet sweeps out equal areas and equal times. So it's, yeah, the areas are just equalling the same amount of areas being swept through time. So the square of the period of revolution of a planet is proportional to the cube of the length of the major axis. of its orbit all right that's quite interesting we'll get to that in the applied project yeah so basically the period of the revolution so again how long it takes a planet go around the sun and then it's proportional to the cube of the length of the major axis so the cube the major axis is going to be this length like that and then the cube of it so basically the square of the period And then you, it's proportional to the cube of that major axis, very interesting. So, anyways, going further. So in his book, Principia Mathematica of 1687, again, Saraz Ignatins everywhere, so he was able to show that these three laws... They're just consequences of two of his own laws, basically the second law of motion, that F equals MA, law, and the law of universal gravitation. So basically, he can actually derive these, which he wants Kepler derived just from observations, astronomical, yeah, basically astronomical data from this Tycho Brigh guy, and then Newton says, hey, I can actually just prove that just by mathematics. Alright, so that's quite fascinating stuff. Alright, so let's go further. So proof of Kepler's first law. planet revolves around the sun in an ellipse. So in what follows, we prove Kepler's first law. So law 1, a planet revolves around the sun in an elliptical orbit with the sun at one focus. And now, recall from my early video where I showed an ellipse is defined as the curve, where the sum of distances from any point on the curve to two fixed point called FOCi, plural, and then focus for a singular, yeah, is constant. So you have here, so all these distances right here. uh this pf one so this p right there uh pf1 pf2 uh these are just constant yeah so those are always constant yeah so basically any any point you go you go here to here is it's going to be the same uh lengths when you sum them up doesn't matter where you go even if you go here from here and all the way here so you're going to have one long one short but it always be constant, you get this curve like that, and there's the sun at one of the focus, and the question is, what's this other focus? Right, and then there's the equation, x squared, just when it's at the origin, x squared, x squared divided by a squared, plus y squared divided by b squared, and then the major axis is over here, and that length is 2a there, so that's a, that's a, that'd be 2a, like that, and then the minor axis is that thinner one, that's going to be 2b. Like that, so that's the 2A and 2B, all right, that's an ellipse for you there, and that's at the origins, yeah, that's at the A, 0, negative, A, 0, and then 0, 0, 0,000. All right, let's continue further. All right, so, since the gravitational force of the sun on a planet is so much larger than the forces exerted by other celestial bodies, we can safely ignore all bodies in the universe, except the sun and one planet revolving about it. All right, yeah, so basically, this is much stronger than anything else, so we can ignore, yeah, pretty much most things. But in this case, we're going to ignore everything. So, we use a coordinate system with the sun at the origin, and we let R equal, yeah, this vector R equals R of T be the position vector of the planet. Yeah, so basically, yeah, that's going to be like, if you have the, you make it at the origin, so it's going to be like this would be the setup there. But anyways, that would be the origin. Anyways, going further. So equally well, R could be the position vector of the moon or a satellite moving around the Earth or a comet moving around a star. Yeah, so again, you could use the same idea there. Yeah, so for the moon around the Earth, and then the Earth around the Sun, you could just use the, for those cases, you could use, just change the position vector, or just change the origin so that the Earth is the one that's, fix the origin and so on. Alright, yeah, but for this case, we have the sun at the origin, and then we let R equals RFT be the position vector of the planet. And here, just for completeness, we made some space here, let's just draw that out, just for completeness again, so we have the origin like this. Let's see, there's the origin, this the X, Y, and then this is the sun right here, this is the sun, shiny, Sun like that, and that's at the origin, and then the position vector, let's say we have a planet. It's going to be somewhere around there. Yeah, it's going to be across there. And this is going to be R vector of T. Yeah, so R of T. And there's a planet. So there's Earth right there. Let's draw the Earth like that, all right. And Antarctica, and... Okay, there it is. So that's Earth, and it's moving around. Alrighty, and then, yes, that's pretty interesting that. So anyway, so the velocity vector is V equals the derivative of the position vector. So remember, again, where the curve is, this is just going to be, let's just say it's like this, it's moving like this. Yeah, so then that's the curve. This would be the position vector, I mean, the velocity, V is equal to the derivative of that, and the acceleration vector is going to be A equals to well, R cubed, so it's going to be, let's say that we have A like that. And it's curving inwards there, or better yet, I think it should be pointing back to the sun, it's accelerating back in there. At least that's where the forces, yes, it's going to be like that. double prime, and then I'll double check by graphing it maybe later. But anyways, I just want to illustrate some of the terms we're using. So, because the Kagos book didn't have that. So anyways, so going further, so yeah, the velocity vector is V equals R prime, and the acceleration vector is A equals to R double prime. So the second derivative of that, or the first derivative of the velocity. All right, so, yeah, we use the following laws of Newton. So his second law of motion basically states, well, if you have a force vector, put it like this, this is F like that, the vector equals to, we'll just have a constant mass mass, mass, times acceleration, and acceleration is a vector, basically if you have an object like here, and then if it's, if you have an object of mass M, is a mass M, and if it accelerates by A, in order for it accelerate, you need to have an applied force in that direction. So in order for it accelerate, there needs to be a force applied to this mass in the direction, and then where it's accelerating. So that is the second law of motion. All right, and now the law of gravitation basically states that if you have two bodies, you're going to have, again, force between two masses, so f equals 2, it's going to be negative of the gravitational constant times by capital M is basically the bigger mass, call that the sun for, in our case, and then lower mass, planet, m, and then divided by r cubed. So this is a distance between them, cubed, and that's R, and then basically times it by, it's in direction of the position vector, R, like that, and where basically you have, if you have the sun, for example, I'll make a big, big one right here, so this is M, that's a sun, let's say you have a lower one, M, that's a planet, this force, and it's going to be negative, so let's say position vectors like this, between, from the sun to the planet, like that, So then, and then the absolute value is going to be, well, this is, yeah, this is R. And also note that R, right here, lowercase R is just going to be, I mean, just R without the vector is the absolute value of that. And then the force is going to be negative. So the force is going to be negative of the direction there. So that's the force of gravity, it's going to be negative. So in other words, it's going to be attracting to it. Yeah, so it depends where you're drawing your position vector from, but they both experience a force inwards. So the sun, the planet, this experience a force inwards to the sun, and the planet's experiencing a force of basically inwards to it, but it's also, yeah, it's also most smaller, so it's actually being thrown over there, and so on. Yeah, so basically, yeah, this was accurate. This is going inwards like that, that's acceleration like that, towards inside there. All right, so yeah, basically like that. And now what I'm going to do is I'm going to go let U be a position vector in the direction there. So this equals to R, yeah, the R vector divided by the absolute value of R, the length of R, and that just equals to R over R, lowercase R. Like that, and this is just basically equals to unit vector. in, yeah, unit vector, like, fix that up, in direction of R. In other words, if you have R like this, let's say that's R, the unit vectors is going to be, well, that's going to be, like that, you, because you're dividing by its magnitude, so you always get 1, and then where this length right here is equal to 1. So basically just, yeah, we're just making it a unit 1, so then we're going to have basically a non-1 value multiplied by 1. So we're going to look at this right here, and then basically rearrange it, we're going to get R is equal to move this R to the other side, and yeah, we're going to just move that over to the other side, and we're going to get a R times it by unit vector, like that. So that's pretty much what R is, and we're going to replace that here. So we're going to get this right here, F is equal to negative. This is the gravitational constant times by mass of the sun, or just the bigger planet, and then lower one, M. So M is the planet, this is the sun, and it's going to be divided by, well, this is R times U, so then now that cancels out, and we're just going to left with this R squared. And this is U vector, just make it like that. All right. Alright, now we have this equation, and yeah, basically here is the terms again. So we're, F right here is the gravitational force on the planet. M and capital M are the masses of the planet in the sun, and G is the gravitational constant. So we first show that the planet moves in one plane. So basically, it moves in one plane, and then later we'll show it's an ellipse. Like that, basically, it doesn't go in 3D, like, all over the place. It just goes into a flat plane. So by equating the expressions for F and Newton's two laws, we find that, and basically combine these two. So just F right here, so if we have the force on this planet, F equals M.A equals to this setup here, right here. So whenever we have a force, you're going to get the acceleration there. So let's just equate these. So we're going to have the force is equal to the first law, the second law, F equals MA, and then gravitational one is going to be equals to negative capital, or first to do the G. G, M, lower M, divided by r squared and then this is going to be the u vector u vector you vector like that well actually before you get to the u vector we'll just use it the normal setup it goes to this thing uh the r cube and then r vector like up anyways we'll just follow along it and then basically what happens all we do is just cancel em so we do so we have acceleration is equal to negative g m over R cubed and then R vector. In other words, the acceleration is parallel to the position vector. as I already showed earlier, has this basic common sense. All right, yeah, going for this. So, yeah, so since A is a scalar multiple of R, again, as I just stated, then they are parallel to each other. So, yeah, it's parallel, but it's also the reverse direction. So A is equal to the secondary of R there, but it's going backwards in there, but it's parallel. So again, this position vector, this is parallel to this one. So I'll just leave it like on. So that's the parallel to it. And then now going further, so thus by my earlier video on the cross product, if you take the cross product of the position vector R and A, the acceleration, because they're parallel, the cross product isn't equal to zero. So the cross part of recall from our video is linked to that, and there is this setup right here, a screenshot, or I mean the thumbnail. So the cross product corollarily, non-zero vectors A and B are parallel, if and only if the cross product is equal to zero. And now, keep this in mind, because if the R and cross-product is 0, well, it's also the zero vector, because this is a vector. So it's going to be bold like that. And also, basically, yeah, they can be parallels, it's going to be on the same plane as well. All right. Well, actually not necessarily, because if A and B are just both changing as well, it could be all over the place. But yeah, so basically this is zero. So now what we'll do now, and we're going to show that it's on the same plane. To do that, now we're going to use Formula 5 from our earlier video on the differentiation rules for vector functions, and this is using the derivative. So recall this one here is the screenshot from them. And if we have this, we're going to use a derivative, basically it's like the, it's like the, it's like the multiplication, yeah, the multiplication, the product rule set up for normal derivatives. Notice how it's like cross product, you take the derivative, then cross the regular V, and then plus U cross derivative of the V. And the important part about this is because we're going to take, so if we have, for example, if we did a cross product of R and V, we're going to eventually get an R cross A, which equals zero. So we're going to take the derivative of V, and then we're going to get something where we can cancel. And we'll show you that as I walk you through this. So I'm going to make this bigger. So before I write this one right here, we're going to look at, well, we know that R cross A, because they're parallel equals to zero vector. So that's this part right here, so it's a zero vector. And we also know, or we also have, well, we know that the acceleration is equal to the derivative of the velocity vector. and this just equals to, well, D over DT of V vector. Just for complete to put it all there, and then we also know that V, yeah, and then we also know that velocity is this equal to the derivative of the position vector, which equals to, again, D over DT of the position vector, just write it out like that format, and that's because we're going to use this setup right here, and we're going to look at, so, D over DT, of, and then instead of you, we're going to do R cross V vector. So now this is going to start making some sense. So now this is going to be, well, using this rule, the derivative of the first one, to derivative of R. R, prime, cross with V prime. Yeah, I mean, just V, and then we add, well, R cross with V prime. Like that. All right. So, no, we know that, well, this one here, R prime, is just this, it's just V. So this is just V equals to V vector. And now we also know that, well, this derivative right here, this just equals to A. Yeah, so then this equals to V cross V plus, then R cross A. All right, and we already know that this is zero. This one is zero because there's a parallel. We also know that V is parallel to itself, so that's also zero. So then this whole thing is zero. So this equals to zero plus zero vector. In other words, we get a zero vector. All right, so that's a zero vector. So thus, what this means is, well, so thus, now, this is a derivative. So if we take the integral, so thus, R cross V. Well, this just equals to, well, we'll take the full integral like this. Yeah, it's going to be just integral of this whole thing there, so it's going to be D over DT. I'm going to put a bracket, because we always need a DT at the end for some reason. Anyways, it's for notation D over DT, and then this is going to be a, I'll put this in this bracket like this, R cross V. and then this is going to be DT, like that. So the integral of a derivative is going to be just that whole setup inside. And where this part here is equal to 0, so basically this equals to the integral of the 0 function, DT, and yeah, the 0 one, we're just going to get a constant. Basically, the derivative, yeah, integral of 0 is going to be a constant because the derivative of a constant goes to 0. So this becomes H, like that. where that's just a constant integration, this equals to constant vector. Yeah, put the constant vector like that. All right, and now the assumption is, well, yeah, just fix that up, constant vector. So basically, we assume that this constant vector is not equal to the zero vector, that is, R and V are not parallel. Yeah, so basically, because if it was equal to zero, then you're going to have this setup where it's parallel. and you basically have a, yeah, we're going to have vectors like this parallel to each other. And so, we're assuming they're not parallel, so, and that makes sense, because if you have orbiting the Earth like that, the velocity vector is going to be tangent to this curve, and yeah, that's not going to be parallel to this, in other words, not going further away vertically, like, just straight up ahead, it's not going to go, just getting bigger and bigger orbit, it's just going tangentially around. All right, so that makes sense, so it's not going to be equal to 0. So this means that the vector R equals R of T is perpendicular to H for all values of T. Because remember, the cross product, is like this. The cross product is always perpendicular to the two vectors. So the parallel vector is always going to be perpendicular to H, so the planet always lies in the plane through the origin perpendicular to H. Yeah, so basically what this means is, well, if you have, this is the, this is going to be constant H, like that, and let's say the velocity vector is going to be like V, like that, and this is going to be position vector R. So it's always going to be parallel. I mean, it's always going to be on the same plane there. So it lies in the plane through the origin perpendicular at there. So position vector is always going to be like that. So in regards to what the orbit is, it's always going to be on a flat plane. like that, because this is always going to be constant. So it's always going to be the same height. Yeah, same height, but also same direction. So it's always going to be pointing…

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