You. May be aware that I'm a huge fan of the YouTube channel three blue one brown run by Grant Sanderson, grant makes excellent videos about math and mathy aspects, of other topics so I'm letting him take over my channel for today grant, take, it away a week. Ago I put out a tweet showing a peculiar, place where an ellipse arises, but, what I didn't mention is that this arbitrary, seeming construction, is actually, highly relevant to a once lost lecture by Richard Feynman on why, planets orbit, in ellipses, the. Construction, itself starts by drawing a circle and then choosing some point within that circle that's not at its center what, we'll call an eccentric, point and then, draw a whole bunch of lines from this eccentric point upto the circumference somewhere, and then, for each one of those lines rotate. It 90 degrees about, its midpoint and when, you do that for all of the lines an ellipse, emerges, in the middle out. Of context, this is a mildly, pleasing, curiosity, but, there's a much deeper form of satisfaction, on its way once you understand, the full story surrounding this front. And center of that story is Richard, Fineman who's, famous in a number of dimensions to, scientists, he's a giant of 20th century physics winner. Of the Nobel Prize for his foundational, insights in quantum, electrodynamics among. Many other things to, the public he's a refreshing, contradiction, to the stereotypes, about physicists, a safecracking. Bongo, playing mildly, philanderers, nonconformist. Who's heavily Brooklyn accented, voice you've probably heard either relaying, some bit of no-nonsense pragmatic. Wisdom about the only sensible way to view the world or, else some wry joke told through a crooked smile but. Two physics students he was an exceptionally, skillful teacher both, for his charisma and his uncanny ability to make complicated, topics feel natural, and approachable, many. Of the lectures he gave to a Caltech, freshman course are immortalized, in the now famous Fineman, lectures whose three volumes you can find for free online, but. Not all of the lectures he gave made, it into this collection one. In particular, a guest lecture given on march 13th 1964. Entitled. The motion of planets around the Sun survived. Only as an unpublished partial, transcript, with a smattering of notes buried in the office of one of fineman's colleagues until it was eventually dug up by Caltech archivist, Judith Goodstein despite. The absence of some crucial blackboard drawings to follow what Fineman was actually saying her, husband David eventually reconstructed. The argument of the lecture which, the two of them published in a book titled fineman's, lost lecture conveying, both the lecture itself and the surrounding story, in a really beautiful way. Here. What I'd like to do is give a more animated and more simplified, retelling, of the argument, that Fineman was presenting, the, lecture. Itself is about why planets and other astronomical, objects, orbit, the Sun in ellipses. It, altom utley has to do with the inverse square law the fact that the gravitational, force pulling an object towards, the Sun is inversely. Proportional to the square of the distance between that object, and the Sun but. Why, how. Exactly does that law give rise to an, ellipse of all shapes of. Course the gravitational, attractions, between different, planets and moons and comments and all of that means that no orbit is a perfect, ellipse but come on to a very good approximation this. Is the shape of an orbit, you. Could solve, this analytically setting. Up the appropriate differential, equation, and seeing the formula, for an ellipse pop out but. Fineman's, goal with this lecture was to do something special and to not rely on any heavy mathematical, machinery in fact, let's listen to him articulate, his own goal I am. Going to give what I will call an elementary, demonstration. My, elementary, does not mean easy to understand. Elementary. Means. That. Nothing very, little is required, to know ahead of time in order to understand, it except to have an infinite amount of intelligence. There. May. Be a large number of steps that are very hard to follow but. Each step does not require, already knowing calculus, already knowing Fourier, transforms. And so on. Yeah. That's, all just a little infinite intelligence I think you're up to that don't you I've. Done what I can to simplify, things down further from his original lecture, but that's not to say that a good deal of focus won't still be required, first. Things first we, need some definition, of an ellipse otherwise, there's just no hope of proving that that's the shape of an orbit some. Of you might be familiar with the classic way of constructing, an ellipse using two thumbtacks, and a piece of string use.
The Thumbtacks to fix the ends of a small string in place and then, pull that string taut with a pencil and try, to trace out a curve while keeping, that string taut it's. Similar to how you might use a single, thumbtack to construct a circle where, the fixed length of the string guarantees. That every point you trace out is a constant, distance from the thumbtack but. In this case with, two thumbtacks. What. Property, are you guaranteeing about. Each point that you trace out, well. At, every point the, sum of the distances, from that point to each of the two thumbtacks, will, be the full length of the string right, so. The defining, property, of this curve is that when you draw lines from any point on the curve to these two special thumbtack, locations, the, sum of the lengths of those lines is a constant, namely, the length of the string, each. Of these points is called a focus, of your ellipse collectively, called foci and fun. Fact the word focus comes, from the Latin for fireplace, since, one of the first places where ellipses were studied was for orbits around the Sun a sort, of fireplace. Of the solar system, sitting, at one of the foci for a planet's orbit making. Up a little bit of terminology for ourselves let's, call this constant, some of the distances, from any point on the ellipse to the two foci the. Focal, sum of that ellipse will. Get to the orbital mechanics in, just a moment but first I want you to turn back to that construction that I showed at the very beginning which, is going to come up again later in the story, remember. We take all of these lines from an eccentric, point of the circle to its circumference, and rotate, each of them 90 degrees about its center but what. On earth does this have to do with the constant, focus on property, I just described you. Could just take my word for it that this emergent shape is an ellipse but I think you'll be much more satisfied in the end if we take a little time right now for, a brief sidestep, into geometry, proof land, first. Off there are really only two special. Points in this diagram, there's the eccentric, point from which all the lines emerge and the, center of the circle so a, reasonable, guess would be that each of these is a focus. Of the ellipse given. The defining, property of an ellipse you know that you're gonna want to look at the sum of the distances, from these two points to well. Something, also. If you're doing any geometry problem, involving a circle you'll, very, likely want to draw a radius, of that circle at some point and to, use the fact that this radius has a constant, length no matter where you draw it I mean, that's what defines a circle so you're probably going to need to incorporate that fact somewhere, with. Those two thoughts in the back of your mind let's, limit our attention, to just one of these lines touching, some point P on the circle remember. What happens in our construction you, rotate this line from the eccentric point 90. Degrees about its center and the, geometry enthusiasts. In the room might fancifully, call this a perpendicular, bisector of. The, original line now. Take a moment to think about the sum of the distances, from, our proposed focus points to, any point Q, along. This perpendicular bisector. The. Key insight, here is that you can find two congruent, triangles, and use, them to conclude that the distance, from Q to the eccentric, point is the same as the distance from Q to P, so. That means that adding, the distances, to each focus, is the, same as adding, the distance from the center to Q then Q to P and there. Are two key things that I want you to notice here first. At the, point where this perpendicular, bisector. Intersects. The radius that. Sum is clearly. Just the radius of the circle and since. That radius is constant no matter where we draw it. The focal sum at that intersection point stays. Constant, which, by definition means. That it traces out an ellipse specifically. An ellipse whose focal, sum is equal, to the radius of the circle isn't.
That Me and. Second. Because the sum of these two lengths at every, other point, on that perpendicular bisector. Is larger. Than the radius meaning. The sum of the distances, to the foci from those points is bigger, than, the ellipses focal sum all, other. Points of this line have, to lie outside the ellipse, what. That means and this is going to be important is that, this perpendicular, bisector, the line that we got after our special 90-degree rotation, is tangent. To the ellipse, so. The reason that all of the lines that we drew earlier make this ellipse appear from nothingness is because we're basically drawing a whole bunch of the tangent, lines to that ellipse the. Reason, that that's going to be important as you'll see later is that this tangency, direction, is going to correspond, to the velocity, of an orbiting object, okay. Geometry. Poofiness done on to some actual physics and orbital mechanics. The. First fact is to use Kepler's, very beautiful, second, law which. Says that as an object orbits, around the Sun the, area, it sweeps out during, a given amount of time like one day is going, to be constant no matter where you are in the orbit for. Example maybe you think about a comet, whose orbit is really skewed close. To the Sun it's getting whipped around really quickly so it covers a larger arc length during a given time interval but. Farther away it's moving slower so, it covers a shorter, arc length during, that same time and, this. Trade-off between the radius, and the arc length balances. Out in just such a way that the swept out area is the same a quick way to see why this is true is to leverage conservation. Of angular momentum. For. Any tiny little time step delta-t, the. Area, swept out is, basically. A triangle right in. Principle you should think of this as a very small sliver, for a tiny time step but I'm going to draw it nice and thick so that we can better see all of its parts the. Area, of this triangle is one-half, base times height right that, base is the distance to this and what, about the height this little linked here how do you find that well. It's going to be the component of the object's velocity perpendicular. To, the line of the Sun what I'll call V perp multiplied. By the small duration of time, so. The full area is one-half. Times the radius times, V perp times, delta T now. Conservation, of angular momentum with, respect to a given origin, point like this son tells. Us that this radius times, the component of velocity perpendicular, to, it remains, constant so long as all the forces acting on the object are directed, towards that origin, well. Specifically, it says that this quantity times, the mass of the object stays, constant, but I mean the mass of the orbiting object isn't going to be changing, so. Our, expression, for the area swept out depends. Only on, the amount of time that has passed delta. T, historically. By the way this went the other way around, Kepler's, second law is one of those empirical, facts that led us to an understanding, of angular momentum and I, should emphasize this. Law does, not assume that orbits, are ellipses, heck, it doesn't even assume the inverse square law the, only thing needed for this equal area property, to hold is that the only force acting on an object is, directed, straight towards the Sun this.
Is A fact that Fineman spent a lot more time showing recounting, an argument, by Newton from his Principia but. It kind of distracts, from our main target so I figure assuming, conservation, of angular momentum, is good enough for our purposes here, I'll be it at some loss of element arity. At. This point despite my suggestive, drawings we don't know the shape of an orbit for all we know it's some wonky non elliptical, egg shape the. Inverse square law is going to help us pin down that shape precisely, but, the strategy is a little indirect, before. Showing the shape of the path traced out by the orbiting, object what, we're gonna show is the shape traced out by the velocity, vectors, of that, object, here. Let me show you what I mean by that as the. Object orbits it's velocity, will be changing, right it's, rotating, always tangent, to the curve of the orbit and it's, longer at points where the object is moving quickly and shorter. At points, where it's moving more slowly what. Will show is that if you take all these velocity, vectors and collect. Them together so, that their tails sit, at a single point their. Tips actually. Trace out a perfect, circle now. This is an awesome fact if you ask me the, velocity spins, around getting faster and slower and various angles but, evidently the, laws of physics cook things up just right so, that these trace out a perfect, circle and the. Astute among you might have a little internal light bulb starting to turn on at the sight of this circle with an off-center point. But. Again we have to ask why. On earth should this be true, Fineman. Describes being unable to easily, follow Newton at this point so instead, he comes up with his own elegant line of reasoning to explain where this circle comes from he. Starts by looking at the orbit whose shape we don't know and slicing. It into little pieces which. All cover the same angle, with respect to the Sun all. Right now think about the amount of time that it takes for the orbiting object to traverse one of these equal angle slices, and how that time changes, as you go to a bigger slice, well.
By Kepler's second law that, time is proportional to the area of the slice right, and, because these slices have the same angle, as, you get farther away from the Sun not, only does the radius increase but, the component. Of arc length perpendicular. To that radio line goes, up in proportion. To the radius so. The, area, of one of these slices and hence the time that it takes the object to traverse it is proportional. To the distance away from the Sun squared. In. Principle, by the way we're only going to considering, very small slices so there won't be any ambiguity, in what I mean by the, radius, and the little. Arc length will basically be a little straight line all. Right now think, about how the inverse, square law comes into play at. Any given point the force that the Sun imparts, on the object, is, proportional to one divided by the radius squared but. What does that really mean what. Force is is, the, mass of an object times, its acceleration the. Amount, that its velocity changes, per unit, time this. Is enough to give us a super useful bit of information about, how, the velocity of the orbiting object changes, as it goes from the start of one slice to the start of the next that. Change in velocity is, the acceleration, times, the change in time right, what. That means is that this change to the velocity is proportional, to the change in time divided. By the radius squared, but. Since, the time that it takes to traverse one slice is proportional. To the radius squared these, terms cancel out so. The change in velocity as, it traverses a given slice is actually, some constant, that doesn't, depend on the slice at all here. Unpacking, what I mean by that if you, look at the velocity at the start of a slice and, then you look at the velocity at the end of that slice and directly. Compare, those two vectors by joining their tails and you look, at the difference, between them the little vector joining, their tips this. Difference has the same length no matter which slice of the orbit you were looking at, so. As you, compare these velocity, vectors at the start of each slice they'll, be forming some kind of polygon, whose, side lengths, are all the same, also. Since, the force vector is always, pointing, towards the Sun as, you go from the start of one slice to the next that, force vector and hence the acceleration, vector is turning. By a constant, angle in geometry. Lingo, what, this implies is, that all the external, angles, of our polygon, are, going to be equal to each other. I know. That, this is a little tricky but hang in there remember, all you need to follow along is infinite.
Intelligence, It's. Worth reiterating just, to make sure it's clear what's happening, with this velocity diagram, the. Change from one vector to the next this little difference vector joining. The tip of one to the tip of the next always. Has the same length that was the consequence, of the perfect cancellation, between mixing Kepler's, second law with the inverse square law and because. Those constant, length change vectors rotate. By a constant, angle each time it. Means that they form a regular polygon, and as. We consider finer and finer slices, of the original orbit based, on smaller and smaller angles, for those slices the. Relevant regular polygon defining. The tips of these vectors in the velocity diagram approaches. A perfect, circle isn't, that neat. Hopefully. At this point you're, looking at the circle you're looking at the eccentric point and you're just itchin to see how this gives rise to an ellipse the way that we saw earlier but it's. A little weird right I mean, we're looking at this diagram in velocity. Space so. How do we use that to make conclusions about the actual orbit, what. Follows is tricky. But clever, step. Back and consider what we know we. Don't know the specific shape of the orbit only the shape that the velocity vectors, trace but. More, specifically, than that we, know that once the planet has turned an angle theta degrees, off the horizontal, with respect to the Sun that. Corresponds. To walking theta degrees, around our circle in the velocity diagram since. The acceleration, vectors rotate, just as much as the radius vectors, this. Tells us the tangency, direction, for each point of the orbit, whichever. Vector, from that velocity, diagram touches. The point theta degrees around the circle that's, the velocity vector, of our orbiting, object, and hence, the tangency direction, of the curve in fact. Let me just start drawing all these vectors as lines, since. All we're going to need to use is the information, they carry about the slope of the orbit curve the, specific, magnitude, of each velocity, will not be as important. Notice. What, I'm not saying is that the angle, of the velocity, vector at this point is Theta degrees, off, vertical, no no no the angle I'm referencing in the velocity diagram is, with, respect to the circles, Center, which, is almost certainly a little different from where the velocity, vectors are all rooted so. The question is what. Special, curve satisfies. The property that, the tangency, Direction the slope for, a point theta, radians, off the horizontal, is given.
By This vector, from, a special eccentric, point of the circle to, a point theta degrees, around. That circle from the vertical, okay. Is the question clear well. Here's, the trick first. Rotate. The whole circle set up 90, degrees and then, take, each of those individual. Velocity, directions, and rotate, them 90 degrees, back the other way that way they're oriented just, like they were before it's. Just that they're rooted in a different spot ah-ha, we've, spotted our ellipse but. We still have a little bit of thinking ahead of us to really understand, how this little, emergent, ellipse is related. To the astronomical, orbit. Importantly. I didn't just rotate these lines about any point I rotated, each one of them about its center, which, means we can leverage the geometric, proof we saw a few minutes ago and, this. Is probably the moment where you kind of have to furrow your brow and think back okay wait a minute what was going on in that proof again well. One, of the key points was that when you have two lines one, from the center of the circle and one from the eccentric, point both, to a common, point on the circles circumference, the. Perpendicular. Bisector, to, the eccentric. Line is tangent. To the ellipse and, what's. More the, point of tangency is where, that perpendicular bisector. Intersects. The, radial, line from the center what. That means is, that the point of our little, ellipse which is theta degrees off the horizontal with, respect to the circle center has, a tangent, slope perpendicular. To the eccentric, line and because. Of the whole 90 degree rotation, thing this, means that it's parallel to the velocity vector, that we need it to be so. This. Little emergent curve inside, the velocity, diagram has, exactly, the tangency property, that we need the orbit to have, and hence, the shape of the orbit must be an ellipse QED. Okay. Pat yourself on the back because, there is no small amount of cleverness required, to follow this first. There was this peculiar, way of constructing, an ellipse which requires some geometry savviness, to properly prove and then, there's the pretty clever step of even, thinking to ask the question, about what, shape the velocity, vectors trace out when you move their tails to the same spot and showing. That this as a circle requires, mixing, together the inverse square law with, Kepler's second law in another sly move but. The cleverness doesn't end there showing, how this velocity diagram with, vectors rooted at an off-center point, implies. An elliptical, orbit brings. In this very neat 90 degree rotation, trick I just. Love this watching. Fineman do physics even elementary physics is like watching Bobby Fischer play chess. Thanks. Again to grant and you should definitely go check out his videos on three blue one Brown.
2018-07-25