Why Are Orbits Elliptical? | Intuitive Proof
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- Опубликовано: 11 авг 2018
- Elliptical orbits can seem counterintuitive to many, but this video pulls back the curtain on Kepler's First Law.
Music: Composer - Jake LaVallee, Owner of uncopyrightedmusic.net 
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A nice, clear explanation - thank you.
Actually another way of seeing it ! Loved it
Brilliantly simple and powerful!
For anyone interested in the details about how to get to the equations in the video, here is the derivation. You start by acknowkedging that the central force F only depends on the displacement vector between the orbiting mass m and the center of mass of the orbit, not on the position of the center of mass or of the orbiting mass with respect to the origin. With this, you can claim that radial component of the central force F is equal to mk/r^2, where m is the orbiting mass, k is a constant of proportionality, and r is the modulus of the displacement vector, also equal to the radial coordinate of the orbiting mass. The angular component of the force is 0.
According to the second Newtonian law of motion, the sum of the forces is divided by the mass of the forces act on is equal to the net acceleration of the mass. In this case, the sum of the forces is simply F itself, because F is the only force taken into consideration. The radial component of the net acceleration is given by r'' - r·φ'^2, where each prime symbol represents a derivative with respect to time, and φ is the angular coordinate of the orbiting mass. Therefore, k/r^2 = r'' - r·φ'^2. The angular component of the net acceleration is equal to r·φ'' + 2r'·φ'. Therefore, r·φ'' + 2r'·φ' = 0. The acceleration components are derived by differentiation the vector in polar coordinates. In summary, the system of equations to be solved is k/r^2 = r'' - r·φ'^2 and r·φ'' + 2r'·φ' = 0.
If you multiply the second equation by r, then the equation is r^2·φ'' + 2r·r'·φ' = r^2·(φ')' + (r^2)'·φ' = (r^2·φ')' = 0. This implies r^2·φ' is a constant with respect to time. In fact, this quantity is equal to modulus of the angular momentum divided by the mass of the orbiting mass, which you can denote as L/m. You can exploit this to express r·φ'^2 with respect to L/m. r·φ'^2 = r^4·φ'^2/r^3 = (r^2·φ')^2/r^3 = (L/m)^2/r^3. This is precisely what the video did to eliminate the angular coordinate as a variable of the equations, noting that the angular momentum L is conserved, and thus, a constant. Substituting r·φ'^2 = (L/m)^2/r^3 in k/r^2 = r'' - r·φ'^2 and rearranging simplifies the problem to merely solving the one equation r'' = k/r^2 + (L/m)^2/r^3 with respect to r, which is precisely what was presented in the video if you let k = -GM. For the sake of generality, you may as well call q = (L/m)^2, since the constants are arbitrary, so r'' = k/r^2 + q/r^3.
In the special case that q = 0 and k < 0, this results in an equation of free fall directly toward the center of mass, while q = 0 and k > 0 results in direct repulsion. If q > 0 and k < 0, then the equation describes an elliptic orbit. Solving for r as a function of t cannot be done using a closed form expression or an elementary function, but by performing a change in variables from t to φ, you can easily prove that r = A/[1 + e·cos(φ - θ)], where A and θ are constants, and e is the eccentricity of the ellipse. Rather than solving the equation, the video simply presents a graph of the potential energy with respect to r to demonstrate the oscillatory motion.
Props for writing it all out! Personally I would use the Lagrangian, but either way it can be helpful to see the mathematical reason it's an ellipse, so thanks!
Adiabadic True, the Lagrangian is "superior," but I appealed to the more familiar Newton's laws for people who aren't as deeply invested in physics could understand
Nicely explained, thanks
Well done. Same explanation for inductor / capacitor tank oscillating. The resistance in LC circuit makes it die off over time, but same for planets over much longer time.
Interesting, good stuff
Amazing
Why u stopped bro go on
i could not follow after the part you said.. that the radius oscillates... pls explain
Think about it in the extreme example of a comet. As it goes straight by sun it gets ever more pulled toward sun and gets literally sling shot around then goes away from sun slowing down but this slowing down makes it more and more effected by sun's gravity to nudge it back then it goes back to sun and gains speed and gets slung around again and this keeps repeating / oscillating over and over again... Think of the momentum vs gravity visually playing a game of forces...
can you please explain how did you got those equations
Thank you for your response . Can we please contact through other mode i have some more question over this topic
The equations are derived entirely from the Newtonian laws of motion and from classical mechanics, by exploiting the conservation of linear momentum and angular momentum.
Thank you sir
Noiceeeee
this vid will potentially make a little curious kid not knowing anything about linear algrebra become even more confused. i mean,
narrator: "we can avoid a lot of math"
also narrator: *proceeds to dish out a lot of math anyways lol
curious kid: *will likely tune out and look for other, more intuitive videos lol
3:04 - "when the planet is far away:."
well what causes it to "go far away from the sun" in the first place?? which in turn, is what causes the elliptical pattern to begin with?
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so to you, curious little kid.. here goes:
imagine you're swinging on a swing set! you push off the ground, and zoom forward in a big arc. that arc is kind of like a planet's orbit around the Sun. but why isn't it a perfect circle, like circles we draw on paper?
here's the cool thing: perfect circles are actually pretty rare in space! planets, like our swing, are moving in two ways at once:
- they're falling inwards: the Sun's gravity pulls on them, just like gravity pulls you down when you're on the swing.
- they're also moving sideways: This keeps them from crashing into the Sun, similar to how you move forward on the swing.
now, if that sideways movement is just the right amount, the planet ends up going in a perfect circle. BUT that's like catching the swing perfectly at the top - not very likely! usually, the sideways movement is a little more or less than needed for a circle.
if the sideways movement is too weak, the planet falls in closer to the Sun before swinging back out, making an egg-shaped orbit (ellipse). this is what happens to Mercury, the planet closest to the Sun.
if the sideways movement is too strong, the planet swings farther away before looping back, making a flatter ellipse. this is how Pluto orbits, way out beyond the other planets.
so, the shape of a planet's orbit depends on how fast it's moving sideways and how strong the Sun's gravity is pulling it in. It's like a cosmic balancing act, and most planets end up doing an elliptical dance around the Sun!
now you ask, "well what causes it to ' move further away from the sun' in the first place which in turn causes the elliptical path?" several factors: gravity of other planets, formation process: during the formation of a solar system, collisions and interactions between celestial bodies can influence the final orbits of planets, non-gravitational forces - in rare cases, things like solar radiation pressure or the drag from interstellar dust can have a slight impact on a planet's orbit.
remember, even though they're not perfect circles, these ellipses are pretty close. that's why we often draw them as circles in pictures - it's easier to understand! and who knows, maybe someday you'll learn more about physics and math to explore these orbits even deeper!
This is a nice video, but I feel like the title is rather clickbaity. You're promising a *proof* of why orbits would specifically be an ellipse, as opposed to any other non-circular shape. But all you're proving is that the radius oscillates. Not that it oscillates in exactly the right way to trace out an ellipse -- you didn't even show that one oscillation of radius takes the same amount of time as one full revolution of theta. This 'proof' is equally supportive of the idea that the radius wiggles back and forth 10 times per revolution, making a kinda wavy circle shape. Or a superellipse, Cartesian oval, or Cassini oval. So like I said, nice video, I appreciate the new point of view, but at the same time I'm disappointed based purely on the title/intro promise of the video.
He has taken a more intuitive way of explaining this. If a perfect circle, the radius would not change - would not oscillate. But it is more likely that an oscillation would be setup - same in electronics where oscillations can occur even not designed to do so. He is expanding the feedback loop of two opposing forces - the momentum (want to go straight with same speed) and the gravity (want to pull objects together) and the amount of overplay like over correction that keeps the eliptical orbit happening. Really good intuitive understanding.