Second kepler law explanation

[kiwiak]

Ok, i get that line joining object and center of object it orbits sweeps equal areas, but actually WHY?

Is there some intuitive explanation?

Edited January 18, 2014 by kiwiak

[KvickFlygarn87]

Because physics, deal with it.

[kiwiak]

Because physics, deal with it.

I cant deal with it.

[FuzzyLlama]

Physics? Sounds more like trig and calc to me.

EDIT: Wikipedia has the equation if you want to see why the areal velocity is always equal as a body orbits. http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Second_law

[Kasuha]

Trivial observation is, when something is moving straight at constant speed through space, the triangle it draws over constant time period relative to certain fixed point is always the same, too. The reason is, the height of the triangle which is the distance of the point from the line of the trajectory remains constant, and the side which is represented by distance traveled over constant time remains constant, too. There's no deep meaning for that, it's just an observation.

So I guess someone noticed this holds for objects moving in gravitational field as well and now it is used as the second law because it is way easier to remember than mathematical expressions which lead to it.

[Blue]

It's just like Newton's simple way of explaining Force was equal to mass times acceleration. F = ma (which, when examined, summarizes both Newton's Second and Third Laws of Motion.) But, in conjunction with the helpful explaination by Kashua, don't reificate Kepler's Laws too much. "The map is not the territory", but the map is a way of explaining the territory.

[Smidge204]

As mentioned, a point moving in a straight line relative to another point will also sweep equal area in equal time.

A point moving in a perfect circle about another point will, intuitively, also sweep equal area in equal time.

A line (e=infinity) and a circle (e=0) are just two extremes, and the "equal area in equal time" rule applies to all curves 0<=e<=infinity.

=Smidge=

[AlexMBrennan]

As mentioned, a point moving in a straight line relative to another point will also sweep equal area in equal time.

Since the area in that case is infinite I don't think that's a good way of going about proving the law.

A line (e=infinity) and a circle (e=0) are just two extremes, and the "equal area in equal time" rule applies to all curves 0<=e<=infinity.

=Smidge=

You have proven two trivial cases (well, maybe), and then simply assert that all other cases are similar. By the same logic, 9 is a prime because 7 and 11 are primes.

As for a derivation, this is the first hit I got on Google.

[Claw]

Ok, i get that line joining object and center of object it orbits sweeps equal areas, but actually WHY?

Is there some intuitive explanation?

I'm not sure exactly what you're looking for, but perhaps another way of looking at it is to start from the other end. The fact that an object (like a ship) sweeps an equal area in equal time around a body (like a planet) is the result of how the object orbits.

During an orbit: When the object is further out from the body, it's speed is slower. When the object is closer to the body, it's speed is faster. (Hopefully you've seen this while playing the game.)

So, when the object is further away AND is traveling slower, it just so happens that the area covered is equal to when the object is closer AND going faster. The "area rule" is a way of using math to match those relationships for a given object.

[Kasuha]

Since the area in that case is infinite I don't think that's a good way of going about proving the law.

It's not way of proving the law, it's way of illustrating the law. The proof is mathematical and is irrelevant to the topic.

And you don't get infinite areas if you select your reference point in finite distance. Because in case of straight line it works with any point in space.

[K^2]

Kepler's Second law is just conservation of angular momentum. Angular momentum of the planet is given by Ér². While the triangle the line r sweeps in some short amount of time t has height t and base vt = Ért. So the area is tÉr²/2. Since Ér² is a constant, the area of the triangle is also a constant.