Kepler's second law.

[ollypenrice]

I like Kepler's second law, the one which states that equal areas are swept out of the plane a planet's elliptical orbit in a given time. I've come across it in many books, as you do, and currently in Allan Chapman's look at enlightenment astronomy. However, I've never come across an explanation, oddly enough, and would like a competent person to confirm or refute my interpretation, which is that the areas remain the same because there is a proportional exchange of kinetic and potential energy throughout the orbit. When the planet is at aphelion the component of potential energy is at its highest and that of kinetic is at its lowest, then vice versa at perihelion.

And then a second question. How, if at all, does this relate to the motion of a pendulum? Here the period remains the same though the swept area dimishes over time as the system loses energy but, again, we have a proportional exchange of kinetic and potential energy. Does the proportional exchange of these two forms of energy generate the consistency of the period?

Intuitively I sense a connection between orbital behaviour and pendulae and a passage I read in Sagan, but no longer have available, seemed to suggest that there were pendulum-second law equivalences.

Thanks,

Olly

[michael.h.f.wilkinson]

Curiously, Kepler derived his second law from flawed physics (as it turned out later). He assumed from Aristotelian mechanics, that there had to be a force driving the planets in the direction of their orbit, rather than a centripetal force. By assuming the force worked within the orbital plane of the planets, he assumed it diminished as 1/R, rather than 1/R² as in the case of gravity. This combination results in the same correct law of equal areas which, as we understand it now, is due to conservation of angular momentum: a shorter radius means a higher angular velocity is needed. The product stays the same. There is also an exchange between kinetic and potential energy, but conservation of angular momentum is what explains Kepler's second law most easily

 

Regarding the pendulum: note that the decrease in amplitude is due to dissipation of energy. A frictionless pendulum keeps on going

[vlaiv]

This got me thinking and the following is what I came up   (guess it does not explain the law but gives interesting mathematical insight).

Sine and Cosine are related by sin² + cos² = 1 (a constant thing) - this defines circle/"circular" motion.

Kinetic energy is form of v², and potential is in form of 1/r²

Also first derivative of second order polynomial is first order polynomial, and integral of first order polynomial is second order polynomial. Also differential and integral of sin and cos interchange.

As surface is integral, and speed is differential - all of these combine in such way that produces above law. Simply things cancel out because of all of these relations.

So we might think of energy as being a sort of circle or conics where one coordinate represents kinetic component and other coordinate potential energy, while radius being total energy. So period for pendulum will be defined as speed along the "energy circle".

In case of pendulum it is set by gravity constant and length of pendulum arm - or rather total amount of energy on energy circle. This gives rather interesting idea - is there a constant of energy transfer so to speak? If we have short pendulum arm - it will swing faster, because total energy that changes from kinetic to potential is less than in case of longer pendulum arm? Is there such thing as energy transfer constant - like amount in J per second? Or if it also depends on gravity (stronger gravity, faster oscillations) can we express gravity thru energy transfer speed, and if so what would it mean for gravity constant to observe it in such a way?

Sorry if I just added confusion instead of giving an answer

[cloudsweeper]

Pendulum:

The period depends only on length (proportional to root of length) for a given value of gravitational field strength.  This comes from maths and the assumption that the angular amplitude is small, so sine theta equals theta (radians).  So as this amplitude diminishes in time, the period is still the same.

But the reduction in amplitude occurs because there is not  a direct exchange between kinetic and potential energies, since energy is being lost against air resistance and other friction.

So, No - I wouldn't think that energy exchange generates constant period T.  As is known, T is shown by principles of SHM to depend only only on length (for a given g and small theta).

And you still have energy exchange at high amplitude oscillations under which T is not constant.

Doug.

[ollypenrice]

1 hour ago, michael.h.f.wilkinson said:

Curiously, Kepler derived his second law from flawed physics (as it turned out later). He assumed from Aristotelian mechanics, that there had to be a force driving the planets in the direction of their orbit, rather than a centripetal force. By assuming the force worked within the orbital plane of the planets, he assumed it diminished as 1/R, rather than 1/R² as in the case of gravity. This combination results in the same correct law of equal areas which, as we understand it now, is due to conservation of angular momentum: a shorter radius means a higher angular velocity is needed. The product stays the same. There is also an exchange between kinetic and potential energy, but conservation of angular momentum is what explains Kepler's second law most easily

 

Regarding the pendulum: note that the decrease in amplitude is due to dissipation of energy. A frictionless pendulum keeps on going

Thanks Michael but I'm still not all the way there! I understand the description provided by the conservation of angular momentum but am thinking about it in a different way. If I throw a ball up into the air its energy is all kinetic as it leaves my hand and all potential is it momentarily hovers before returning to my hand. At half distance it's presumably about 50/50. Now if I visualize this it seems closely to resemble the behaviour of a planet in orbit, especially if I observe with my eye in the plane of the orbit. It goes from the perihelion end very quickly and slows to an apparent standstill at the aphelion end before accelerating again under gravity to the aphelion end. And, again, when I look at a pendulum I feel there has to be a similarity: the pendulum goes as far as it can against gravity, stops, reverses, accelerates...etc.

So my intuition is screaming at me that the vertically thrown ball, Kepler's planetary orbits and the pendulum all have something in common. If they don't I'm really going to struggle to believe it!!!

At the back of my mind is the thoroughly 'unhistorical' question, why did Gallileo not see this?  (The answer may well be 'because there is nothing to see and Olly has concocted the entire plot himself...')

Olly

[George Jones]

I don't know anything about analysing Kepler's second law in terms of conservation of energy (I might look into this later).

For central forces (not just inverse-square gravitation forces), Kepler's second law is equivalent to conservation of angular momentum. In order to keep angular momentum the same: when a planet is closer, it must move faster (to compensate for the smaller "lever arm"); when a planet is farther, it must mover slower (to compensate for the larger "lever arm" ). A mathematical treatment of this faster/closer and slower/farther produces Kepler's "equal areas" second law.

[vlaiv]

You are right, those three things are connected as you said (pendulum, thrown rock and planet in orbit), and connecting point is kinetic / potential energy exchange, and in this particular case, it is caused by gravity in all of those. Similar thing happens for example with spring - oscillatory motion, but it is caused by spring constant stuff. Here it is gravity at play.

Only difference between rock, pendulum and planet is constraints - planet having none, rock being constrained by earth surface (otherwise it would fall all the way to the other side of earth and back again and again and again :D) , and pendulum being constraint at arm pivot point and by arm itself.

As to why Galileo did not see this, hm, we can only speculate

[cloudsweeper]

@ollypenrice - wrote:   It goes from the perihelion end very quickly and slows to an apparent standstill at the aphelion.

That's not so, Olly.  The planet is fastest at the perihelion, and slowest at the aphelion, but it's still hurtling through space and doesn't approach anything that could be described as a standstill in the way a pendulum does.

Tricky stuff, this!

Doug.

 

[vlaiv]

Yes, standstill would be the edge case where ellipsis minor axis tends to 0 (case of rock thrown straight up)

[George Jones]

Yes velocity is never zero, but the rate of change of distance is zero at aphelion and perihelion, i.e., distance (but not position) is "at a standstill" at aphelion and perihelion.

Very roughly, at aphelion and perihelion, the orbit is approximated by a circle, so, at these positions, distance has zero instantaneous change, but position has non-zero instantaneous change.

[ronin]

Wouldn't Keplers second law just be a consequence of general relativity?

The planets are orbiting the sun but not "parallel" to the suns bending of space-time if we use a 2D sheet analogy. Therefore they would follow the conic section of an ellipse. Would have expected that the equal area bit is a conseqeunce of that. I also suspect that Kepler could "see" that the area was equal but unable to prove why and so it became a law where it is stated what happens without at the time the absolute proof of why.

Afraid that for planets orbiting things like stars it is GR that has to be used not Newtonian.

[Tiki]

3 hours ago, ollypenrice said:

If I throw a ball up into the air its energy is all kinetic as it leaves my hand and all potential is it momentarily hovers before returning to my hand. At half distance it's presumably about 50/50. Now if I visualize this it seems closely to resemble the behaviour of a planet in orbit, especially if I observe with my eye in the plane of the orbit. It goes from the perihelion end very quickly and slows to an apparent standstill at the aphelion end before accelerating again under gravity to the aphelion end. And, again, when I look at a pendulum I feel there has to be a similarity: the pendulum goes as far as it can against gravity, stops, reverses, accelerates...etc.

 

Olly

 

Perhaps what you are looking for is the notion of 'effective potential'. 'Effective potentials' are quite intuitive. Try a google search.

[ajk]

Instead of a pendulum fixed in one axis going back and fourth, allow it to move in two dimensions and it will trace an ellipse like an orbit. The ball thrown into the air and caught is just a portion of this motion, your hand (and the Earth) preventing the rest.