Kepler's Laws of Orbits Explained

[marlsda]

Simple geometric and logical proofs of Kepler's Laws and other fundamentals of celestial mechanics are presented at Orbits Explained, a new website featuring a new mathematical device called the hododyne at http://www.orbitsexplained.com . Welcome teachers and students and readers and scientists! Come and see the new a priori methods.

 

[yevaud]

A site that explains Vis-Visa? Very good. <div class="Discussion_UserSignature"> <p><em>Differential Diagnosis:  </em>"<strong><em>I am both amused and annoyed that you think I should be less stubborn than you are</em></strong>."<br /> </p> </div>

 

[Saiph]

Interesting, but I still don't think it's an a priori proof.<br /><br /><blockquote><font class="small">In reply to:</font><hr /><p>The a priori proof presented in Orbits<br />Explained begins with Newton's demonstration that equal areas<br />are swept in equal times as a planet moves past the Sun,<br />regardless of the relationship between distance and gravitational<br />force. <p><hr /></p></p></blockquote><br /><br />Except that keplar observed the equal areas in equal times relationship and Newton posited that as conservation of angular momentum, based upon observations of rotational and linear systems. So you don't end up with a priori here. <div class="Discussion_UserSignature"> <p align="center"><font color="#c0c0c0"><br /></font></p><p align="center"><font color="#999999"><em><font size="1">--------</font></em></font><font color="#999999"><em><font size="1">--------</font></em></font><font color="#999999"><em><font size="1">----</font></em></font><font color="#666699">SaiphMOD@gmail.com </font><font color="#999999"><em><font size="1">-------------------</font></em></font></p><p><font color="#999999"><em><font size="1">"This is my Timey Wimey Detector.  Goes "bing" when there's stuff.  It also fries eggs at 30 paces, wether you want it to or not actually.  I've learned to stay away from hens: It's not pretty when they blow" -- </font></em></font><font size="1" color="#999999">The Tenth Doctor, "Blink"</font></p> </div>

 

[kmarinas86]

The underlying "knowledge" behind an a priori is an underlying tautology:<br /><br />A implies B. A <i>is</i> because A <i>exists</i>, therefore B.<br /><br />Tautologies have a basis on ideas, but not necessarily reality. They can be proven <i>in theory</i>, but without prior direct observation, reality may beg to differ.

 

[marlsda]

Thanks Yevaud. I am grateful for the comment. -David Marlin

 

[dragon04]

Now it's time to gladly show my ignorance, or perhaps my lack of formal scientific training.<br /><br />Can someone give me an example of an a priori proof? <div class="Discussion_UserSignature"> <em>"2012.. Year of the Dragon!! Get on the Dragon Wagon!".</em> </div>

 

[marlsda]

Thanks Saiph. The a priori claim stems from an approach that does not allow Kepler's observations nor Newton's interpretation of them. In Principia, Newton actually gives the geometrical a priori proof of "equal areas equal times" without using observations. His geometrical proof is retold in Chapter 5 of the "Orbits Explained" site. The next step is to assert that "equal areas equal times" dictates that radius is inversely proportional to tangential velocity. Finally, apply the radius and tangential velocity to the hododyne and let it spin. The elliptical orbit, Hamilton's Hodograph and all its features are generated in a priori fashion.

 

[doubletruncation]

Hi Marlsda,<br /><br />I enjoyed your website, I found it quite interesting, and it was fun to go back and look at some mechanics stuff that I haven't thought about in a long while. I must confess though, that I had skepticism reading the website for the following reason: Bertrand proved in 1873 that the only central power-law forces that give closed orbits are the 1/r^2 law, and Hooke's law (force is propotional to r, used for springs). Check out "Classical Dynamics" by Jose and Saletan (page 88-92) for a proof. So it seems impossible to me that you could conclude that the planets orbit in ellipses using only the assumption that they are moving in a central force. It seems that you would either have to assume elliptical orbits, or you would have to assume a 1/r^2 law. Reading through your website, I think I agree with what you have written until chapter 11. In that chapter you show that the distance and tangential velocities can be plotted using a hotodyne. It seems to me, though, that you are implicitely assuming that the angle CGA would correspond to the orbital angle of the planet when it is at a distance GA. That assumption would not be valid in a general central force law, to get the angle as a function of distance you would have to integrate the force equations. To put it another way, while you can use the hotodyne to plot the radius and the tangential velocities in what looks like an ellipse, the orbit of a general object will not be on that ellipse. I still think the hotodyne is a neat way to visualize the orbit of a planet - but only if you specify before-hand that the force law is 1/r^2, in which case the angle CGA will work out to be the orbital angle of the planet. Also, as a point that you may have already appreciated, chapter 5 is essentially a proof that angular momentum is conserved in a central-force law (angular momentum is just radius times the tangential velocity). <div class="Discussion_UserSignature"> </div>

 

[marlsda]

Hello Doubletruncation,<br />I am truly grateful that you spent considerable time and effort analyzing my attempt at a priori proofs for orbits. I see that you are not convinced that it can be achieved without assuming an inverse square law of force or an elliptical path. Is it acceptable to state that the position of a planet can be represented on the rotating radius of a circle much like an airplane blip on a rotating radar radius on a circular radar screen? If so the planet position can be represented on the rotating hododyne segment AB. And the segment AG must at least qualify to be the radius to the planet since it satisfies the requirement that it varies in inverse proportion to segment HB which can represent tangential velocity. The general object that you speak of must be at position G and therefore on an elliptical path because no other location for the general object will satisfy the requirement that an inversely proportional segment will still exist within the hododyne. Thus the planet is on an ellipse and its angle to the Sun, or to perihelion, is self evident. If this is acceptable to you, then the Inverse Square Force Law follows a priori instead of being a prerequisite so that this is, in a sense, another method (Chapter 21 of website) in agreement with Bertrand's finding regarding closed orbits. <br />I will find "Classical Dynamics" as you suggested. Again, thank you very much and feel free to counter my attempted answers. - d.m.<br />

 

[doubletruncation]

I think if we take a step back and think about what you're trying to do, it should be clear that it's not possible. As I understand it, what you want to do is prove that the planets must follow elliptical orbits and follow a 1/r^2 force law assuming only that the planets are acted on by a central force. If that were true, it would be impossible for any central force other than 1/r^2 to exist (because just assuming a central force implies that the force law must be 1/r^2). However, you can write down any function of r that you please and solve newton's equations and you will get something that may or may not be an elliptical orbit. Imagine, for example, taking a spring with a ring at one end and marble on the other. Affix a post to a table and slide the ring at the end of the spring over the post. You should then be able to spin the spring around the post freely, and also pull on the marble to let it oscillate on the spring. You'll find that you can have the marble "orbit" the post, but the force on the marble due to the spring is f ~ -r (and always directed toward the post) rather than f ~ 1/r^2. In fact, I should clarify, that even assuming that the orbit is elliptical you can't prove in general that the force law is 1/r^2, you'll have to also assume Kepler's third law.<br /><br /><i>Is it acceptable to state that the position of a planet can be represented on the rotating radius of a circle much like an airplane blip on a rotating radar radius on a circular radar screen?</i><br /><br />I don't think this is a perfect analogy. Just because you can construct the machine where segment AG and HB are inversely proportional doesn't mean that the angle has anything to do with the orbital angle of the planet. Imagine the planet is somewhere on it's orbit and that it has instantaneous radial (vr) and angular (vtheta) velocities. After an infinitesimal time, dt, it will move to a new radial position that is at r + vr*dt and a new angular position that is at theta + vtheta*dt. (Note <div class="Discussion_UserSignature"> </div>

 

[marlsda]

Every text on dynamics that I have seen recruits the inverse square law of force as a given in the derivation of the planetary laws. It is true that if one starts with delta thetas, delta radial velocities, and an unknown force law, one can not arrive at the elliptical orbit using calculus. It requires a different approach. Plato's proof that the square root of two is irrational does so indirectly by ruling out the possibility that it is a odd integer divided by an even integer, or even by odd, or odd by odd. The hododyne is a generator of closed orbits and includes all possible orbital curves by varying the ratio of the lengths of its two spinning segments. Every planet in a Sun's gravitational field must be capable of being represented by a unique hododyne of appropriate segment proportions. Because of the inverse proportion between tangential velocity and radius, the path of a planet can not be represented by anything that is in contradiction to the hododyne since a hododyne's sole function is to generate inverse proportions. So there must be a valid hododyne for any given orbit. Lastly, a planet can not logically jump randomly from one location on its path to any other, it must proceed directly along continuously on the curve that the hododyne traces, an ellipse. It is impossible to state this using calculus yet it is evident on inspection of the hododyne and hodograph. I am aware that I am claiming as a result of this that central forces in closed orbits must be inverse square to distance a priori. (If given the ellipse, the inverse square law follows.) So, l really must look into the marble orbiting on a spring to see if I can reconcile the action and results with my scheme. Thank you once again.