Why does Kepler's Third Law exist?

[sophiecentaur]

That would come centuries later - requiring computers.

 

[sophiecentaur]

Probably. But the relevant point here is that Kepler's laws do nothing to predict the arrangement of the planets.

This is a bit late to the party dude but I think I've sussed out the OP's real question. The way that the tables are presented of planets, their orbits and 'that coefficient' may look as if the relative positions of the planets is somehow related to Kepler's law. But this is putting the cart before the horse.
Fact is that you could take our Sun and a different set of planets and they would also fit on that straight(ish) line. So why isn't there a planet half way between Mars and Earth, sitting on that line? Such a body would / could destabilise the whole set up because of its interaction with, particularly, the nearest planetary orbits. The planets were, of course, formed from bands of dust and rock and gas, which separated out into massive spheres. The presence of massive Jupiter (so the theory goes) prevented all the rock in the Asteroid belt from ever merging together so an extra planet wouldn't have formed where the Asteroid belt sits.
It's all due to gravitational effects but it's a many-body problem and Kepler was only looking at two bodies at a time and making many assumptions.

 

[gmax137]

@sophiecentaur I was about to make a similar post, but more simply: to the OP, do you understand that K3 applies to any other solar system as well?

K3 describes the relationship between period and radius but it does not determine either.

 

[Vanadium 50]

Are we discussing Kepler's Law or Bode's Law?

 

[strangerep]

I have researched the internet but can't find a reason why [Kepler 3] exists.
Is it somehow a consequence of some type of gravitational balance, if not is there some other mechanical reason?

Since no one else has mentioned this...

A short answer is that K3L follows from a property of the Lagrangian known as Mechanical Similarity. The examples section in that Wiki page mentions Kepler 3, but not by that name. Landau & Lifschitz vol1 p22 has more detail.

In essence, if some of the variables in a Lagrangian are homogeneous (meaning that, e.g., for some constant (where might be different for different variables), there are cases where the homogeneities in the different variables can combine to result in merely multiplying the Lagrangian by a constant factor. This doesn't change the equations of motion, but does correspond to the existence of similarly shaped orbits of different size and energy. This symmetry doesn't commute with the Hamiltonian, hence is not associated with a conserved quantity.

One can therefore regard Kepler 3 as arising because of the ways that kinetic energy and gravitational potential energy scale under independent rescalings of space and time.

There are other cases where mechanical similarity is useful. L&L give examples.

Kepler 3 is interesting because it does not derive from an underlying conservation law, but rather this less well-known feature of scaling similarity.

 

[Martyn Arthur]

ah...the equations are beyond me (for now at least) but I'm giving it a go.
Sorry if I am off track, but I am endeavoring..
My question was "is there a physical reason why all bodies rotating under gravity enjoy the same relationship between T and r".
So every planet, or indeed anybody rotating [indeed moving] with constant acceleration has constant momentum.
The relationship between T and r is thus a determinant of the relationship between the gravitational constant, the body's momentum and the body's distance from the Sun.
This relationship between T and r is a constant value which is obeyed by all bodies rotating with constant acceleration.
Kepler's equations, the third in particular, are a mathematical calculation 'proving' that the relationship is a consequence of the foregoing.
So the relationship is a "universal" constant but one which, unlike the speed of life, has an underlying mathematical explanation.
Thank, hopefully Martyn

 

[PeroK]

ah...the equations are beyond me (for now at least) but I'm giving it a go.
Sorry if I am off track, but I am endeavoring..
My question was "is there a physical reason why all bodies rotating under gravity enjoy the same relationship between T and r".
So every planet, or indeed anybody rotating [indeed moving] with constant acceleration has constant momentum.
The relationship between T and r is thus a determinant of the relationship between the gravitational constant, the body's momentum and the body's distance from the Sun.
This relationship between T and r is a constant value which is obeyed by all bodies rotating with constant acceleration.
Kepler's equations, the third in particular, are a mathematical calculation 'proving' that the relationship is a consequence of the foregoing.
So the relationship is a "universal" constant but one which, unlike the speed of life, has an underlying mathematical explanation.
Thank, hopefully Martyn

The gravitational force is proportional to mass, as is acceleration, according to Newton's laws of gravity and motion respectively. The motion of bodies under the force of gravity, whether parabolic projectile motion or elliptical orbits, is therefore independent of the mass of body.

That's quite fundamental. And I'm surprised if it has not been emphasised in the texts from which you are studying.

 

[strangerep]

ah...the equations are beyond me (for now at least) but I'm giving it a go.
Sorry if I am off track, but I am endeavoring..
My question was "is there a physical reason why all bodies rotating under gravity enjoy the same relationship between T and r".

Did you even try to read the little bit of Landau & Lifschitz that I referenced? I guess not.

So every planet, or indeed anybody rotating [indeed moving] with constant acceleration has constant momentum.

False.

The relationship between T and r is thus a determinant of the relationship between the gravitational constant, the body's momentum and the body's distance from the Sun.

Not the body's momentum directly, but rather it's energy, i.e., $m v^2 /2$.

Kepler's equations, the third in particular, are a mathematical calculation 'proving' that the relationship is a consequence of the foregoing.
So the relationship is a "universal" constant but one which, unlike the speed of life, has an underlying mathematical explanation.

Here's a cut down version of the scaling symmetry I tried to explain in my previous post.

Kinetic energy $K$ is a function of $v^2$, which dimensions of $L^2/T^2$ (length squared over time squared). Gravitational potential energy $V$ is a function of $1/R$, i.e., inverse length.

Now imagine that you rescale all lengths by a factor $\alpha$ and time by a factor $\beta$ (both are just +ve real numbers), i.e., $$ r ~\to~ r' = \alpha r ~,~~~~ t ~\to~ t' = \beta t ~.$$ Under this rescaling, $$K ~\to~ K' = \alpha^2 K/\beta^2 ~,~~~~ \text{and}~~~ V ~\to~ V' = V/\alpha ~.$$ Question: what value must $\beta$ have such that $K$ and $V$ scale by the same overall factor. This is answered by putting $$\alpha^2 /\beta^2 = 1/\alpha ~,~~~~ \Rightarrow~~ \alpha^3 = \beta^2 ~.$$ That gives Kepler's 3rd law! Under these rescalings, $$T^2 ~\to~ \beta^2 T^2 ~,~~~~ R^3 ~\to~ \alpha^3 R^3 ~.$$ Multiplying the Lagrangian $L = K-V$ by a constant factor does not change the equations of motion. I.e., any spacetime rescaling such that $\alpha^3 = \beta^2$ must result in a new solution of the equations of motion, a different (but valid) orbit.

 

[Vanadium 50]

ah...the equations are beyond me

"And he said, Go, and tell this people, Hear you indeed, but understand not; and see indeed, but perceive not."

You want an explanation of a mathematical fact without using mathematics. This is impossible. You don't need to be an expert in mathematics to understand youwon't get what you want.

 

[brainpushups]

@Martyn Arthur, if you can find access to, or purchase for yourself, the book Physics, The Human Adventure by Horton and Brush you'll find a nice discussion of Newton's synthesis of Kepler's laws that uses only algebra (but only addresses circular orbits) in Chapter 11. Further, the discussion of the historical development is far more detailed than most texts and will offer some good insight into the nature of theory in physics which you may find beneficial. The text is written at the undergraduate level for non-science majors and is well worth a read.

 

[strangerep]

You want an explanation of a mathematical fact without using mathematics. [...]

I think you might have jumped to an overly harsh assessment.This all depends on what level of math the OP can handle. After all, he marked the thread as I-level. We'll have to wait for him to give us a clearer indication of what level of math he can handle.

 

[Vanadium 50]

Well, he's been rejectiong all the derivations and pointers to books.

 

[sophiecentaur]

Well, he's been rejectiong all the derivations and pointers to books.

Yes; only the very occasional genius is in a position to define just how they're going to learn a subject. Entering the Tour de France requires the possession of a bicycle.

 

[Demystifier]

Ah so the relationship is a determinant of gravity and law 3 calculates the relationship in the discussion. I think I am right in saying then that it is purely the nature of gravity, how it functions, that is the cause of the relationship.
If so thank you very much for your patient help in this.
Thanks
Martyn

Yes.

 

[DaveC426913]

You want an explanation of a mathematical fact without using mathematics. This is impossible.

With respect, it's a fact of nature - it's worked since before math was invented.

Nature is the territory; math is only the map. I think the OP gropes for an explanation in terms of the territory, not the map.For example, this is a "explanation" of orbital velocity that involves no math:

 

[Martyn Arthur]

Thank you for the refernce to the book,

Physics, the Human Adventure: From Copernicus to Einstein and Beyond Hardcover – 1 Mar. 2001​

I am studying with the OU and it doesn't have any copies, I would like to read it but its too costly on amazon (I am a pensioner :-( )
But this dialogue has hugely helped me to start venturing into a domain about which I knew nothing.
You guys have been so patient and helpful that I say thank you again.
Assume I have no maths, not too far from the truth, pending progression on my course.
Please just look at it at a basic level absolutely, and forget please all equations.
Gravity, it is either a consequence of the interaction between masses or the curvature of space.
Is it fair to say that this is still in the melting pot?
Kepler's law has identified a universal relationship of planets, with other planets and the Sun.
Is the reason by Kepler 3 for this relationship either;
a. demonstrated to be a relationship by reference to know facts of gravity etc
b. an undefined reason;
For example, without known data, the apparent 'coincidental' distance between the Moon and the Sun without physical explanation that facilitates the eclipses (that have enabled us to learn so very much).

 

[sophiecentaur]

Please just look at it at a basic level absolutely, and forget please all equations.

If you stipulate that no equations can be used, all you can say is that two masses attract and that the attractive force relates to the separation (further apart = less). You cannot demand an 'explanation' with more depth. Your "basic level" needs to take you back to Science as it was at least a thousand years ago.
Kepler uses Maths so you can't appreciate those laws without maths.

The coincidental similarity (not equality) between the angles subtended by Sun and Moon is not 'significant'. There are many other planets and their moons which do not demonstrate such perfect eclipses. Of all the Jovian moons, can you think of one that would exhibit the same behaviour? If so, which one and what about the others?

 

[topsquark]

Assume I have no maths, not too far from the truth, pending progression on my course.
Please just look at it at a basic level absolutely, and forget please all equations.

I think it's great that you are trying to stretch yourself and I sympathize that your level of Mathematics isn't all that advanced. But you need to understand that without Mathematics we can only talk about generalities. There is no way we can describe Kepler's third in any quantitative way because what it talks about depends on the Mathematics; Kepler's third law is an equation, after all.

Keep being curious. Keep asking questions. What you don't understand look up and read about. Learn how to use the Mathematics. When you have questions about what you've read or questions on new topics go ahead and ask about them. Rinse and repeat.

-Dan

 

[Vanadium 50]

forget please all equations.

Kepler's Third Law is an equation.

 

[DaveC426913]

Kepler's Third Law is an equation.

The inverse square law of light is an equation too, but it can be intuited with a model and some geometry. The variables in the equations (equatia? equatiae?) represent real world phenomena.

 

[sophiecentaur]

Keep being curious. Keep asking questions.

That's easy to say but we all know that learning by just asking questions seldom gets us very far. That is not much further than pop Science on the TV which may be 'enough' to enjoy the edited bits of Science but it never allows one to make valid predictions or conclusions.

To know the appropriate questions to ask, you must first have learned a significant amount. Q and A is very high risk.

 

[topsquark]

That's easy to say but we all know that learning by just asking questions seldom gets us very far. That is not much further than pop Science on the TV which may be 'enough' to enjoy the edited bits of Science but it never allows one to make valid predictions or conclusions.

To know the appropriate questions to ask, you must first have learned a significant amount. Q and A is very high risk.

I agree. It sounds to me like he's at the beginning of all of this. A teacher in an organized course is the best thing for him. I guess I'm assuming he already knows that and would be taking a course at some point so I didn't bring that up.

-Dan

 

[Vanadium 50]

The inverse square law of light is an equation too, but it can be intuited with a model and some geometry.

You mean $A = 4\pi r^2$? Looks like an equation to me!

 

[DaveC426913]

You mean $A = 4\pi r^2$? Looks like an equation to me!

It is.
But the phenomena itself can be intuited with the use of a visual aid, like so:

Sure there's technically a bit of simple math there, but 99% of the meaning of the equation is embodied in the geometry of the diagram.

 

[PeroK]

I am studying with the OU ...

I strongly suspect the OU physics course will include some equations.

 

[sophiecentaur]

A teacher in an organized course is the best thing for him.

I think you could be being a bit more polite than I. I would suggest that "best' should be replaced with "essential" for whoever wants PF advice. It is mostly hard stuff and very little of it avoids Maths.

Even @DaveC426913 's diagram is 'maths' - just dressed up in a more friendly way.

 

[DaveC426913]

Even @DaveC426913 's diagram is 'maths' - just dressed up in a more friendly way.

Agree. I just think it's the level of maths the OP is looking for.

 

[Martyn Arthur]

I continue to be profoundly grateful for your patience of you all, the recent posting reassured me that I am not being regarded as a 'nuisance'.
May I please approach this from a different perceptive, forgetting Kepler and equations completely?
Separately [as a comparison] the relationship between the distances between Earth, the Sun, and the Moon are, without any 'apparent' reason such that eclipses are caused to occur.
It could then follow that the explanation for the two of the two minor bodies relative to the sun is a simple consequence of the compound gravitational forces acting in accord with the gravitational constant.
There is a gravitational constant, it is what it is, and we do not seek further.

Likewise is it the case that the defined relative orbiting situations, in a particular mode/sequence, are a physical result of the gravitational constant et al.

From my perspective, objectively / simply of trying to understand.
I am trying to pitch this in the most basic format. is the specific relationship between orbital relationships defined by science [then as defined by Kepler, not forgetting ever Newton, or is it just the way it is?
Martyn Arthur...thanks ongoing for the patience and understanding of you guys,
Martyn Arthur [agan]

 

[DaveC426913]

It could then follow that the explanation for the two of the two minor bodies relative to the sun is a simple consequence of the compound gravitational forces acting in accord with the gravitational constant.
There is a gravitational constant, it is what it is, and we do not seek further.

That our Moon often eclipses the Sun is a matter of coincidence - there's no deeper factors*. There's lots of Moons, and they inhabit a wide range of distance from their primaries.

*(I mean, except for the usual limits on multiple body systems: a moon can't get too far from its primary or it won't have a stable orbit, and it can't get too close or it will disintegrate.)

 

[sophiecentaur]

Separately [as a comparison] the relationship between the distances between Earth, the Sun, and the Moon are, without any 'apparent' reason such that eclipses are caused to occur.

You are seeing something significant here but is it really? Should there be an "apparent reason" why two sixes turn up together? There is a perfectly reasonable explanation of the way the Solar System was formed in a rough disc shape (do some Googling about it). That means that most of the objects orbit round the Sun near to its equatorial plane. Also, most of the planets rotate (and their moons) with axes much the same as the axis of those orbital planes. To get an eclipse, all that's necessary is for a large (or just large enough) object to pass between the Sun and another object. That will cause the object (or part of it) further away to be briefly in shadow. Solar Eclipses are very impressive because the Sun is a major part of our visual experience and the eclipse is total (but not always). Every so often, other objects go across the Sun (from our POV). Transits of Venus are quite common.