# Inverse square law and Kepler's third law

Reshma
The inverse square law for gravitation was deduced from Kepler's third law.
So how was the inverse square law for electrostatics(Coulomb's law) deduced?

## Answers and Replies

vincentchan
The inverse square law for gravitation was deduced from Kepler's third law.
WRONG, it is the other way around....
So how was the inverse square law for electrostatics(Coulomb's law) deduced?
Does it matter?
Gauss' Law if you like...

Reshma
vincentchan said:
WRONG, it is the other way around....

GET THIS:
Kepler (1571-1601) worked out empirical laws governing planetary motions.
Tycho Brahe (1546-1601) compiled extensive data from which Kepler was able to derive the three laws of planetary motion that now bear his name.

Newton(1642-1727) showed that his law of gravitation LEADS to Kepler's laws.

Does it matter? Gauss' Law if you like...
Explain it :grumpy:

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kanato
Coulomb experimentally measured the forces exerted on charged objects with a variety of charges and distances between them, and came up with the force law.

Reshma
Here is the deduction according to me:

Centripetal force is given by:

$$F_c = m\omega^2r$$

Angular velocity is: $$\omega = \frac{2\pi}{T}$$

Hence acceleration is :
$$a_c = \omega^2r = r(\frac{2\pi}{T})^2 = \frac{4\pi^2r}{T^2}$$

By Kepler's Law: $$T^2 \propto r^3$$

Replacing the denominator:

$$a_c\propto \frac{r}{r^3} \propto \frac{1}{r^2}$$

This acceleration is same as gravitational acceleration $$a_g$$. Since forces varies as acceleration: $$\vec F\propto a_g$$

$$\vec F\propto\frac{1}{r^2}$$

P.S.--->Someone please let me know if this is right.

Thank you.

Reshma
What is the analogy for electrostatics?

maverick280857
Interestingly Kepler was Tyco Brahe's assistant. Tyco Brahe's extensive calculations led Kepler to the laws that we now know as Kepler's Laws of Planetary Motion.

You might be interested in seeing this: http://www.glenbrook.k12.il.us/gbssci/phys/Class/circles/u6l3b.html [Broken]

Also (please correct me if I'm wrong), Newton began the notion of calculus and used it to prove the two Shell Theorems that we use so frequently in electrostatics and gravitation without bothering much about them. He also discovered that with the sun at one focus, the force required to keep planets in an elliptical orbit was purely radial and varied as the inverse square of this radius. This obvious looking proposition has a very interesting and mathematically englightening proof (in fact many) in older "terse" texts on mechanics.

In other words you can start out with a general force function (without assuming anything) and derive that it must be inverse square for the kind of motion that there is (on paper of course--because in reality there is some deviation).

Finally you might be interested to know that (source = Krane), extensive laboratory experiments reveal that the deviation of electrostatic forces from inverse square dependence is far less than that of graviational forces. In othe words, if you write either force function as proportional to $r^{-(2+ \delta)}$ then $\delta = 10^{-4}$ for gravitational forces and $\delta = 10^{-16}$ for electrostatic forces...which suggests that electrostatic forces are truer inverse square forces than are gravitational forces.

I apologize for some of the content in this post is not germane to the present discussion, but I thought I'd throw it in nevertheless.

Cheers
vivek

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maverick280857
Reshma said:
Here is the deduction according to me:

Centripetal force is given by:

$$F_c = m\omega^2r$$

Angular velocity is: $$\omega = \frac{2\pi}{T}$$

Hence acceleration is :
$$a_c = \omega^2r = r(\frac{2\pi}{T})^2 = \frac{4\pi^2r}{T^2}$$

By Kepler's Law: $$T^2 \propto r^3$$

Replacing the denominator:

$$a_c\propto \frac{r}{r^3} \propto \frac{1}{r^2}$$

This acceleration is same as gravitational acceleration $$a_g$$. Since forces varies as acceleration: $$\vec F\propto a_g$$

$$\vec F\propto\frac{1}{r^2}$$

P.S.--->Someone please let me know if this is right.

Thank you.

While it is conceptually not wrong to show the connection between Kepler's Law and inverse square fields as you have done, this normally isn't the way it is proved (see for example, Central Orbits in Dynamics treatise). Kepler's Law holds only approximately by the way.

A more rigorous proof would involve considerations of polar orbits and deriving an equation of orbit and then showing that the angular momentum constancy, zero tangential force (or equivalently purely radial force), center of force = focus, orbit = elliptical lead to the inverse square field. For circular orbits however, your proof is acceptable iff you can show separately that Kepler's Third Law holds in the form you have used.

Cheers
vivek

Reshma
Interestingly Kepler was Tyco Brahe's assistant. Tyco Brahe's extensive calculations led Kepler to the laws that we now know as Kepler's Laws of Planetary Motion.

Kepler was actually concerned with the specific problem of planetary motion in the gravitational field of the sun. A more precise statement of his third law would therefore be: the square of the periods of the various planets are directly proportional to the cube of their major axes.

Also (please correct me if I'm wrong), Newton began the notion of calculus and used it to prove the two Shell Theorems that we use so frequently in electrostatics and gravitation without bothering much about them. He also discovered that with the sun at one focus, the force required to keep planets in an elliptical orbit was purely radial and varied as the inverse square of this radius. This obvious looking proposition has a very interesting and mathematically englightening proof (in fact many) in older "terse" texts on mechanics.

Yes Newton had discovered the gravitation law but delayed its publication till he found a substantial proof to support them which are of the shell theorems.

In other words you can start out with a general force function (without assuming anything) and derive that it must be inverse square for the kind of motion that there is (on paper of course--because in reality there is some deviation).

Well, from the information I've gathered, Kepler's laws were largely based on rigorous observations rather than mathematical proof till Newton invented Calculus. Same with electrostatics.

Finally you might be interested to know that (source = Krane), extensive laboratory experiments reveal that the deviation of electrostatic forces from inverse square dependence is far less than that of graviational forces. In othe words, if you write either force function as proportional to $r^{-(2+ \delta)}$ then $\delta = 10^{-4}$ for gravitational forces and $\delta = 10^{-16}$ for electrostatic forces...which suggests that electrostatic forces are truer inverse square forces than are gravitational forces.

The deviation can also be largely due to the fact that gravitational forces operate over extremely large distances unlike the electrostatic force.

Homework Helper
Reshma said:
GET THIS:
Kepler (1571-1601) worked out empirical laws governing planetary motions.
Tycho Brahe (1546-1601) compiled extensive data from which Kepler was able to derive the three laws of planetary motion that now bear his name.

Newton(1642-1727) showed that his law of gravitation LEADS to Kepler's laws.

Explain it :grumpy:

What does that represent...?

Daniel.

vincentchan
dextercioby said:
What does that represent...?
that's mean Kepler died at age of 30...

Homework Helper
Nope,Kepler died much older.And besides,the 3-rd law was disovered ~1616... Daniel.

snbose
Reshma said:
The inverse square law for gravitation was deduced from Kepler's third law.
So how was the inverse square law for electrostatics(Coulomb's law) deduced?

Coulomb's law is based on experiment. It was not deduced. Just like Newton's force law, P=mf.

It is true that Johannes Keplar and Tycho Brahe's research preeceded Newton's discovery of the inverse sqare law,but Newton 'historically' arrived at the inverse square law independently.
It was then it was found to be aligned with "Keplar's Laws".

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Homework Helper
And that is Johannes Kepler...(sic).And that "foce law" is F=dp/dt...

Daniel.

maverick280857
Why precisely gravitational field isn't as true an inverse square field as the electric field cannot be purely because of large distances of operation. The electric field is closer to ideality so to say. Firstly we cannot precisely determine why the gravitational field is nonideal and say that THAT is the truth. In principle, we could attribute it to several things (in classical physics minus relativity one could attribute it to the nonideal shapes of sources that set up the fields...if you consider only the earth then the field isn't perfectly inverse square--there are deviations because of its oblate spheroidal shape and the like).

So I wouldn't attribute it to one thing alone.

maverick280857
Reshma said:
Well, from the information I've gathered, Kepler's laws were largely based on rigorous observations rather than mathematical proof till Newton invented Calculus. Same with electrostatics.

You would probably know that Kepler's third law isn't completely accurate? If you tend to think otherwise you might want to read AS Ramsey or Kleppner.

Homework Helper
What do you mean it is not completely accurate...?Sure,for a 2 body system (isolated,no perturabations to trajectories) it should take into account the movement around the common center of mass (usually that is achieved by putting the reduced mass in the equations)...

What's new...?

Daniel.

Homework Helper
maverick280857 said:
You would probably know that Kepler's third law isn't completely accurate? If you tend to think otherwise you might want to read AS Ramsey or Kleppner.
Kepler's third laws would be completely accurate if there was only one planet. Newton's Universal Law of Gravitation flows directly from Kepler's third law.

AM

Homework Helper
Andrew,two-body-system interracting through Newton's gravity force... Daniel.

Reshma
Reshma said:
Kepler (1571-1601) worked out empirical laws governing planetary motions.
dextercioby said:
What does that represent...?

vincentchan said:
that's mean Kepler died at age of 30...

I apologise to Dextercioby and Vincentchan for the typo.
The correct year is: 1571-1630

Reshma
maverick280857 said:
Why precisely gravitational field isn't as true an inverse square field as the electric field cannot be purely because of large distances of operation. The electric field is closer to ideality so to say. Firstly we cannot precisely determine why the gravitational field is nonideal and say that THAT is the truth. In principle, we could attribute it to several things (in classical physics minus relativity one could attribute it to the nonideal shapes of sources that set up the fields...if you consider only the earth then the field isn't perfectly inverse square--there are deviations because of its oblate spheroidal shape and the like).

So I wouldn't attribute it to one thing alone.

I did not say that is the only reason. It could possibly be one of the reasons.

Reshma
Andrew Mason said:
Kepler's third laws would be completely accurate if there was only one planet. Newton's Universal Law of Gravitation flows directly from Kepler's third law.

AM

YES, I've deduced it in this post! Reshma
maverick280857 said:
You would probably know that Kepler's third law isn't completely accurate? If you tend to think otherwise you might want to read AS Ramsey or Kleppner.

Hi Vivek, thanks for all the replies!

I have "Introduction to Newtonian Attraction" by AS Ramsey. Can you please point out that particular chapter(because I did not find much references made to Kepler's laws in this book)?

I hate to say this, but after prolonging this discussion this far I haven't received a single clear-cut answer :grumpy:

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Reshma
Also note, Kepler's third law is rigorously true for the electron orbits in the Bohr atom (Source: Classical Mechanics: Herbert Goldstein).

Homework Helper
Reshma said:
Also note, Kepler's third law is rigorously true for the electron orbits in the Bohr atom (Source: Classical Mechanics: Herbert Goldstein).
In the Bohr atom the force on the electon is a central force that varies as 1/r^2. That gives you Kepler's third law. But the Bohr atom is not an accurate model of physical reality.

AM

maverick280857
Andrew Mason said:
In the Bohr atom the force on the electon is a central force that varies as 1/r^2. That gives you Kepler's third law. But the Bohr atom is not an accurate model of physical reality.

AM

Ah finally someone hits the nail on the head. You see its like this: you start out with an assumed model of the motion. Then you use some other "laws" or "ideas" to show that something (which you're trying to show) holds 'rigourously'. But the assumption need not be correct in real life as Andrew has so rightly pointed out. As for reality, there are perturbations in orbits due to several reasons. In a crude way perhaps it is right to say that a perturbation term (which cannot be correctly described in our world) gets added to the differential equation of every point motion referred to an inertial or noninertial frame. For all practical purposes in undergraduate studies, this term equals zero.

Now, regarding the book reference, to be honest I haven't even heard of this book by AS Ramsey. I was referring to Mechanics by AS Ramsey parts I and II (sections on Central Orbits/Polar Orbits).

Cheers
vivek