# Inverse square law and Kepler's third law

1. Feb 25, 2005

### 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?

2. Feb 25, 2005

### vincentchan

WRONG, it is the other way around....
Does it matter?
Gauss' Law if you like...

3. Feb 25, 2005

### Reshma

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:

Last edited: Feb 25, 2005
4. Feb 25, 2005

### 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.

5. Feb 25, 2005

### 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.

6. Feb 25, 2005

### Reshma

What is the analogy for electrostatics?

7. Feb 25, 2005

### 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

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

8. Feb 25, 2005

### maverick280857

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

9. Feb 26, 2005

### Reshma

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.

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.

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.

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

10. Feb 27, 2005

### Reshma

11. Feb 27, 2005

### dextercioby

What does that represent...?

Daniel.

12. Feb 28, 2005

### vincentchan

that's mean Kepler died at age of 30...

13. Feb 28, 2005

### dextercioby

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

Daniel.

14. Feb 28, 2005

### snbose

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".

Last edited: Feb 28, 2005
15. Feb 28, 2005

### dextercioby

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

Daniel.

16. Mar 1, 2005

### 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.

17. Mar 1, 2005

### maverick280857

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.

18. Mar 1, 2005

### dextercioby

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.

19. Mar 1, 2005

### Andrew Mason

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

20. Mar 1, 2005

### dextercioby

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

Daniel.