Proof of inverse square law for gravitation?

[mark2142]

Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law ($T^2=\frac {4\pi r^3}{GM}$)?

Thank you.

 

[topsquark]

Newton arrived at "there is a force that drives a planet around the star by examining kepler's laws but how did he arrive to inverse square law by kepler's third law ($T^2=\frac {4pi r^3}{GM}$)?

Thank you.

Actually, $K = \dfrac{4 \pi ^2}{GM} r^3$.

The simplest way (there may be a more elegant method I'm not familiar with) is to consider a test mass with mass m in a circular orbit about a mass M. The centripetal force on the mass will be
$F = m r \omega ^2 = \dfrac{4 \pi ^2 mr }{T^2}$
where $\omega$ is the angular frequency of the orbit and r is the radius of the orbit.

Plug in $T^2$.

Once you know it's an inverse square law you can go on to make a more general model of the orbits that validates Kepler's Laws.

-Dan

 

[mark2142]

Plug in .

Hi Dan, what is K?
Do you mean $K=T^2$ ?

 

[topsquark]

Hi Dan, what is K?
Do you mean $K=T^2$ ?

Sorry. Yes, K was supposed to be $T^2$. I originally wrote the post just using a constant of proportionality before I noticed that you had listed the full form for Kepler's Law. Apparently a K survived the edit!

-Dan

 

[phyzguy]

You can read how he originally came to the idea of the inverse square law. He realized that the Earth's gravity was what was holding the Moon in its orbit. Since he understood the concept of centripetal force, he was able to calculate that the gravitational acceleration required to keep the Moon in its orbit is about 3600 times smaller than the gravitational acceleration at the Earth's surface. He also knew that the radius of the Moon's orbit is about 60 times larger than the Earth's radius, and thus the inverse square law would give the right force. Here's what he actually said:

“In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Kepler's rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly. All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more then than at any time since.”​

 

[mark2142]

Actually, .

Where did you get $K=T^2= \frac {4pi^2}{GM} r^3$ from? Did Kepler knew the exact law or just $\frac {T^2}{a^3}=3$? According to my knowledge this law is derived from universal law of gravitation and Newtons second law. And I am asking where do you get universal law of gravitation from?

mass will be

where is the

Second eqn is alright.

 

[mark2142]

You can read how he originally came to the idea of the inverse square law. He realized that the Earth's gravity was what was holding the Moon in its orbit. Since he understood the concept of centripetal force, he was able to calculate that the gravitational acceleration required to keep the Moon in its orbit is about 3600 times smaller than the gravitational acceleration at the Earth's surface. He also knew that the radius of the Moon's orbit is about 60 times larger than the Earth's radius, and thus the inverse square law would give the right force. Here's what he actually said:

Yes I read this about it in "Six easy pieces" by feynman but there was a line that says we can derive inverse square law from keplers third law and I want to know how?

 

[Ibix]

Yes I read this about it in "Six easy pieces" by feynman but there was a line that says we can derive inverse square law from keplers third law and I want to know how?

As noted upthread, it's just equating gravity with the centripetal force. For a circular orbit centripetal force is $F=m\omega^2r=m\left(\frac{2\pi}T\right)^2r$ which implies $T^2=\frac 1F4\pi^2mr$. Kepler's third law says that this is proportional to $r^3$, so we get $F\propto\frac m{r^2}$. The constant of proportionality is the gravitational parameter, $GM$, which you determine from observation of a particular system.

 

[mark2142]

As noted upthread, it's just equating gravity with the centripetal force. For a circular orbit centripetal force is $F=m\omega^2r=m\left(\frac{2\pi}T\right)^2r$ which implies $T^2=\frac 1F4\pi^2mr$. Kepler's third law says that this is proportional to $r^3$, so we get $F\propto\frac m{r^2}$. The constant of proportionality is the gravitational parameter, $GM$, which you determine from observation of a particular system.

Thank you! So simple.
Was proportionality constant $4pi^2/GM$ in kepler's third law determined after finding Newtons law of gravitation according to this:

$\frac {GMm}{r^2}= mr\frac {(2pi)^2}{T^2}$
$T^2=\frac {4pi^2}{GM}r^3$

So, $K=\frac {4pi^2}{GM}$
Yes?

 

[topsquark]

Thank you! So simple.
Was proportionality constant $4pi^2/GM$ in kepler's third law determined after finding Newtons law of gravitation according to this:

$\frac {GMm}{r^2}= mr\frac {(2pi)^2}{T^2}$
$T^2=\frac {4pi^2}{GM}r^3$

So, $K=\frac {4pi^2}{GM}$
Yes?

It is certain he knew the value of K, or to be precise GM, but Newton was the first to use gravitation to estimate the mass of the Sun.

FYI: LaTeX tip: To write $\pi$, use \pi.

-Dan

 

[phyzguy]

It is certain he knew the value of K, or to be precise GM, but Newton was the first to use gravitation to estimate the mass of the Sun.

I don't think this is true. He had no way to do this, because he had no idea of the value of G. This is why the Cavendish experiment, which determined G for the first time, was referred to as "weighing the sun".

 

[topsquark]

I don't think this is true. He had no way to do this, because he had no idea of the value of G. This is why the Cavendish experiment, which determined G for the first time, was referred to as "weighing the sun".

You may be right about that. I didn't look up when Cavendish did his experiments. I got the Newton thing form a website so it's validity might be questionable.

-Dan

 

[phyzguy]

The Cavendish experiment was in 1797, 70 years after Newton's death.

 

[topsquark]

The Cavendish experiment was in 1797, 70 years after Newton's death.

Hah! My Dad is turning over in his grave. He was a History teacher.

Thanks for the catch.

-Dan

 

[mark2142]

I don't think this is true. He had no way to do this, because he had no idea of the value of G. This is why the Cavendish experiment, which determined G for the first time, was referred to as "weighing the sun".

So it was not until Newton arrived at his law that we understood what was K in kepler's third law. Later Cavendish determined G in Newtons law of gravitation and weighed the earth.

 

[topsquark]

So it was not until Newton arrived at his law that we understood what was K in kepler's third law. Later Cavendish determined G in Newtons law of gravitation and weighed the earth.

Yes. The important point is that Kepler's Law allowed us to determine that gravitation is an inverse square law, which allowed Newton to solve the central force problem for it. (Well, there are plenty of other good things that come from it, but I'm not an Astronomer! :) )

-Dan

 

[mark2142]

thanks...

 

[vanhees71]

I'm not sure about the exact history, but AFAIK indeed Newton argued with the special case of circular orbits and Kepler's Laws as well as the independence of the acceleration of bodies due to gravitational forces from the mass to deduce that the interaction force must be a central force (from Kepler II, which holds for any central force due to angular-momentum conservation), that it must be proportional to $m_1 m_2$ (independence of the free-fall acceleration from mass) and finally the proportionality to $1/r^2$ due to Kepler III. Finally he found the general solution for the motion of two bodies under this interaction force, proving that also Kepler I holds. At least that's the (hi)story you usually find in textbooks on mechanics ;-)).

 

[Arjan82]

You can read how he originally came to the idea of the inverse square law.

There was apparently quite a quarrel between him and Hooke about who came up with that Idea first:
https://www.bbvaopenmind.com/en/sci...ius-whose-big-mistake-was-confronting-Newton/

also some discussions here:
https://hsm.stackexchange.com/questions/521/robert-hooke-and-the-inverse-square-law

Newton was quite a guy