Newton's Law of Gravitation: Empirical or Theoretical?

Question

Is the Newtons Law of Gravitation, an empirical law of theoretical? They say that Kepler's laws of planetary motion is an emirical law because this law was obtained by generalizing a set of data. Whereas, Newtons law of gravitation is not empirical because he didnt collect any experimental data rather he attempted to account of Kepler laws, he didnt generlize but he 'explained'........

Please shed more light on this..........thanks!

Answer 1

I would call both Newton's and Kepler's laws as empirical.  They encapsulate observations into equations.  But they don't explain why it is so.

Answer 2

Someone explained empirical / theoretical to me like this:

You're watching a black box, from which comes a high-pitched squeaking sound. A cat rushes into the box, and a few moments later, leaves, content, with a tiny, hairless tail hanging from its mouth.

The empirical interpretation is: "Cats are attracted to black boxes by high pitched squeaking sounds. When they leave the box, they will have a tiny, hairless tail hanging from their mouth".

This is accurate, and even useful. You could use it to predict when a cat might go into a box, and even to attract cats into a desired box. As long as the observations were accurate, this interpretation is 100% correct.

The theoretical interpretation is: "Cats like to eat mice, which make high pitched squeaking sounds and have tiny, hairless tails. Sometimes, they can be found in boxes".

This explains much more deeply what is going on, and has much more predictive power. It tells us a lot of (theoretical) information about both cats and mice, as well as their interactions with boxes. However, it -is- purely theoretical, and  could be completely wrong!

Hope that helps!

Answer 3

Kepler's laws are indeed empirical and they are based upon the observational data obtained by Tycho Brahe. However, Newton's law of universal gravitation may be derived from Kepler's laws. It is just this approach that Newton adopted - starting with

Kepler's second law (k2 for short). The element of area dA swept out by the radius vector, of an orbiting planet, in time dt is approximately: -

dA = ½r(r + dr)dθ

Hence, in the limit (calculus): -

dA = ½r²dθ

If k2 is written as: -

r²dθ/dt = constant = C or d(r²dθ/dt)/dt = 0

In this approximation the planets and the sun are considered to be point masses.

Now, from k1. The equation of an ellipse in polar coordinates with one focus as the origin is: -

1 - e.cosθ = (a(1 - r²))/r

Where a is the semi major axis and e is the eccentricity of the ellipse.

If we differentiate twice with respect to time and use K2 (constant form above) to eliminate dθ/dt, we obtain: -

(C²/r²).e.cosθ = - a(1 -e²).d²rdt²

A central conservative field of force has the potential equation: -

F = -dV/dr

From Newton's second law of motion it may be shown that the equation may be written down as: -

F(r) = m((d²r/dt²) - r.(dθ/dt)²)

.......= -(mC²/(a.(1 - e²)r²).(ecosθ  +(a.(1 - e²)/r))    

Hence, using K1 (from the above equation form): -

F(r) = -mC²/(a(1 - e²)r²)

If we now relate the constant C to the orbital period of the planet: -

dA/dt = 0.5C

With

A =√(πa²(1 - e²))

and this gives

√(πa²(1 - e²)) = 0.5CT

Hence, from the F(r) equation: -

F(r) = -4π²a³m/(T²r²) = -Bm/r²

Where B = 4π²a³/T²

Newton's great contribution was to realise that the forces between  the earth and the sun  were the same.  Thus, for the sun (mass M): -

B/M = G

and for the earth (mass m constant B’)

B'/m = G

So that for the Sun,  B = GM

Or

F = -GMm/r²

This is how Newton derived his famous equation using Kepler's empirical laws and the equation of a central conservative force field.

I hope this brief proof is of some assistance.

Answer 4

F is proportional to inverse r^2 is theoretical. He took a guess (an "ansatz") at the relation and this expression worked!