The Litmus Test

Note: this post uses the wrong formula for gravity. The correct formula can be found here:

http://oddbitsandsquarepegs.blogspot.pt/2017/07/an-odd-thing-newton-does.html

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The litmus test of any formula designed to describe orbits is that it has to reproduce Kepler's Second Law of Planetary motion. It has to yield a constant value for T^2/r^3.

Now, consider the formula arrived at for the capacitor model of gravity:

F = Kq1m2/r^2 + Kq2m1/r^2 - kq1q2/r^3

K and k are constants, q1 is the charge of body 1, m1 is the mass of body 1, q2 is the charge of body 2, m2 is the mass of body 2, r is the distance between the bodies.

Due to the quick tapering off of the inverse cube law, the final term can be ignored for normal orbits. This gives us:

F = Kq1m2/r^2 + Kq2m1/r^2

Now consider a planet of mass m2 and charge q2 in orbit around the sun with mass m1 and charge q1.

Since the sun makes up the vast majority of mass and charge in our solar system (more than 99%) and it is charge that pulls on mass, we see that we can also remove the second term of the above equation in which the planet is pulling on the sun. In our solar system, it is the sun that pulls planets to itself, not the other way around. Where there are vast differences in size, the gravitational pull of the smaller body can be ignored. (This is a premise in the capacitor model, or else, small bodies would be repelled by the repelling force.):

F = Kq1m2/r^2

Assuming that the orbit is a near perfect circle, the centripetal force Fc = m2*v^2/r.

Since F must be equal to Fc we get:

m2*v^2/r = Kq1m2/r^2

Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*r) / T:

v^2 = (4 * pi^2 * r^2) / T^2

Substitution of the expression for v^2 into the equation above yields:

(m2 * 4 * pi^2 * r^2) / (r * T^2) = Kq1m2/r^2

By cross-multiplication and simplification, the equation can be transformed into:

T^2/r^3 = (4 * pi^2) / Kq1

Since q1 is the charge of the sun and therefore equal for all planets, and all the other terms to the right are constants, we have arrived at an expression where T^2/r^3 is a constant, precisely as required by Kepler.

The equivalent proof for Newton's law can be found here: http://www.physicsclassroom.com/class/circles/Lesson-4/Kepler-s-Three-Laws

Johannes Kepler

By Unidentified painter - Unknown, Public Domain, Link