iSoul In the beginning is reality

Circular orbits

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions:

  1. Each planet orbit the Sun in a spatially circular path.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is constant.

Let’s follow the exposition given in Elements of Newtonian Mechanics by J.M. Knudsen and P.G. Hjorth (Springer, 1995), starting on page 6. Because the speed is constant, the acceleration follows the equation for uniform circular motion derived previously:

a = / Rs,

in which v is the speed and a is the acceleration. We have from the definition of speed:

v = S / T = 2πRs / T.

Elimination of v from these equations leads to

aRs / , or

a ∝ S / .

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

.

Combining this with the equation for acceleration yields

a ∝ 1 / Rs², or

a ∝ 1 / , or

a ∝ 1 / T4/3, or

a ∝ 1 / Rt4/3.

Inserting the first acceleration into Newton’s second law leads to:

Fm / Rs²,

with force, F, and mass, m. The force is directed toward the Sun, with a magnitude inversely proportional to the square of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite force from each planet toward the Sun. Which is to say:

FM / Rs².

for mass, M. The combined law of gravitation is thus:

F = GmM/Rs²,

for some constant G.

Now consider gravity at the surface of the Earth. Newton showed that the gravitational force of a body would be the same if the mass were concentrated at its center, so the above equation can be used with the radius of the Earth, ρ²:

F = GmM/ρ² = mg,

with g as the acceleration of gravity on Earth. Then

g = GM/ρ².

If the known values of G, M, and ρ are inserted into this equation, the result is g = 9.8 m/s².


Let us now modify this derivation so that distance is the independent variable instead of time. This is better called a law of levitation since it is naturally directed toward the smaller mass. We have then three propositions:

  1. The Sun orbits each planet in a temporally circular path.
  2. The Sun is at the center of vass of its orbit.
  3. The pace of the Sun is constant.

Because the pace is constant, the prestination follows the equation for uniform circular motion derived previously:

b = / Rt,

in which u is the pace and b is the prestination. Again, distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

We have from the definition of pace:

u = T / S = 2πRt / S.

Elimination of u from these equations leads to

bRt / , or

bT / , or

bT / Rs².

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

, or

Rt².

Combining this with the equation for prestination yields

b ∝ 1 / Rt1/3, or

b ∝ 1 / T1/3, or

b ∝ 1 / S1/2, or

b ∝ 1 / Rs1/2.

Inserting the latter prestination into Newton’s second law in the form of surge, Γ, gives:

Γ ∝ ℓ / Rs1/2,

with surge, Γ, and vass, ℓ. The surge is directed away from the Sun, with a magnitude inversely proportional to the square root of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite surge toward each planet from the Sun. Which is to say:

Γ ∝ L / Rs1/2,

for vass, L. The combined law of levitation is thus:

Γ = HℓL/Rs1/2,

for some constant H. Since this is a function of the inverse square root instead of the inverse square, it is more sensitive to changes in Rs, as with a body in radial motion.

Consider levity at the surface of the Earth, using the above equation with the radius of the Earth, ρ:

Γ = HℓL/ρ1/2 = nh,

with h as the prestination of levity on Earth. Then

q = HL/ρ1/2.

The values for H, L, and ρ may be inserted into this equation to determine the value of h.

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