Circular orbits

Knowing, space & time / By /

*** Revised from June 2017 ***

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the length space circumference S, and the duration space circumference T, which is known as the period. Distinguish the length and duration versions of the radius: R, for base, and Q, for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, 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 from the perspective of the Sun toward each orbiting planet:

  1. Each planet orbits the Sun in a circular path with radius R in 3D space.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is a constant, v.

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 = v² / R,

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

v = S / T = 2πR / T.

Elimination of v from these equations leads to

a ∝ R / T²,

or a ∝ S / T²,

or a ∝ S / Q².

Kepler’s third law states that the orbital period (temporal circumference), T, and the semi-major axis, A, are related as T² ∝ A³. For circular orbits A is R and this becomes

T² ∝ R³,

or T² ∝ S³.

Combining this with the equation for acceleration yields

a ∝ 1/R²,

or a ∝ 1/S²,

or a ∝ 1/T4/3,

or a ∝ 1/Q4/3.

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

F ∝ m / R²,

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, there is an equal and opposite force of levitation that is:

F ∝ M / R².

for mass, M. The combined law of gravitation (or gravitation-levitation) is thus:

F = GmM/R²,

for some constant G.

Now consider gravity for (falling) objects 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 from the perspective of the Earth, toward each transiting celestial body:

  1. Each Sun orbits each planet in a circular path with period T in 3D time.
  2. Each planet is at the center of vass for their orbiting Sun in 3D time (see here).
  3. The pace of the orbiting Sun in 3D time is a constant, u.

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

b = u² / Q,

in which u is the pace and b is the relentation. Again, distinguish the stantial and temporal versions of the radius: R for space and Q for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, u.

We have from the definition of pace:

u = T / S = 2πQ / S.

Elimination of u from these equations leads to

b ∝ Q / S²,

or b ∝ T / S²,

or b ∝ T / R².

Kepler’s third law interpreted for 3D time states that the orbital spatial circumference, S, and the semi-major time axis, C, are related as S² ∝ C³. For circular orbits C is Q and this becomes

S² ∝ Q³,

or S² ∝ T³,

or R² ∝ T³.

Combining this with the equation for relentation yields

b ∝ 1/Q²,

or b ∝ 1/T²,

or b ∝ 1/S4/3,

or b ∝ 1/R4/3.

Inserting the latter relentation into Newton’s second law in the form of release, Y, gives:

Y ∝ n / Q²,

or Y ∝ n / T².

with release, Y, and vass, n. The release is directed away from the Earth, with a magnitude inversely proportional to the square of the Sun’s period around the Earth in 3D time.

Because of Newton’s third law, there is an equal and opposite release toward each celestial body from the Earth. Which is to say, there is an equal and opposite release of gravitation that is:

Y ∝ N / Q²,

or Y ∝ N / T²,

for vass, N. The combined law of levitation (or gravitation-levitation) is thus:

Y = LnN/Q²,

or Y = LnN/T²,

for some constant L.

Consider levity near the surface of the Earth as it rises to meet nearby bodies, using the above equation with their period P:

b = LnN/P² = nb,

with h as the relentation of levity near the Earth. Then

b = LN/P².

The levity of bodies near the Earth, b, is determined from the values for L, N, and P.

So the switch from 3D space to 3D time results in an exchange of space and time variables.