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 *R _{s}* and

*R*. Then

_{t}*S*= 2π

*R*and

_{s}*T*= 2π

*R*. Also,

_{t}*R*=

_{s}*R*, and

_{t}v*R*=

_{t}*R*, with velocity,

_{s}u*v*, and allegrity,

*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:

- Each planet orbit the Sun in a spatially circular path.
- The Sun is at the center of mass of each planet’s orbit.
- 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* = *v²* / *R _{s}*,

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

*v* = *S* / *T* = 2π*R _{s}* /

*T*.

Elimination of *v* from these equations leads to

*a* ∝ *R _{s}* /

*T²*, or

*a* ∝ S / *T²*.

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

*T²* ∝ *R _{s}³*, or

*T²* ∝ *S³*.

Combining this with the equation for acceleration yields

*a* ∝ 1 / *R _{s}²*, or

*a* ∝ 1 / *S²*, or

*a* ∝ 1 / *T ^{4/3}*, or

*a* ∝ 1 / *R _{t}^{4/3}*.

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

*F* ∝ *m* / *R _{s}²*,

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:

*F* ∝ *M* / *R _{s}²*.

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

*F* = *G**mM*/*R _{s}²*,

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* = *G**mM*/*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:

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

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

*b* = *u²* / *R _{t}*,

in which *u* is the pace and *b* is the modulation. Again, distinguish the spatial and temporal versions of the radius, *R*, by using *R _{s}* and

*R*. Then

_{t}*S*= 2π

*R*and

_{s}*T*= 2π

*R*. Also,

_{t}*R*=

_{s}*R*, and

_{t}v*R*=

_{t}*R*, with velocity,

_{s}u*v*, and allegrity,

*u*.

We have from the definition of pace:

*u* = *T* / *S* = 2π*R _{t}* /

*S*.

Elimination of *u* from these equations leads to

*b* ∝ *R _{t}* /

*S²*, or

*b* ∝ *T* / *S²*, or

*b* ∝ *T* / *R _{s}*

*²*.

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

*T²* ∝ *R _{s}³*, or

*T²* ∝ *S³*, or

*R _{t}*

*²*∝

*S³*.

Combining this with the equation for modulation yields

*b* ∝ 1 / *R _{t}^{1/3}*, or

*b* ∝ 1 / *T ^{1/3}*, or

*b* ∝ 1 / *S ^{1/2}*, or

*b* ∝ 1 / *R _{s}*

*.*

^{1/2}Inserting the latter modulation into Newton’s second law in the form of *surge*, *Γ*, gives:

*Γ* ∝ ℓ / *R _{s}^{1/2}*,

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* / *R _{s}^{1/2}*,

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

*Γ* = *H**ℓL*/*R _{s}^{1/2}*

*,*

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 *R _{s}*, 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* = *H**ℓL*/*ρ ^{1/2}* =

*nh*,

with *h* as the modulation of levity on Earth. Then

*q* = *H**L*/*ρ ^{1/2}*.

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