Orbit Distance With Time Relationship

The orbital distance (r) of any planet from the Sun has a definite relationship with the time it takes to circumnavigate the star. Considering the time (comparatively in P days) taken for a planet's 'year' and the distance from the central star either in millions of kilometres (mkm) or AU (Astronomical Units), a constancy is apparent.

Orbital Period (P days)

Mercury

• P = 88, P² = 7744

Venus

• P = 224.7, P² = 50490

Earth

• P = 365.25, P² = 133408

Mars

• P = 686.9, P² = 471832

Asteroid Belt
   
Jupiter

• P = 4331.87, P² = 18765098

Saturn

• P = 10760.27, P² = 115783411

Uranus

• P = 30684.65, P² = 941547746

Neptune

• P = 60189.55, P² = 3622781929

The Asteroid Belt is spread over a wide distance

Planetary distance (r) from the Sun (mkm, AU)

The respective planetary values below are for r, r³, AU, AU³

Mercury

• 57.9, 194105, 0.387, 0.058

Venus

• 108.2, 1266723, 0.723, 0.378

Earth

• 149.6, 3348072, 1.000, 1.000

Mars

• 227.9, 11836764, 1.523, 3.533

Asteroid Belt

Jupiter

• 778.3, 471455918, 5.203, 140.852

Saturn

• 1427, 2905841483, 9.539, 867.978

Uranus

• 2871, 23664622310, 19.190, 7066.835

Neptune

• 4497, 90942871470, 30.060, 27162.324

According to Kepler's third law

The respective planetary values below are for P²/r³, P²/AU³

Mercury 

• 0.0399, 133517

Venus

• 0.0399, 133571

Earth

• 0.0398, 133408

Mars

• 0.0399, 133550

Asteroid Belt                            

Jupiter

• 0.0398, 133226

Saturn

• 0.0398, 133394

Uranus

• 0.0398, 133235

Neptune

• 0.0398, 133375

Average

• 0.0398, 133410

In both cases the constancy is clearly demonstrated and shows a relationship between a 2D quantity (function²) and a 3D quantity (function³).

• A planet's 'year' will be completed after P days and it will travel through 360°. Relative to Mercury as the innermost planet and consequently the fastest moving, each of the other seven planets will move through less than this full circle. In the time taken for one complete orbit of Mercury (360°), the planet Venus will have moved only 88/224.7 * 360 = 140.99° and Earth by 88/365.25 * 360 = 86.735° and so on.
  

Note: the ellipticity of a planet's orbit is ignored
for the sake of this empirical argument

Orbital Period (P days), Orbital (Circular) Angle (θ°)

Mercury

• P = 88, θ = 360.00°

Venus

• P = 224.7, θ = 140.99°

Earth

• P = 365.25, θ = 86.735°

Mars

• P = 686.9, θ = 46.12°

Asteroid Belt
       
Jupiter

• P = 4331.87, θ = 7.313°

Saturn

• P = 10760.27, θ = 2.944°

Uranus

• P = 30684.65, θ = 1.032°

Neptune

• P = 60189.55, θ = 0.526°

The following values are derived from P and θ:

• θ² = 129600.00, AU³ = 0.058, θ² * AU³ = 7516.80

• θ² = 19878.18, AU³ = 0.378, θ² * AU³ = 7513.95

• θ² = 7522.96, AU³ = 1.000, θ² * AU³ = 7522.90

• θ² = 2127.05, AU³ = 3.533, θ² * AU³ = 7514.87

• θ² = 53.48, AU³ = 140.852, θ² * AU³ = 7532.76

• θ² = 8.667, AU³ = 867.978, θ² * AU³ = 7522.77

• θ² = 1.0658, AU³ = 7066.835, θ² * AU³ = 7531.83

• θ² = 0.2767, AU³ = 27162.324, θ² * AU³ = 7515.82

In all cases the constancy is clearly demonstrated and again shows a relationship between a 2D quantity (function²) and a 3D quantity (function³).