Accreting & Decaying Orbits & Orbitals

(1) x = e ^ a is the equation for an Archimedean spiral.

(2) x = h * e ^ a is the physics equation for discrete

displacement in the spiral arc.

(3) A = e ^ x ,

(4) x = e ^ n

Both immediately preceding equations describe an accreting orbit. (4) represents a series of discrete, natural orbitals, similar to mosquito coil in configuration, beginning from the centre of the orbital. (3) represents the same orbit curve, contiguously, rather at full orbital intervals.

The decaying, discrete, natural orbital direction is

(5) x = e ^ - n [ max ].

(6) d = W/m * g ,

h * e ^ n ,

(7) h [o ] * e ^ n = W/m * g ,

(8) h = W * e ^ n / m * g ;

(9) C ~ x ,

(10) x = Pi * d ;

(11) x = e ,

(12) e / Pi ~ C / Pi ,

(13) ~ d ,

(14) e = Pi * ( Pi / 3 ) rad ,

(15) = ( Pi / 3 ) rad, according to the Pi ^ n rule

(16) x = h [o] * ( b * e ) ^ a ,

In nature,

(17) b >= 1

Conservation of work also translates to conservation of displacement. That dictates that an orbits lowest beginning with a higher than h [o], as the height, will asymptote by the necessity of conservation of displacement.

The asymptote will mark the outermost orbital, linear range or distance. The trajectory orbit will decay in a perfect mirror of the orbit's accretion. It will then repeat its earliest motion, all where there is not resistance to the motion.

Retro-power braking, such as retro-rocket braking or atmospheric air-breaking, will produce the result

(18) 0 < b < 1 ,

(19) x = h [o] * ( b * e ) ^ - a [ max ] .

The theory posed is applicable to all orbits and orbitals of every scale, in physics.

All orbits are elliptical. They are of the lowest and highest velocity orbital path (conventional elliptical orbit) when b = 1. In summary, there is a decaying elliptical orbit, an accreting, elliptical orbit and a stable elliptical orbit. They have different qualitative and quantitative orbital biases - proper rational bias, unitary rational bias and improper rational bias.

© Sydney Matinga 2026