Inverse motion

• Inverse motion
• Motion of planets Uranus a Neptune
• Period H
• Resonance of outer planets

Inverse motion

Let us have line r moving with period R around centre. With regard to line r bodies P and Q move inversely if r is oriented axis of motion of these two bodies, i.e. when R=[P,Q].
Bodies open and then close angle with regard to r with synodic period (P,Q).

Let us assume, it is not necessary that this motion run continuously. Let the condition of inverse holds at every instant t = k*T, where k is whole number and T time interval (period).

It holds at every instant t:
Lp-Lr = Lr-Lq,
where Lp,Lq and Lr are longitudes of bodies P,Q and Lr is longitude of line r.
So
frac(T/P)-frac(T/R) = frac(T/R)-frac(T/Q), i.e. 2*frac(T/R) = frac(T/P)+frac(T/Q)

Motion of planets Uranus a Neptune

Planets Uranus and Neptune exert approximate inverse motion with regard to places of conjunctions of Jupiter and Saturn. During time t = (J,S) = 19.859 y positions of outer planets change in average by angles:

Planet         Longitude
Jupiter        Lj= frac((J,S)/J)*360° = 242.7°
Saturn         Ls= frac((J,S)/S)*360° = 242.7°
Uranus         Lu= frac((J,S)/U)*360° = 85.1°
Neptune        Ln= frac((J,S)/N)*360° = 43.4°

Differences of longitudes:
Lj-Lu = 242.7*360°-85.1*360° = +157.6°
Ln-Lj = 43.4*360°-242.7*360° = -199.3*360° = +160.7°

Deviation is c. 160.7*360°-157.6*360° = 3.1°. Angle 3.1° during 19.859 years corresponds to 360° during c. 2320 years.

Inverse motion U and N with regard to (J,S) is modulated by period H, H is c. 2320 years.

Period H

Difference (Ls-Lu) and -(Ls-Ln) is d = Ls-Lu+Ls-Ln = 2*Ls-Lu-Ln = 2*(J,S)/S -(J,S)/U-(J,S)/N = (J,S)*(2/S-1/U-1/N).
Deviation from full angle during (J,S): (1-d) = 1 - (J,S)*(2/S-1/U-1/N).

During 1 year:
h = (1-d)/(J,S) = 1/(J,S) - (2/S-1/U-1/N) = 1/J-1/S-2/S+1/U+1/N
h = 1/J-3/S+1/U+1/N.

So
1/H = 1/J-3/S+1/U+1/N.

In degrees:
d*360° = 19.859*(2/29.457-1/84.020-1/164.770)*360°
d*360° = 0.991433 * 360° = 356.916°
(1-d)*360° = -3.084° (= 157.601° - 160.685°)
h*360° = -3.084° /19.859 let  =  0.1553°/year.
h = 0.1553/360 = 0.00043139 full angles / year.
Period H:
H = 1/h = 1/0.00043139 = 2318.1 years.

Resonance of outer planets

Resonant ratio of orbital periods of Uranus and Neptune is 1:2 (N/U =1.961); period of inequality I = (U, N/2), approximately 4200 years.
Observer moving with period I gets periods of outer planets J',S',U',N':

1/J' = 1/J-2/N+1/U = 11.8953 y
1/S' = 1/S-2/N+1/U = 29.6636 y
1/U' = 1/U-2/N+1/U = 2/U-2/N = 85.722 y
1/N' = 1/N-2/N+1/U = 1/U-1/N = 171.444 y

For this observer N':U' is exactly 2/1. Ratio S'/J' is approximately 5:2 and U'/S' approximately 3:1.
Period of inequality J-S: (S'/5,J'/2) = 2362 y
and period of inequality S-U: (U'/3,S'/1) = 778 y.
Value of period H (1/H = 1/J-3/S+1/U+1/N) remain the same: H = 2320 y.

Sidereal periods of outer planets fulfil equation:
3/J-8/S-2/U+7/N = 0.

Our observer therefore realizes:
5/S'-2/J'=1/H (=5/S-2/J+3/U-6/N=1/J-3/S+1/U+1/N)
3/U'-1/S'=3/H (=5/U-4/N-1/S =3/J-9/S+3/U+3/N)

It holds:
1/H = 1/J- 3/S+1/U+1/N
3/H = -1/S+5/U-4/N
5/H = -1/J+1/S+9/U-9/N
7/H = 4/J-13/S+9/U

And for synodical periods:
1/H = 1/(J,S)-2/(S,U)-1/(U,N)
3/H = 4/(U,N)-1/(S,U)
5/H = 9/(U,N)-1/(J,S)
7/H = 4/(J,S)-9/(S,U)

Generally m^2/P-n^2/Q = k/H, tedy P*Q/(Q*m^2-P*n^2) = H/k.
Bohr's quantization of atoms: 1/T = c*R*(1/m^2-1/n^2).