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Vladimir Ladma

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Stable resonance

Linear resonance is assumed to be stable, if
∑a(i) = 0.

Let us have n-periods P(i): {P0, P1,..., Pn} in system P. For observer from the system Q, whose period of motion with regard to the system P is M, these periods appear to be (P(i),M), i.e. {(P0,M), (P1,M)..., (Pn,M)}.

Synodical periods observed from the system Q are equal to synodical periods observed inside of P:
((P(i),M), (P(j),M)) = (P(i), P(j)).
But the same does not hold for axial periods:
[(P(i),M), (P(j),M)] != [P(i), P(j)].

For any constant a(i) is ∑(a(i)/(P(i),M)) = ∑(a(i)/P(i)) - ∑(a(i))/M. So in case of stable resonance (∑(a(i))/M= 0) we get
∑(a(i)/(P(i),M)) = ∑(a(i)/P(i)); so, stable resonance is independent on selection of reference system.

E.g. observer moving with any period M (with regard to stars) will get also period H of resonance 1/J-3/S+1/U+1/N = 1/H:
1/(J,M)-3/(S,M)+1/(U,M)+1/(N,M) = 1/H, because 1/J-3/S+1/U+1/N is stable (1-3+1+1=0).

Stable resonances of J-S-U

Period of 320 years has far less popularity than 180 years period of outer planets. But accuracy, with which it covers synodical periods of outer planets, is not so small.

All periods of simple stable resonances J-S-U ( e.g. 1/J-2/S+1/U, 1/J-3/S+2/U, 2/J-3/S+1/U, ...) are divisors of this period about 320 years. (Synodic "resonance" 1/P-1/Q is a trivial case of stable resonance, 1-1=0.)

It holds:
7/J-23/S+16/U = 0 (stable resonance, c. 28600 years).

1. 317.742 16*(J,S)
2. 158.871 8*(J,S), (J,-S/3,U/2)
3. 105.914 3*(J,-S/2,U)
4.  79.435 4*(J,S), Gleisberg period
7.  45.392 (S,U)
8.  39.718 2*(J,S)
9.  35.305 (S,N), Bruckner period
14. 22.696 2*W,  Hale period
16. 19.859 (J,S)
23. 13.815 (J,U)
25. 12.710 (J/2,-S/3,U), (J,N)?, conjunctions of inner planets?

Spectrum contains some known periods (Gleisberg, Bruckner,.., see ratio 1:4:9:16:25).

Observer in the world of synodic periods

Let us imagine an observer in the fictive space of synodical periods. His motion does not modulate orbital periods, but synodical periods.

Observing of outer planets

Let H is period of stable resonance, 1/H=1/J-3/S+1/U+1/N (c. 2320 years). Observer moving with this period (in the world of combined synodical periods) will get the following values of ((J,S),(S,N)), ((J,S),(U,N)) and ((J,U),(U,N)):

 1/((J,S),(S,N))-1/H = 1/J-2/S+1/N-1/H	   = 1/S-1/U = 1/(S,U)
 1/((J,S),(U,N))-1/H = 1/J-1/S-1/U+1/N-1/H = 2/S-2/U = 2/(S,U)
 1/((J,U),(U,N))-1/H = 1/J-2/U+1/N-1/H	   = 3/S-3/U = 3/(S,U)

Therefore for this observer it holds:

1/((J,U),(U,N)) : 1/((J,S),(U,N)) : 1/((J,S),(S,N)) = 1 : 2 : 3

Observing of inner planets

Let H is period of stable resonance, 1/H = 1/M-4/V+2/E+1/R (c. 5.504 years). Observer moving with this period (in the world of combined synodical periods) will get the following values of ((M,V),(V,R)),((M,V),(E,R)) and ((M,E),(E,R)):

 1/((M,V),(V,R))-1/H = 1/M-2/V+1/R-1/H	   = 2/V-2/E = 2/(V,E)
 1/((M,V),(E,R))-1/H = 1/M-1/V-1/E+1/R-1/H = 3/V-3/E = 3/(V,E)
 1/((M,E),(E,R))-1/H = 1/M-2/E+1/R-1/H	   = 4/V-4/E = 4/(V,E)

Therefore for this observer it holds:

1/((M,V),(V,R)) : 1/((M,V),(E,R)) : 1/((M,E),(E,R)) = 2 : 3 : 4