Resonances

• Linear resonance
• Order of linear resonance
• Trivial resonance
• Simple resonances
• Basic resonances

• Resonances of solar tides
• Earth-Mars-Jupiter
• Beats of inner planets

Linear resonance

System of periods T(i) is in resonant state, if such whole numbers a(i) exist, that it holds:
abs(∑(a(i)/T(i)) < α,
where i=1..n, α is a small number.

We write period of resonance:
(T1/a1,T2/a2,T3/a3,...) = 1/(a1/T1+a2/T2+a3/T3...) = 1/α

Order of linear resonance

Order of linear resonance is usually defined as number:
k= ∑(|ai|).
This definition is quite simple. But in some cases it has no sense. E.g. in musical resonances multiples of number 2 have a special meaning (octave identity),...

Trivial resonance

Simplest type of orbital resonance - resonance 1:1.
In Solar system:

• fixed rotation of planetary satellites (Moon, galilean satellites of Jupiter, ...)
• motion of bodies in Lagrange points (Jupiter-Trojans, Dione-Helene, ..)

Simple resonances

a/ (Q,P) = n*P ,
where P,Q are sidereal periods of two bodies.
Hence P/Q = (n+1)/n.
Circular orbits do not exist (H.Scholl). Elliptical orbits should be stable as well as unstable (J.D.Hadjidemetriou).

b/ (Q,P) = P/n ,
where P,Q are sidereal periods of two bodies.
Hence P/Q = n+1.

Basic resonances

a/ I= (Q/q,P/p) = (P,Pa),
so called resonance of eccentricity (resonance of type e, resonance of e-type).
Symbols P,Q represent sidereal periods, Pa is anomalistic period, (P,Pa) is period of line of apsides.

Period of inequality I is equal to period of rotation of elliptical orbit (line of apsides) of one (generally smaller) body.

Conjunctions appear at places of greatest possible distance of bodies.

1/ near the pericentre of inner body:

• Enceladus - Dione (1:2)
  (1.370218 days/1, 2.736915 days/2) = -1065.087 days).
• asteroids of Hilda group - Jupiter (2:3)
  

2/ near the apocentre of outer body:

• Titan - Hyperion (3:4)
• Neptune - Pluto (2:3)

b/ I= (Q/q,P/p) = [(P,Pn),(Q,Qn)],
so called resonance of inclination (resonance of type i, resonance of i-type).
Symbols P,Q represent sidereal periods, Pn,Qn draconic periods. (P,Pn) and (Q,Qn) are periods of lines of nodes, [(P,Pn),(Q,Qn)] is period of motion of their axis.

Period inequality I is equal to axial period of lines of nodes.

Bodies meet in a furthest possible distance from intersection of their orbital planes.

• Mimas - Tethys (1:2)
  Both bodies move on orbits inclined more than 1° with a small eccentricity (Tethys practically on circle).
  (0.9424218 days, 1.887802 days/2)= -[-0.98468,-4.9841] y = 1.645 y
• Jupiter - asteroid Thule (4:3)
  Asteroid orbits have large inclinations.

c/ I= (Q/q,P/p) = R/r ,
so called synchronizing (Laplacean) resonance.
P,Q,R are sidereal periods of three bodies and p,q,r whole numbers.
Special case: r=1, p=q-1.
I.e. 1/R-q/Q+(q-1)/P = 0,
Hence (R,Q/(q-1)) = (Q,P/(q-1)) and
(Q,P) = q*(R,P)
where q>1. (For q=1 it is trivial resonance of 2 bodies).

Period inequality I of two bodies is a whole number divisor of orbital period of the third body.

Bodies avoid each other.

• Io-Europa-Ganymed (q=3)
  (1/I-3/E+2/G=0)
• Miranda-Ariel-Umbriel (q=3)
  (1/M-3/A+2/U=0)
• Venus rotation-Venus-Earth (q=5)
  (1/Vr-5/V+4/E=0; Special case - one period is rotational.)

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Resonances of solar tides

One of suggested models of solar activity is Wood model of influences of planet Jupiter, Earth and Venus on Sun. Configuration E-V-Sun-J or Sun-V-E-J repeats with period approximately 22 years.

E.g. configurations:
1727.87, 1748.63, 1793.40,	(20.76+44.77+44.77 y)
1838.17, 1882.94, 1903.70,	(44.77+20.76+44.77 y)
1948.47, 1993.25, 2038.01,	(44.78+44.76+20.77 y)

Let ((E,J)/5 ,(V,J)/3) = 2*W.

Then 1/2W = 3/V-5/E+2/J = 1/22.13505 let, W = 11.06753 years.

Line of conjunctions V-E-J move (backward) with period:
P= (W,J) = (11.06753, 11.861983) = 165.25 years.

Some authors identify solar period with axial period of Jupiter-Neptune, [J,N] = 22.13075 y = 2*11.06538 years. This period would equal to the period mentioned above only in case 6/V-10/E+3/J-1/N = 0 (unstable resonance).

Earth-Mars-Jupiter

During period of inequality Earth-Mars (1:2), i.e. during (E,R/2) = 15.7712 years, Jupiter gets 4/3 of its orbit (4/3 J= 15.8087 years).
Hence 3/J-8/R+4/E = 0 (unstable resonance, c. 1781 years).

Beats of inner planets

From integer ratios of pairs Mercury-Earth (E/M=4/1) and Venus-Mars (R/V=3/1) we get beats:
R1 = (E/4,M) = (1.0000/4, 0.2408) = 6.575 years,
R2 = (R/3,V) = (1.8808/3, 0.6152) = 32.82 years,
where R2/R1 is c. 5/1.
From this follows: 1/M-5/V-4/E+15/R = 0 (unstable resonance, c. 298 years).