Resonance

Introduction

Linear resonance

System of periods T(i) is in resonant state, if such whole numbers a(i) exist, that it holds:

 |∑a_i/T_i| < α 

where i=1..n, α is a small number. The value 1/α  is a period: T = (T₁/a₁,T₂/a₂,T₃/a₃,...) = 1/(a₁/T₁+a₂/T₂+a₃/T₃...) = 1/α

E.g. for periods J=11.862 and S=29.457 years with a1=2, a2=-5 we get: α = 2/J -5/S = 0.001133, i.e. T = 1/α = (J/2,S/5) = 883 years.

Order of resonance

Order of linear resonance is usually considered as number: 

 k= ∑|ai| 

So e.g. ratio 2:1 is of order 3, 3:2 of order 5 and 4:3 of order 7.
This definition is simple, but it servers rather for orientation only.  It does not suffice to judge quality of resonance (see comparison with music – quality of musical intervals and so on).

Resonance and music theory

While the music lacks a law of gravitation, (but binding of sensitive tones is similar to gravitational bond ...) research of musical simple ratios (consonant intervals and tuning systems, ...) is ahead of astronomy and celestial mechanics.

Considering musical resonances multiples of number two have special meaning (octave identity). Tuning of certain ratios cause mistuning of others (tuning is always a compromise). Music is alternation of stable (resonant) and unstable (temporary, chaotic) groupings. Music systems (eg 12-tone) are constructed in such a way, that stable formations could occur.

Types of resonances

Let P denote planets or asteroids and M moons;  P, M are orbital periods, Pr, Mr rotational periods.

According to combination of P,Pr,M and Mr we can distinguish 10 types of resonances

Nonlinear resonance

Nonlinear resonance depends not only on periods, but also on amplitudes of the partial movements.

Resonance of orbital periods

Resonance of orbital periods

The simplest case of resonance is integer ratio of two orbital periods:

 P/Q = n 

Trivial case of such resonance is resonance 1:1, e.g. in Solar system:

fixed rotation of planetary satellites (Moon, Galilean satellites of Jupiter, ...)

motion of bodies in Lagrange's points (Jupiter-Trojans, Dione-Helene, ..)

A.M.Molčanov (y.1968) analyzed the existence of simple ratios in the Solar System using a statistical analysis. S.F.Dermott (y.1969) attempted to express ratios of orbital periods by integers. Any nonlinear oscillation system gets -according to Molčanov - into synchronized movements mode (due to evolution) even under the influence of very weak ties.

Resonance with orbital period

In some cases motion of one body is so pronounced that it completely controls the motion of another body. For example, Jupiter affects the motion of thousands of asteroids and comets.

Let P,Q are sidereal periods of two bodies. We distinguish two elementary cases:

 (Q,P) = r∙P 

A/

Hence P/Q = (r+1)/r.  E.g. for  P=J a Q=A, where J is period of Jupiter and A period of asteroid, (see Effect of Jupiter).

Stable circular orbits do not exist for integer r (H.Scholl). Elliptical orbits should be stable as well as unstable(J.D.Hadjidemetriou).

 (Q,P) = P/r 

B/

Hence P/Q = r+1.

Rotation of resonant line

Integer ratios in the relation of periods (of Solar system bodies) usually are not realized exactly, cycles are generally modulated by other cycles. This more complex case is usually included under the term secular resonance.

In the case of motion of 2 satellites around the center two types of resonances are distinguished:

If one of the bodies have greater eccentricity, so called resonance of excentricity occurs (e.g. in pairs Enceladus-Dione, Titan-Hyperion, asteroid Hilda-Jupiter).

If one of the bodies have greater inclination of orbit so called resonance of inclination appears (e.g. Mimas-Tethys).

Both types (including their possible combinations) appear to be only different manifestation of the same principle: Conjunctions of bodies always occur near the point where are the respective orbits farthest.

Laplace‘s resonance

So called Laplace’s resonance is in case of 3 satellites the most significant form of resonance (Io-Europa-Ganymed, Miranda-Ariel-Umbriel). In this case, there are never conjunction of all three bodies at once. (A similar relation seems to tie rotation of the Venus with orbital periods of Venus and Earth.)

Laplace‘s resonance

Let P,Q,R are sidereal periods of three bodies and p,q,r integers.

So called synchronizing (Laplace's) resonance is defined by relation:

 I= (Q/q,P/p) = R/r 

Resonance disturbs regular motion of moons. Strong interaction of moons makes difficulties in calculations (Wargentin, Lagrange,Laplace, Souillart).

Special case: r=1, p=q-1.  Tj. 1/R-q/Q+(q-1)/P = 0 and hence (R,Q/(q-1)) = (Q,P/(q-1)) and  (Q,P) = q∙(R,P)  where q>1.

 1/I-3/E+2/G=0 

Period of  inequality I of two bodies is an integer divisor of orbital period of the third body.

Bodies avoid each other., e.g.:

(Special case - one period is rotational.)

Other resonances

Resonance of Uranian sattelites

In system of Uranian sattelites (Miranda - Ariel -Umbriel - Titania) the following relation (unstable resonance) appear two times:

 4/P-8/Q+3/R=0 

1/ for Miranda,Ariel and Titania (M/4,-A/8,T/3)  2/ for Ariel, Umbriel and Titania (A/4,-U/8,T/3).

 Similar relation holds also for Earth, Mars and Jupiter (E/4,-R/8,J/3) ≈ 1782 years.

Power resonance

   (V,E) = 13.227 years 
   (E,R) = 13.633 years 
   (V',-E'/2,R') = 0

   (R,S) = 3.3678 years 
   (V,R) = 3.357 years

   (J,S) = 88.830 years (J,U) = 30.438 years 
   (J,N) = 22.157 years (S,U) =177.050 years 
   (S,N) = 88.420 years 
   (J',-S',N') = 0

Stable resonance

Linear resonance is assumed to be stable, if

 ∑a_i = 0 

(in terms of previous denotation).

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) 

(The same does not hold for axial periods).

For any constants 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. period H of resonance 1/J-3/S+1/U+1/N = 1/H will be determined also by observer moving with any period M (with regard to stars):
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).

Trivial case of stable resonance is synodic "resonance" 1/P-1/Q=0, 1-1=0 (for P=Q).