System of periods T(i) is in resonant state, if such whole numbers a(i) exist, that it holds:
where i=1..n, α is a small number. The value 1/α is a period: T = (T_{1}/a_{1},T_{2}/a_{2},T_{3}/a_{3},...) = 1/(a_{1}/T_{1}+a_{2}/T_{2}+a_{3}/T_{3}...) = 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 linear resonance is usually considered as number:
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).
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.
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
Examples of resonance typesNonlinear resonance depends not only on periods, but also on amplitudes of the partial movements.
The simplest case of resonance is integer ratio of two orbital periods:
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, ..)
Types of arrangementLet us split the observed ratios of orbital periods by type of arrangement.
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.
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:
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).
B/
Hence P/Q = r+1.
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.
Resonance of excentricitySo called resonance of eccentricity of the orbit (resonance of type e, resonance of e-type).
The symbols P,Q represents sidereal periods, P_{a} is anomalistic period, (P,P_{a}) the period of rotation of the line of apses.
Period inequality I is equal to the period of rotation of elliptical orbit (line of apses) of one (usually smaller) body.
Conjunctions appear at places of greatest possible distance of bodies.
1/ near the pericentre of inner body:
Enceladus - Dione (1:2)2/ near the apocentre of outer body:
Titan - Hyperion (3:4) Neptunee - Pluto (2:3)(During 64 days Titan runs 4x and Hyperion 3x, their conjunction line turns with a period c. 18 years.)
So called resonance of inclination of the orbit (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.
Laplace Pierre Simon de , [laplas] 1749-1827, French mathematician, physicist, astronomer and politician, known for his treatment on celestial mechanics. He tried to create a general theory of mechanics, which describes the motion of celestial bodies, including all anomalies. Intervened in mathematical analysis and probability theory, suggested integral transformation of differential equations and probabilistic calculation method, now known as the "Monte Carlo". He introduced the generating functions. He is the author of hypothesis about the origin of the solar system from a rotating nebula (Kant-Laplace theory). |
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.)
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:
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.
Period of inequality I of two bodies is an integer divisor of orbital period of the third body.
Bodies avoid each other., e.g.:
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.)
In system of Uranian sattelites (Miranda - Ariel -Umbriel - Titania) the following relation (unstable resonance) appear two times:
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.
(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
Linear resonance is assumed to be stable, if
(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:
(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:
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).
Examples of stable resonancese.g.. (J,-S/2,U) (J,-S/3,U/2) (J/2,-S/3,U) i.e. 1/J-2/S+1/U, 1/J-3/S+2/U, 2/J-3/S+1/U, ...
1813.55, 1830.63, 1847.74, 1864.84, 1881.92, 1898.99, 1916.05, 1933.12, 1950.21, 1967.28, 1984.38
To synchronize these conjunctions it must hold: 17/E-33/R+16/S=0 (beats c. 415 years).
Difference
(L
_{S}-L_{U}
)
a
-(L
_{S}-L_{N}
)
is d =
L
_{S}-L_{U}+L_{S}-L_{N}
=
2∙L
_{S}-L_{U}-L_{N}
= 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 = 1/J-3/S+1/U+1/N.
So
In degrees:
d∙360° = 19.859∙(2/29.457-1/84.020-1/164.770)∙360° = 0.991433 ∙ 360° = 356.916° (1-d)∙360° = -3.084° (= 157.601° - 160.685°) h∙360° = -3.084° /19.859 years = 0.1553°/year. h = 0.1553/360 = 0.00043139 full angles / year. Period H: H = 1/h = 1/0.00043139 = 2318.1 years.
Let us assume fictive space, where only synodical periods can be perceived (motion of bodies is hidden).
E.g. we can hear a crack (AB) during conjunction of two bodies (A and B); during conjunction of three bodies (A,B,C) three cracks (AB, AC, and BC).
(Similarly was world observed by ancient astronomers...)
Motion of an observer in the fictive space of synodical periods does not modulate orbital periods, but synodical periods.
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:
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 years 1/S' = 1/S-2/N+1/U = 29.6636 years 1/U' = 1/U-2/N+1/U = 2/U-2/N = 85.722 years 1/N' = 1/N-2/N+1/U = 1/U-1/N = 171.444 years
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 fulfill equation:
For synodic 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 m2/P-n2/Q = k/H, so P∙Q/(Q∙m2-P∙n2) = H/k. For comparison Bohr's quantization of atoms: 1/T = c∙R∙(1/m2-1/n2)
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
Values L=(3L_{J}-8L_{S})-(2L_{U}-7L_{N}), where L_{J},L_{S},L_{U},L_{N} are longitudes of planets in selected moments oscillates approximately around 120˚:
In conjunctions J-S is (3L_{J}-8L_{S})/5 = L_{J} = L_{S}, in conjunctions U-N is (2L_{U}-7L_{N})/5 = L_{U} = L_{N}.
Opposition J-UYear | 3 L_{J}[˚] | 8 L_{S} [˚] | 2 L_{U} [˚] | 7 L_{N} [˚] |
(3L_{J}-8L_{S})-(2L_{U}-7L_{N}) [˚] |
1810,46 | 124 | 230 | 83 | 309 | 254 – 134 = 120 |
1824,28 | 322 | 139 | 215 | 157 | 183 - 58 = 125 |
1838,09 | 141 | 51 | 334 | 7 | 90 – 327 = 123 |
1851,90 | 300 | 320 | 80 | 219 | 340 – 221 = 119 |
1865,70 | 98 | 229 | 185 | 73 | 229 – 113 = 116 |
1879,52 | 272 | 142 | 302 | 287 | 131 - 15 = 116 |
1893,33 | 108 | 49 | 72 | 142 | 58 – 290 = 128 |
1907,15 | 306 | 325 | 204 | 356 | 341 – 208 = 133 |
1920,97 | 127 | 230 | 325 | 208 | 257 – 116 = 141 |
1934,77 | 287 | 146 | 72 | 59 | 142 - 13 = 129 |
1948,58 | 84 | 49 | 176 | 268 | 35 – 269 = 127 |
1962,39 | 257 | 326 | 291 | 115 | 291 – 176 = 115 |
1976,21 | 91 | 230 | 61 | 323 | 221 - 97 = 123 |
1990,02 | 290 | 148 | 193 | 172 | 142 - 21 = 121 |
2003,84 | 113 | 52 | 315 | 22 | 61 – 293 = 128 |
2017,65 | 275 | 328 | 63 | 234 | 307 – 189 = 118 |
2031,45 | 71 | 233 | 167 | 88 | 198 - 80 = 118 |
2045,26 | 242 | 146 | 281 | 302 | 95 – 339 = 116 |
2059,08 | 74 | 55 | 50 | 157 | 19 – 253 = 126 |
2072,90 | 274 | 326 | 183 | 11 | 307 – 172 = 136 |
2086,71 | 98 | 238 | 306 | 223 | 220 - 83 = 138 |
We will try to look for some period h that can have for inner planets meaning similar to period H of outer planets.
Period hLet 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: