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Passage Through Resonance
Abstract
The effect of both gravitational and tidal torques on the spin rate of the planet Mercury is used as a model for a passage through resonance problem. Both the resonance and nonresonance aspects of the spin rate are studied by means of the method of averaging. On the basis of this analysis, it is concluded that the present 3:2 resonant state is due to nonresonant capture which evolved into resonant capture as the orbital eccentricity increased.
We present a mechanism for capture into resonance in perturbed two-frequency Hamiltonian systems. When an isolated attractor of the averaged system passes through a resonance on a time scale which is asymptotically slower than that on which the damping works, it transfers its domain of attraction to the resonance.
The mathematical cases studied in the current work lie between two extremes: when the equation theta-double dot = 0 is perturbed to a nonintegrable Hamiltonian system, and when the perturbation contains substantial friction. The problem is to determine the possible 'final values' of the angular velocity omega and to show how their number increases as the friction goes to zero. The application motivating the undertaking is to planetary spin resonance, in which the periodic orbit of a planet or satellite about its primary is given and the spin of the planet on its axis, assumed normal to the plane of the orbit, is studied.
The existence and properties of invariant manifolds for a certain class of finite-dimensional maps are examined. The theory of these manifolds is applied to the method of averaging and to a problem in celestial mechanics (i.e., the rotation of Mercury).
J. Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364–373; 26 (1974), pp. 638–669] studied resonance for a spinning reentry vehicle using a model system of ordinary differential equations with slowly varying coefficients. He and L. Lewin [SIAM J. Appl. Math., 35 (1978), pp. 738–754] gave formal multiple-time-scale expansions and numerical results to give a description of a mechanism for capture in sustained resonance. J. Sanders [SIAM J. Math. Anal., 10 (1979), pp. 1220–1243] studied these equations more rigorously using the method of averaging, but still could not prove the existence of sustained resonance. This paper continues the study of these equations using higher order averaging and the Melnikov method and shows rigorously that capture in sustained resonance does take place for some initial conditions. The Melnikov method measures the opening of a saddle connection for a small perturbation in terms of an integral. Since it has usually been applied to perturbations which depend periodically on time,...
Among the numerous cases of resonnances encountered in the solar system, many of the satellite to satellite and spin-orbit couplings can be explained as the result of an evolution driven by tidal forces. Models of such evolution are described and it is shown that they all lead to a similar type of reduced equations.
It is proposed to solve these equations in form of perturbation of a simpler “restricted tidal problem”. Such an approach gives a simple picture of the capture as well as conditions for a permanent capture into a resonant situation, as it is the case of the major satellite resonances and the Mercury rotation. In some other cases, involving higher order resonances, it is possible to have only temporary captures. This has probably happened many times to the Moon, inducing a lengthening of the evolutionary history of the lunar orbit as given by the classical secular deceleration theory.
This paper continues the mathematical study of spin/orbit coupling which was begun in earlier articles by the second author. The equation studied is
[(q)\ddot] = eP(t,q) + eT(t,[(q)\dot] )\ddot \theta = \varepsilon P(t,\theta ) + \varepsilon T(t,\dot \theta )
whereP andT are periodic int and the angle . No specific form forP andT is assumed but only the conditions necessary to analyze the system near resonance or a ground state. The behavior of an averaged system of these equations is shown to be reflected qualitatively in the actual equations by rigorous mathematical methods.
A well-known simple model of spin/orbit coupling leads to the equation
[(q)\ddot] = ef(t,q,[(q)\dot] )\ddot \theta = \varepsilon f(t,\theta ,\dot \theta )
periodic int and the angle . A mathematical study of this equation is carried out emphasizing the near-resonance behavior and the mechanism of resonance capture; the results are complementary to the author's previous global study showing how to locate the active responances. It is shown that the capture process is to a surprizing degree independent of the functional form off, making some of the detailed hypotheses of previous studies unnecessary.
In this paper some results on the Method of Averaging are presented. We give new conditions for the existence of attractive invariant tori together with some information on their domain of attraction. The results are based on invariant manifold theory and error estimation. An application to the rotational motion of Mercury is given.
The second-order nonlinear differential equation for the rotation of Mercury implies locked-in motion when the period is within the range where e is the eccentricity and T is the period of Mercury's orbit, the time t is measured from perihelion, and lambda is a measure of the planet's disiortion. For values near 2T/3, the instantaneous period oscillates about 2T/3 with period (21lambdae/2)T.
RECENTLY, Peale and Gold1 have shown that the non-synchronous rotation of Mercury is likely to be a consequence of tidal friction. They point out that in an eccentric orbit the spin of an axially symmetric planet will not relax to the orbital mean motion, but instead will approach a final value which is somewhat larger. The final spin rate will be somewhere between the mean orbital angular velocity and the orbital angular velocity at perihelion. The precise value for the final spin is determined by the condition that the net tidal torque on the planet around each orbit be equal to zero. The spin rate at which this condition is satisfied is determined by the frequency and amplitude dependence of the planet's `Q' (1/Q is the specific dissipation function2). According to Peale and Gold: ``The condition discussed here is based on the supposition that the solar torque exerted on the tidal bulge exceeds that exerted on any permanent deformation from axial symmetry. In the converse case a period of 88 days for the rotation would indeed result.'' This last statement, if true, would imply that Mercury's principal moments of inertia, in the plane perpendicular to its spin axis, differ by less than a few parts in 107. This value is very small when compared with the values known for the Moon-another solid, slowly rotating body.
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DURING the recent inferior conjunction of the planet Mercury in April, 1965, radar observations were obtained by the Arecibo Ionospheric Observatory in Puerto Rico (operated by Cornell University with the support of the Advanced Research Projects Agency under a Research Contract with the Air Force Office of Scientific Research). The system operated at a frequency of 430 Mc/s, with an antenna gain of 56 dB and a transmitted power of 2 MW. The resulting sensitivity was sufficient to obtain significant echoes not only from the nearest part of the planetary disk but also from more distant regions, removed by up to 0.06 of the planet's radius. By using short transmitted pulses of 500 µsec duration, it was possible to isolate the echo power from these more distant regions, and to carry out a Fourier analysis of their spectral composition.
SOLAR tidal friction must be an intense effect for Mercury, and it must be expected that the planet's spin would have relaxed from any original value to one that is under the control of this effect in a time short compared with the age of the solar system. The retarding torque exerted by the Sun on a planet is proportional to 1/r
6 (where r is the distance Sun-planet), a factor which is some 300 times greater for Mercury than for the Earth. For a planet on a circular orbit the final condition would then be one of synchronous rotation like the motion of the Moon with respect to the Earth. Mercury's motion around the Sun takes 88 days and for synchronous rotation the sidereal period would thus be 88 days also. The observed value of 59 ± 5 days differs markedly from this (see preceding communication).
IN a recent communication by S. J. Peale and T. Gold1 the rotational period of Mercury, determined from radar Doppler-spread measurements to be 59 +/- 5 days2, has been explained in terms of a solar tidal torque effect, taking into account the large eccentricity of Mercury's orbit, and the 1/r6 dependence of the tidal friction (r being the Sun-planet distance). They conclude from a very brief discussion that after slowing down from a higher direct angular velocity, the planet will have a final period of rotation between 56 and 88 days, depending on the assumed form of the dissipation function. However, from their discussion it is by no means clear why permanent deformations would imply a period of 88 days as a final rotation state after a slowing-down process. A very nearly uniform rotational motion of 58.65 sidereal-day period, that is 2/3 of the orbital period, may indeed be a stable periodic solution. This rotational motion could have the axis of minimum moments of inertia nearly aligned with the Sun-Mercury radius vector at every perihelion passage. The orbital angular velocity at perihelion (2phi/56.6 days) is close to 2phi/58.65 days, leading to an approximate alignment of the axis of minimum moment of inertia with the radius vector in an arc around perihelion where the interaction is strongest. The axial asymmetry of Mercury's inertia ellipsoid may result in a torque that counterbalances the tidal torque, giving a stable motion with this orientation and with a period two-thirds of the orbital period. It would therefore be possible for Mercury to have a higher permanent rigidity than that permitted by Peale and Gold.
The equations of motion for the rotation of Mercury are solved for the general case by an asymptotic expansion. The findings of Liu and O'Keefe, obtained by numerical integration of a special case, that it is possible for Mercury's rotation to be locked into a 2:3 resonance with its revolution, are confirmed in detail. The general solution has further applications.
The theory of the rotation of the planet Mercury is developed in terms of the motion of a rigid system in an inverse-square
field. It is possible for Mercury to rotate with a period exactly two-thirds of the period of revolution; there is a libration
with a period of 25 years.