Lorentz Andgravitational Resonances on Circumplanetary Particles

Mi ron-sized ir umplanetary dust parti les are subje t to various non-gravitational perturbations, prin ipally solar radiation pressure and ele tromagneti for es, whi h are typi ally a few per ent as strong as the planetary gravity. Individually, these perturbations an ause some orbital evolution, but when the perturbations a t in on ert the ex ursions an be mu h larger. We demonstrate this e e t for a single example, the oupling between resonan es and drag for es. Throughout this work, we emphasize the parallels between satellite-satellite gravitational resonan es and their ele tromagneti ounterparts (Lorentz resonan es). INTRODUCTION A dynami al system typi ally has a set of natural frequen ies at whi h it an rotate or vibrate. When su h a system is for ed at one of these natural frequen ies (or a multiple of it), the amplitude of os illations grows as a result of the umulative e e t of in-phase perturbations; the system is said to be in resonan e. A hild on a swing provides a familiar example of a resonant system. If the swing (initially at rest) is pushed at an arbitrary frequen y or at random times, the amplitude of os illation is likely to remain small; if, however, the swing is pushed on e per period, the os illation amplitude will grow quite large. In an entirely similar manner, harged dust grains os illate wildly near the lo ations of \Lorentz resonan es" whi h o ur at those positions where the ele tromagneti for e sensed by an orbiting parti le (and arising from a planet's spinning magneti eld) has a omponent that mat hes a natural frequen y of the orbit /1/. The abrupt verti al expansion of the jovian ring into its halo and the disappearan e of the halo itself /1,2/ have been as ribed to the a tion of these Lorentz resonan es on orbiting dust grains. Gravitational resonan es o ur when the orbital periods of two obje ts are nearly a simple ratio of integers. Many features in the main saturnian ring system have been su essfully attributed to gravitational resonan es with exterior satellites. For example, the 2:1 resonan e with Mimas de nes the inner edge of the Cassini division, whi h divides the A and B rings, while the sharp outer edge of the A ring o urs at a 7:6 resonan e with the moon Janus. Satellites themselves are often found in resonan es with one another; examples in lude the saturnian pairs En eladus/Dione, Titan/Hyperion and Mimas/Tethys, as well as the jovian triple Io/Europa/Ganymede (see /3/ for a qualitative physi al des ription of these gravitational resonan es). In this paper we wish to illustrate how resonan es ouple with drag for es. This idea is not new; indeed it has been extensively studied in the ontext of satellite evolution where tidal e e ts from the entral body reate small drags on satellite orbits. This problem has been thoroughly treated using Hamiltonian me hani s (see e.g., /4/). The purpose of the urrent paper is twofold. First, we wish to draw parallels between the extensively studied satellite (gravitational) resonan es and their less well known relatives, Lorentz resonan es. Se ondly, we will reprodu e some results of the Hamiltonian theory using the Lagrangian orbital perturbation equations /5/, whi h are written in terms of the orbital elements. The latter quantities provide a physi ally meaningful des ription of an orbit; for orbits on ned to a parti ular plane, the semimajor axis a, the e entri ity e, and the longitude of peri enter ~ ! are suÆ ient. These three elements, respe tively, des ribe the instantaneous size, shape, and orientation of an ellipti al orbit; the Lagrangian equations that des ribe the time rate of hange of su h orbital elements are well suited to visualizing the results of orbital perturbations. The advantage of our approa h is its simpli ity: many non-intuitive e e ts of resonan es, su h as resonant trapping and jumps, will be elu idated. RESONANCE EQUATIONS The problem of determining the perturbing e e ts of one satellite on another is fundamental to elestial me hani s and has been studied for enturies. It is not solvable in losed form, but an approximate solution an be developed as a power series of small quantities. The typi al pro edure ( f. /5/, p. 339) is as follows. First, one evaluates the disturbing fun tion, de ned as the negative of the perturbing satellite's potential, at the position of the perturbed parti le. Next, the disturbing fun tion is written in terms of the orbital elements; this step requires ompli ated power series expansions in e entri ities, in linations, and the semimajor axis ratio. Finally the hanges to the orbital elements an be al ulated with the potential form of Lagrange's planetary equations (/5/, p. 336) whi h relate the time rates of hange of the orbital elements to derivatives of the disturbing fun tion and to instantaneous values of the elements themselves. We pro eed in a similar manner for Lorentz resonan es. Be ause the Lorentz for e due to a magneti eld annot be derived from a potential, we must al ulate the ele tromagneti for e arising from an arbitrary magneti eld and express it in terms of orbital elements, an arduous task whi h requires power series expansions in the parti le's e entri ity and in lination. These for es are then inserted into an alternate form of Lagrange's planetary equations (/5/, p. 327). The results of this al ulation yield, as above, expressions for time derivatives of the orbital elements whi h are fun tions of the instantaneous values of these elements. We plan to submit the details of this al ulation for publi ation in I arus. In both of the above derivations, se ular terms (i.e., those that do not depend on satellite longitudes) as well as periodi terms (with longitude dependen e) appear. Se ular terms are ubiquitous, whereas periodi terms, over long times, average to zero at all but a few resonant lo ations. In this paper we fo us on one of these lo ations as an example: the 2:1 ( rst-order) e entri ity resonan e. Near this lo ation, the resonant argument is given by: = 2 0 + ~ !; (1) where and 0 are the longitudes of the perturbee and perturber, respe tively. At the resonant lo ation (de ned by _ = 0 see gure 1), the perturbed body ompletes approximately two orbits for every one y le of the perturbing for e (the period of an exterior satellite in the gravitational ase or the planetary spin period for Lorentz resonan es). We ignore all periodi terms with di erent frequen y dependen ies (sin e they average to zero), and the se ular perturbations (whi h are small ompared to the strong 2:1 resonant terms). The orbital elements most strongly a e ted by su h a resonan e are the abovementioned a, e, and ~ !. Instead of the semimajor axis a, we use the unperturbed orbital mean motion n _ , whi h is related to the semimajor axis via n2a3 = GMp, where G is the gravitational onstant and Mp is the planetary mass (/5/, p. 131). Writing out the Lagrange perturbation equations to lowest order in e entri ity and in lination, we nd that the e e ts of both the gravitational and Lorentz versions of the 2:1 rst-order e entri ity resonan e an be represented by a set of equations of the following form: dn dt = 3en2 sin (2a) de dt = nA1 sin (2b) d~ ! dt = nA2 e os : (2 ) Here t is time, (always positive) measures the appropriate resonan e strength and the Ai are onstants. The quantity is a ompli ated fun tion of the semimajor axis ratio whi h must be expanded as a power series; a ross the small distan e over whi h the resonan e exerts its in uen e, however, an be treated as a onstant. In the gravitational ase, is rst order in the satellite/planet mass ratio and A1 = A2 = 1. In the Lorentz ase, depends on the parti le's harge-to-mass ratio, distan e from the planet, and the magneti eld strength. For rst-order ele tromagneti resonan es, A1 = A2 = n=n0 1, so the 2:1 resonan e, like gravity, has A1 = A2 1. The dominant ontribution to this resonan e omes from the g32 omponent of the magneti eld (a non-symmetri o tupole term see /2/ whi h gives values for the giant planets). Although we have spe ialized equations (2a) to the 2:1 e entri ity resonan e, the form of the equations for other rst-order e entri ity resonan es (2:3, 3:4, 1:2 et .) is entirely similar only the parameters and the Ai need to be hanged. First-order in lination resonan es (whi h exist for Lorentz for es but not for satellite gravity) and higher-order resonan es are also not too di erent. A ordingly, the general behavior dis ussed below for the 2:1 e entri ity resonan e a tually applies to a wide variety of other types of resonan es as well; that is to say, the trapping and jumps dis ussed below are general phenomena. Fig. 1. S hemati diagram showing the entral planet, the orbiting dust grain, and the 2:1 resonan e. The outermost line represents the lo ation of the perturbing satellite (for a gravitational resonan e) or of syn hronous orbit (for a Lorentz resonan e). A grain drifting through a rst-order resonan e toward this lo ation may be ome trapped while one drifting away from it will experien e a jump. DRAG FORCESSeveral drag for es operate in the magnetospheres of the giant planets. Most large satellites aredriven slowly outward by tidal for es from the primary while small parti les are a e ted by a host ofpro esses /6/ in luding plasma, atmospheri , and Poynting-Robertson drags whi h, for dust grains,operate mu h more rapidly than tidal evolution. Be ause drag for es are typi ally mu h smaller thanmany other orbital perturbations, their e e ts on most orbital elements an often be ignored. Unlikemost other perturbations, however, drag for es systemati ally a e t an orbit's energy and thereforeits size and mean motion. Furthermore, be ause of the limited radial extent of the resonan e zone,we an approximate the fun tional form of the drag rate in this region by a simple onstant _ndrag.The in lusion of drag for es requires that we repla e equation (2a) withdndt = 3en2 sin + _ndrag:(3)RESONANCE TRAPPINGWhen _ndrag < 0, orbits evolve outward: near the 2:1 resonan e, this evolution is toward the perturb-ing satellite (in the ase of gravity) or toward syn hronous orbit (in the Lorentz ase). For this typeof evolution, resonan e trapping, in whi h the evolution in mean motion eases, is possible ( gure1). Clearly trapping an o ur only if the rst term in equation (3) is equal and opposite to these ond for some . Solving equation (3) for sin in this ase and substituting into equation (2b),we nddedt trapped = _ndragA13ne ;(4)whi h is easily integrated yielding:e =e20 2 _ndragtA13n1=2:(5)Linearizing equations (2a) around this solution, we nd that it is stable against small perturba-tions. Note the remarkable fa t that the rate of growth of the e entri ity given by equation (5) isindependent of the resonan e strength . This result an also be obtained from equation (7) below,whi h expresses the onservation of energy in a rotating referen e frame (see /2/). Thus the \squareroot growth" in time (equation 5) is a property shared by gravitational and Lorentz resonan es of alltypes and orders. An example of resonant trapping and the asso iated e entri ity growth is shownin gure 2; for the parameters given in the gure aption, equation (5) redu es to e 0:00145N1=2(N is the number of perturber orbits) in rough agreement with the gure. This behavior holds untile 0:5 at whi h time higher-order e e ts be ome important.JUMPS AT RESONANCEWhen _ndrag > 0, inner orbits evolve away from the perturbing satellite (or from syn hronous orbit).In this ase trapping for low e entri ities is not possible as an be seen from equation (5) whi h im-plies that e entri ity be omes imaginary! Instead we shall nd a di erent behavior at the resonantlo ation.Be ause drag for es are so small, the rst term in equation (3) is usually far greater than the se ond;this fa t allows us to obtain an adiabati invariant. Ignoring the drag term for the moment, we divideequation (2a) by equation (2b) and nd dnde = 3enA1 ;(6)whi h an be integrated to yieldln nn = 3e22A2(7)where n is an integration onstant. Re alling that equation (2a) are a urate to only rst orderin e entri ity, we solve this equation to lowest order in e and nd thatn = n 1 3e22A1(8)is a onserved onstant of the motion (see /7/). Sin e the resonan e zone is traversed qui kly,equation (8) remains approximately onstant during the passage. The half-width of the libratingzone, dn=2, an be rudely estimated by setting the derivative of equation (1) equal to zero, takingn = 2n0 + dn=2 and os = 1, and solving for dn. We nd dn 2n A2=e. Inserting this into equation(6), and negle ting the di eren e between de and e, we nd:de = 2A1A231=3:(9)This ase is displayed in gure 3; using the parameters from the gure aption, we al ulate thejump amplitudes from equations (9) and (6) and obtain de 0:04 and dn 0:012 values smallerthan, but in rough agreement with, the gure.DISCUSSIONLorentz and gravitational resonan es di er primarily in the magnitudes of the resonant strength. For mi ron-sized dust grains around the jovian planets, is orders of magnitude larger in theLorentz ase; thus Lorentz resonan es are more e e tive at trapping dust parti les and are able toindu e larger orbital jumps than resonan es due to a satellite's gravity. Slight additional di eren esarise when Ai 6= 1; most rst-order Lorentz resonan es have Ai < 1 whi h redu es the trapped growthrate (equation 5) and jump amplitude (equation 9). Despite this small di eren e between the twotypes of resonan es, the equations that govern them are remarkably similar and, onsequently, it isnot surprising that orbital behavior at Lorentz and gravitational resonan es is so alike.REFERENCES1. J.A. Burns, L.E. S ha er, R.J. Greenberg and M.R. Showalter, Lorentz resonan es and thestru ture of the jovian ring, Nature 31, 115-119 (1984).2. L.E. S ha er and J.A. Burns, Lorentz resonan es and the verti al stru ture of dusty rings:Analyti al and numeri al results, I arus 96, 65-84 (1992).3. R. Greenberg, Orbit-orbit resonan es among natural satellites, in: Planetary Satellites, ed. J. A.Burns, Univ. Arizona, Tu son 1977, p. 157-168.4. R. Malhotra, Some Aspe ts of the Dynami s of Orbit-Orbit Resonan es in the Uranian Satellite System,Ph.D. Thesis (Cornell University), 1988.5. J.M.A. Danby, Fundamentals of Celestial Me hani s (2nd ed.), Willmann-Bell, Ri hmond, VA, 1984. 6. J.A. Burns, M.R. Showalter and G.E. Mor ll, The ethereal rings of Jupiter and Saturn, in:Planetary Rings, eds. R. Greenberg and A. Brahi , Univ. Arizona, Tu son 1984, p. 200-272.7. F.A. Rasio, P.D. Ni holson, S.L. Shapiro and S.A. Teukolsky, Orbital evolution of the PSR-1257+12 planetary system, Publi ations of the Astronomi al So iety of the Pa i , in press (1992).