179is described by the mean anomaly l or equivalentlythe mean longitude l[l 1v. Collectively thesevariables are called orbital elements. In the Keplerproblem, where a single planet orbits a spherical star,all the elements of the planet except the meanlongitude are fixed, which is why the elements areuseful quantities. We use the masses and a, e, and ifrom the JPL ephemeris DE200 (Table 2).6. B. V. Chirikov, Phys. Rep. 52, 263 (1979); A. J. Licht-enberg and M. A. Lieberman, Regular and ChaoticDynamics (Springer-Verlag, New York, 1992).7. M. Holman and N. Murray, Astron. J. 112, 1278(1996).8. B. Peirce, ibid. 1, 1 (1849); U.-J. Le Verrier, Annales deL’Observ. Imp. de Paris 1, 1 (1855); a modern com-puter algebraic expansion to eighth order is given byC. D. Murray and D. Harper, Expansion of the Plane-tary Disturbing Function to Eighth Order in the Indi-vidual Orbital Elements (QMW Maths Notes, Schoolof Mathematical Sciences, London, 1993).9. F. Moulton, An Introduction to Celestial Mechanics(Dover, New York, 1970), p. 361.10. N. Murray and M. Holman, Astron. J. 114, 1246(1997).11.iiii, M. Potter, ibid. 116, 2583 (1998); A. Mor-bidelli and D. Nesvorney, ibid., p. 3029; for example,our integrations of asteroid 7690 Sackler show that itis in a three-body resonance involving the asteroid,Jupiter, and Saturn.12. J. Wisdom and M. Holman, Astron. J. 102, 1528(1991).13. M. Standish, personal communication.14. J. Wisdom, M. Holman, J. Touma, Fields Inst. Com-mun. 10, 217 (1996).15. A similar survey is reported by G. D. Quinlan, in IAUSymposium 152 (Kluwer, Dordrecht, Netherlands,1992).16. See, for example, S. V. W. Beckwith, A. Sargent, R. S.Chini, R. Guesten, Astron. J. 99, 924 (1990); K. R.Stapelfeldt et al., Astrophys. J. 502, L65 (1998); A.Dutrey et al., Astron. Astrophys. 338, L63 (1998).17. P. Goldreich and S. Tremaine, Astrophys. J. 241, 425(1980).18. J. A. Fernandez and W.-H. Ip, Icarus 58, 109 (1984); R.Malhotra, Nature 365, 819 (1993); Astron. J. 110,420 (1995); N. Murray, B. Hansen, M. Holman, S.Tremaine, Science 279, 69 (1998).19. M. Mayor and D. Queloz, Nature 378, 355 (1995);D. W. Latham, R. P. Stefanik, T. Mazeh, M. Mayor, G.Burki, ibid. 339, 38 (1989); G. W. Marcy and R. P.Butler, Astrophys. J. 464, L147 (1996); R. P. Butler,G. W. Marcy, E. Williams, H. Hauser, P. Shirts, ibid.474, L115 (1997); R. W. Noyes et al., ibid. 483, L111(1997); ibid. 487, L195 (1997).20. We thank B. Gladman and J. Wisdom for helpfulconversations. Supported by NSERC of Canada.28 September 1998; accepted 8 February 1999R EPORTSCompositional Stratification inthe Deep MantleLouise H. Kellogg,1* Bradford H. Hager,2Rob D. van der Hilst2A boundary between compositionally distinct regions at a depth of about 1600kilometers may explain the seismological observations pertaining to Earth’slower mantle, produce the isotopic signatures of mid-ocean ridge basalts andoceanic island basalts, and reconcile the discrepancy between the observed heatflux and the heat production of the mid-ocean ridge basalt source region.Numerical models of thermochemical convection imply that a layer of materialthat is intrinsically about 4 percent more dense than the overlying mantle isdynamically stable. Because the deep layer is hot, its net density is only slightlygreater than adiabatic and its surface develops substantial topography.Several fundamental constraints must besatisfied by a successful model of the dy-namics and thermochemical structure ofEarth’s mantle. The model must producesufficient heat, either by radioactive decayor by cooling of the planet, to account forthe inferred global heat flux. The modelmust be capable of producing the rich va-riety and the observed systematics of geo-chemical signatures in mantle-derived ba-salts (1). The model must be consistentwith inferences from seismic tomographythat some subducted slabs extend to nearthe base of the mantle (2) and that thelowermost mantle is characterized by longwavelength structure (3, 4) and complexrelations between the bulk sound and shearwavespeed (5, 6) [see (7) for an overview].Finally, the model must be dynamicallyconsistent. Here, we present a model that isdynamically feasible and satisfies the essentialgeochemical and geophysical observations. Itdiffers from many previous models by placing aboundary between compositionally distinctmantle regions deep in the lower mantle, ratherthan at a depth of 660 km.The characteristic isotopic ratios of mid-ocean ridge basalts (MORB) and oceanic is-land basalts (OIB) provide evidence for asuite of distinct reservoirs in the mantle (1).These reservoirs and signatures are thought tobe produced by the formation and recyclingof oceanic crust and lithosphere, plus smallamounts of recycled continental crust. In ad-dition,129Xe,3He/4He, and40Ar contents ofthe mantle (8 –10) indicate that the mantle hasnot been entirely outgassed.87Sr/86Sr and143Nd/144Nd isotope ratiosof the crust and MORB have been used toestimate the mass of mantle from which thecrust was extracted, and hence to infer themass of the remaining, less depleted compo-nent. Estimates for the mass of the depletedmantle range from 25% (11) (coincidentallythe mass of the mantle above the 660-kmdiscontinuity) to 90% (1). Similar mass bal-ance arguments are used to determine theamount of mantle that must have been out-gassed to produce the40Ar in Earth’s atmo-sphere (10); these predict a volume of de-gassed mantle of ;50%. Uncertainties arisebecause the K/U ratio of Earth is still underdebate (12) and the lower crust or the unde-gassed parts of the mantle have retained sub-stantial amounts of40Ar (13), or some Armay be recycled.Another fundamental constraint is providedby Earth’s heat budget (14, 15). Of the 44 TW(16) of the present-day heat flux out of Earth, 6TW is generated within the crust by radioactivedecay of U, Th, and K, and 38 TW must beprovided either by generation of heat within themantle and core or by cooling of the planet(17). For example, if Earth had the radiogenicheat production of the average chondritic me-teorite, the total heat production would be 31TW; the remaining 13 TW would be providedby cooling of the planet by 65 K per 109years.Geochemical analyses of basalts, however,show that the source region of MORBs is de-pleted in heat production by a factor of 5 to 10relative to a chondritic silicate value (18). Thus,if the MORB source region made up most ofthe mantle, the mantle heat production wouldbe only 2 to 6 TW, comparable to that