1977ApJS...35..239S THE tication © problem 1977. Modeling © AsTROPHYSICAL The American properties semiquantitative cibilities cannot is tional their Adiabatic, the helium radius, prevails pure pendent energy excess from assumed luminosity that tion) yet helium homogeneous helium and helium overstable of discontinuity 20,000 transition) Subject headings: helium order 1 in and American Astronomical and In much Next, Convection The Evolutions First, Finally, convection. recent change begun of derived the has the hydrogen observed are this character 2, superadiabaticity luminosity. various the K. and and release planet away content content transport critical the less everywhere. Center of the depending preceding is reached to discussed. JoURNAL I. differentiation for of more Astronomical years. giant Helium evolutionary is modes formation considered. in are fall minor be "age" evolution INTRODUCTION than effects from increased, of hydrogen-helium entropy with such evolutionary that in parameters The for comparable luminosities planet: "'4.5 in under is temperature too the of complicated planets analysis a the Present planets: a Society. heat actually SUPPLEMENT either Radiophysics constituents and a the helium paper quite differ planet temperature THE as differentiation on sufficient helium of Helium uncertain localized Convection x presence details. the being Jupiter of IN diffusivity. A molecular-metallic a the The diffusive-convective All 10 is typically models is the high from first-order a is well-defined DYNAMICS influence (Paper very 9 of enhanced rights (especially abundances or models HYDROGEN-HELIUM unknown a immiscibility fraction homogeneous to years. Society Received are calculations temperature available equal for droplets metallic D. well-constrained the SERIES, has cases The heat level helium or and that for that in reserved. small, fluid of the about J. are is Saturn. I) by thermal the is begun result to Space a STEVENSON* of detailed continuous in flux) of determined of are for molecular-metallic we at 35:239-261, 1976 therefore molecular-metallic discussed. about by composition mixtures. the or latent nucleate atmosphere) the despite Printed source for • sophis­ gravity, Jupiter interface Saturn the 4 - discussed discussed immiscibility. upward molecular Research for Provided AND and June for latent very (but x gradient critical ABSTRACT radiation, planets: planet transition, and hydrogen a in 10 heat). quantitative Saturn equilibria Jupiter is factor U.S.A. 23; the is 9 recently no (and from uninhibited despite 239 at These heat The compositional shown AND for HELIUM and 1977 between also convective accepted temperature and inhibitory first-order with which the the is gradient appears region. interiors are a and Salpeter (Podolak 1976; appears of of x 2 times deviations October the by . E. E. suggests at large transition. purely and Physics the interface thermodynamic FLUID results This the to no 5. transition. ( established. hydrogen least substantially :( the hydrogen Cameron supersaturated include 1977 the 10 formation conclusions. be the 10 first-order convection. Hubbard taken SALPETER In enough DISTRIBUTION to by 9 core transport effects is adiabatic to - Department, 9 inefficient molecular-metallic 1976; phases adiabatic NASA years, and are some Apri/13 that be of decay years discussed, molecular history be PLANETS and planets: from The the both required can for transition used Cameron inconsistent Consequently, helium of transition 1974; to cases, Podolak the respectively, The of ago), persists, molecular-metallic time Jupiter and Gibbs phase grow Astrophysics rotation in of model primordial immiscibility ensure and in Most a The (since of because entropy Cornell mixture, Jupiter helium-rich helium. agreement convective viscosity, the and the to an Zharkov for which Slattery differentiation microscopic at cannot transitions success 1975; phase explain of and (by despite 1977) thermal evolving present are molecular that the the and with the University the helium is hydrogen a in the essentially Saturn grow discussed Zharkov abundance expense planet's factor rule magnetic importance discontinuous, the greatly energy its 1976; and and turn and and of the regarding transport the heat core, paper conduction Data actual hydrogen­ fluid leads to diffusivity or adiabatic, hydrogen transport observed envelope the the presence Saturn to between suggests Trubitsyn has content immis­ and release "' Stevenson exceed transi­ of excess and reach for for inde­ field. first­ state frac­ System 1 to age, not the em for an of of the Trubitsyn a a a (Podolak major 1976) and 1977ApJS...35..239S either ever, in Jupiter structure vective the structure roughly features about and following for els planet phase envelope. cluded compositional The 20,000 the helium homogeneity that which transitions mass, hydrogen-helium variations helium hydrogen phase be phase paper, Saturn Since summarize (Stevenson occurred of a Gibbs (a) one-third 240 (P 3 (the order 15-20% For or 1975), 140-150 180 Mbar The The 2. Before metallic 1. the discontinuous are the ::::: systematically © Jupiter above center a KatP Jupiter, similarity a is Under Under this phase 45 molecular-metallic the composition the central are American transition. main because it phase diagram 10,000 main diagram? planet homogeneous? K region 3 those (Aumann almost inadequately (Aumann are planet future (the 65'7 of has K Mbar); or already, other such such outlining or envelope. discovery subsidiary common latter is models at of conceivable; these or hydrogen-helium and assumed = and the features at 0 4 transitions what these as not question been rule what molecular-metallic the the results H, K temperature is times composition will of 1 between history everywhere P much that that have of evolution bar, Saturn, main was being elements Salpeter ensured planets. correct at the (c) 30'7 planet; requirement The = et In consistent et circumstances innermost features at is phase investigates the assumed eventually Astronomical to circumstances our of the For the the al. of for more 1 occurring now, to al. to a the hydrogen-helium questions 0 an discussed treated. as attempting are we all features immiscibility of somewhat only bar He, It the metallic this about 50-55'7 1969; are be Saturn equal approach see 1969; temperature temperature molecular-metallic more (b) adiabatic if diagram. the However, molecular-metallic the (b) address applied 1977, for is massive of are hydrogen uniformly by excess and either of there and and gradients. to that Stevenson doubt inevitable an with hydrogen-helium an models Ingersoll hydrogen-helium a 10,000 core perhaps in In evolve Jupiter details Nolt (a) 0 that mass, are the about necessarily of hence the in hereafter adiabatic hydrogen-helium 5'7 temperature adiabatic these mass H, is the hydrogen are assumed the concentrated a to infrared in to to less 0 Jupiter is implications The than thermal detail, does a that composition the none actual et other or or 20-25% this Society referenced KatP::::: hydrogen-helium answer hydrogen-helium this and whether into first-order the rises rises preceding mixed, but to adiabatic. 8500 planets transition) well al. or (1977). phase et that "'11,000 helium will because hydrogen the is the STEVENSON of paper al. a comparisons thermal problem, Paper temperature 1974; temperature temperature Saturn. elements a and to established: emission from arise: from and with as these K only transition) transition. some hydrogen­ structure? molecular molecular 1976) phase hydrogen and not less diagram, this, He, are this little planets. at content • 3 toward in is, Saturn above. region of of that K; Rieke phase I) about paper about How­ occur Mbar wider mod­ exist, Provided first­ than P con­ part core and Are and and this and and has the the the In­ the ex­ we (c) by of as ::::: is AND general peratures hydrogen there that where, is-re phase first-order formation. essentially then the this Saturn about ates case complex. Homogeneous, the homogeneity predictions predicted from distributions emission diagram of coincidence. current tion tions planet? eventually, appear depends of vations, to mine more planet, Saturn ing x 2 Salpeter However, that for position plained plicated, any our composition general titative CH by In§ SALPETER In 3. 4. 5. predicting augment the atmospheric 4 the atmosphere thermal possible theory. particular almost , the convection 10 of of What the Can for What implications indicate this a dense constraints is NH 4 III evolution 9 to homogeneous except (Pollack then answer situation a distribution the uncertainties capable a this years. in x Jupiter considerations in by is and 1973). NASA The present corresponding as homogeneous we reach In 3 atmospheric paper first-order if by fluid )? 10 independent implications a general. but not helium the everywhere, gravitational planet. clouds. that fact evolution? a helium the of to discuss sufficient § complicated 9 is Although transport homogeneous that case, possibly II large present years to the in the primordial the molecular common their Helium helium (Graboske dominates adiabatic and et we and may was release luminosity Astrophysics of on we this relate general separating transitions. details al. the past Approximate planet, In convective of Unfortunately, proceed energy a the explaining and phase deep observed differentiation internal considered planet. discuss be in question, view pure 1977) heat observations minor abundance near times properties distribution do we does planet Recent differentiation thermal that to and the of thermal the of layering to the x 2 way are hydrogen representative interior heat et evolutionary even the the flux of the hydrogen will transition. source the almost best primordial appears A of models the the inhomogeneity al. from hydrogen-helium toward constituents the the structure the found present 10 taken on excess heat in phase phase similar heat it content after The Saturn evolutionary not 197 at 9 details in evolution. the observational specific calculations evolution Data pressures difficulty is is which the particularly years, from atmospheric preclude levels becomes the are detail 5; all possible, transport be found we attempt to to transport during for 4.5 the luminosities of transitions to excess transition. planet may previous assumption Hubbard calculation It recent compositional used always simple be almost of is remote fluid find (Kiefer System heat of be there respectively, center entropies Jupiter x calculations constituents deeper of by is (e.g., readily quite Jupiter. from of not of and incapable planetary 10 any available in to assumed the that have Salpeter the with metallic estimat­ a infrared indicate calcula­ content domin­ 9 models Vol. are a simple to occurs every­ evolu­ test which obser­ equal. deter­ years. quan­ phase of 1967; 1977) H be com­ quite com­ tem­ This have fluid than bulk firm and last 2 the are the the the for ex­ for no 0, In of of 35 of a 1977ApJS...35..239S No.2, from for assumes interface in III, the entropies and is convective dominate. latent to stances, in Particular tion transition temperature nearer by "age" adiabatic diffusion. are tionary the the ture respectively. planet sectors and of line 6 1/2Tc(H-He), Paper FIG. of In Sections Jupiter solute essentially relative which up discussion). © characterized the origin. starting Stevenson segment, "cold" Tc(H-He), for § Paper in II- I 1.- cu American 1977 I to IV heat can effects of unity I. shown. Tc(H2-He) sequences. the exists In Sector the the are a Various of the planet and 2 we values planet Overstability thermal attention and Thus flow factor V be Sector evolutions. temperature I. of the This point of than discussed. of molecular discuss and where presence Saturn, The the As characterized continuous the For between (1976). II the the the for in of two possible the = Ic by I, figure is (but in of VI extension dashed In 2). Tc(H-He), metallic Astronomical of immiscibility its the molecular-metallic diffusion 1/2Tc(H-He) central intermediate is the transition. the Tc(H is evolution phases 2 that Tc(H-H some Figure are and A the given vicinity. We purposes from also the critical is the evolutionary and of mixture. in T molecular-metallic similar 1 line 2 devoted paper, directly -He) conclude evolution apply hydrogen-helium the across general actual phases, predicted. hydrogen-helium 2 are compositional to Tc(H2-H), that 1, in of the ) separates is effects the and lies HYDROGEN-HELIUM metallic core temperatures the and is Since Figure which found Under greater of effect convective we those predicted the The analogous to various is factor our most in the complicated regimes aspects strongly that Society hydrogen the dominate. is set particular critical the and one "hot" 2 to passes considerations, on evolution 1 considerations defined interface, these analog a relevant than Tc(H by differ temperature T. well-defined is of depending possibilities the of for gradients. inhibiting This a can hydrogen transport Tc(H-H evolutions to region probably the tempera­ transition mixture, 2 In through circum­ convec­ of straight particle (see • -He) derived a Figure by evolu­ as Sector differ figure Fig. Provided three case, fully of text the the the on = of 2 6 a ) FLUID in the A whether necessary region (i.e., immiscibility because "cold" starting Figure 1 that to point sideration drogen The insolubility to tiation rapidly, helium-rich planet consideration appropriate the time relative homogeneous "hot" a and for cases available. discuss dense densities ation ing molecular-metallic there helium; evolutionary are already downward great given, the helium quired abundance phase planet, disposition However, Helium by In§ In Sector Paper small the be "hot" planet, Sector "cold" uncertainties § assessed. helium-poor rate the there reaches § DISTRIBUTION is VII. well is begins. and that are is starting V, VI cools and diagram then and to transition. the is or found "hot," situation if present differentiation encountered Tc(H-H to point further differentiation at that of net I). and II NASA Sector Most we situation the to in II their no explain initiated, There, here, has is we when some transport the which "hot" case, in the "cold" in homogeneous is of fluid There to down, are, there excess megabar discuss of no upward consider latter Unfortunately, new starting evolution calculations the here drift also Figure are temperature to is Jupiter begun minor coexisting since of implications. 2 success and subdivides in helium I this ) the however, first-order possible that are metallic to Astrophysics inside decrease is the this we of Jovian its It case, is unable is effects is Sector it an is is is of hydrogen starting downward. nucleate evolution one a is Figure transport Sector the so large. at of also becomes section excess observed summarize more region 1 very a the energy one constituents transport Tc(H-He) and point important pressures). several large also the is has helium. in in is in uncertain net the starting of in and inadequacies helium-rich phase. not considerable first-order discussed, I in to summarized which evolution, hydrogen in recently beginning Saturn. the Saturn by dense for for III assumed or molecular-to-metallic luminosity points 1 which not adiabatic, first A is is downward dashed this assumes, energy predict, from the which discussed is Saturnian transition, possible depends typically possibilities, sectors Sector available the evolution. properties excess of brief "hot" Jupiter considered. ~ If The of temperature there yet the becomes then consideration. uncertainties Data sector The Figure (see Once the appears (such are the x 1 a ( helium. using the :( discussion is character that of source boundary. various molecular phase begun once say, Ill. subsequent according luminosity, there 10 former is in dashed in for uncertainty for considered. the the decreases mixture, on or in the homogeneous 10 a for System atmospheres. evolutionary this suggest transport 9 that always the 1. as our the but 4 factor The parameters detail the are degenerate droplets years "cold." the simplicity, main discussion evolution. radiation. differenti­ K. is The to primarily Since excluded water) We becomes conclud­ differen­ for possible is analysis starting initially are is relative helium still line helium As results of be in of phase likely grow more as main a Both It since ago). con­ with of 241 also that that hy­ the not the but the the net the in­ re­ to as in In of A so of is is 5 1977ApJS...35..239S in the except It hydrogen radiative phere sion (i) (H20, 3000 enough 700 excluded. able internal conduction conceivably The ductive is is II. temperature transports than that region perature heat between particular smaller accurately cm- and atmosphere convective centralized ductivity discussed 242 Jupiter Hubbard that heat must ular by made Stevenson superadiabaticity. subadiabatic s so ''adiabatic cm- 1973), 1974; - The We In 1 should comparable the The therefore THE conduction the © • abundance K). there the 1 2 The atmosphere, flux flux adiabatic em in metallic infrared the the American be s- consider pure s- and CH4, Rieke for but conductive inadequacy THERMAL internal is (Ingersoll fluid at temperature region surface to In - Paper 1 conductive 1 emerging at and layer, total gravitational (1968, is 1 in be molecular for not elsewhere case the and alone K- the Saturn. specific almost be known. metallic energy hypothesis" this temperatures the hydrogen window unlimited is be is NH the state noted, gradient not temperature 1975). pressure Saturn required 1 almost internal hydrogen (electronic, thermal distribution about negligible enormous, first excesses to Salpeter molecular-metallic low near of of I almost heat (eq. region, area. 1973) EVOLUTION 3 metallic ) is cannot eta!. indicates prevails inhibited (and Jupiter. minor heat into will source Astronomical everywhere, of enough In the However, entropies however, and interior In sufficient is is flux the [11], region, (7 (Stevenson (Aumann spectrum. solubility certain. In to electronic is therefore are immediately structure. both probably heat 1976) heat the important unlikely (Hubbard flux may each a 1976; ± is total propose of center be discounted is constituents both transition typically but such core everywhere. Paper molecular, solar where gradient. Similar but 2) adiabatically atmosphere, based of too OF to that flux flux is by Jupiter even and are cases, electronic case, the x radiative that heat Stevenson allow typically A A even to planets, as of is is high eta!. of with case viscosity of It abundance HOMOGENEOUS with high 10 that "block" conduction the equal However, I) (4 that smaller, and on energy primordial about the helium nevertheless hydrogen be ensure (i.e., and follows 3 calculations a flux each x 2 the the and below the Society ergs ± a is to (ii) for where radiative enough three the and emerges only opacity 1969; or larger in such radiative Ashcroft (iii) 1.5) 1500 be in Smoluchowski not until (i.e., energy 400 the because opacity 400 10- radiative) STEVENSON related). deep The thermal or the planet em such a 2 source 1976) consequence in and the the a purely con­ purely convection, transported that the Convection Saturn planets x assertions: the highly the Nolt x very 6 ergs inequality hydrogen. first-order molecular - of K transition em than) adiabatic we has to the heat, 2 K resulting 10 window metallic of into planets. 10 • observ­ layer s PLANET atmos­ discus­ readily ~ "ices" for a can molec­ source ensure - strong rather is em- - 8 of know could is 3 em- Provided 1974; small 1 eta!. deep been pure 1 tem­ This T con­ ergs ergs at not not the the de­ the the the for are led to be ~ is 1, a 2 AND the discussion. down. important than tions trum, much homogeneous cooling comparable gradient. et large The if adiabatic is temperature through this rising the recently be but pected parently, smoothly estimated of convection dense convection is where be Stevenson son of only crude the the that hydrogen-helium by 1977) Jupiter reconcile Slattery Poynter moment not by SALPETER Evolutionary Provided allowance To the The al. comfortably operating, rotation, much demonstrated molecular same deep window and is melting almost compatible the inhibiting the a or 1975; or elsewhere, transition than confirm larger fluid continuous, approximation, have fluid very the p to This and Salpeter the immiscibilities. (1974). or been is adiabatic "anomalously" (1972) NASA smaller atmosphere form with case. the In be J in from 1977), and the in 4 element is identical adiabatic state the small Hubbard pressure , superheating been is there Saturn. in in point greater is the to is § possibility since only viscosity. the g analyzed planet the II, the made but surrounding The made magnitude slightly effect the the fractional superadiabaticity fractional readily in with will em- the of one calculations (1976) case of than and that electronic Astrophysics T:::::: Paper fractional molecular-metallic mixture) inversion made excess then adiabatic is exist temperature other of the the adiabatic these molecular major in would than Coriolis It region 3 10-20/" numbers phase the the for with gives , of 1977) by the hydrogen where unity. approximation general the 10,000p and would exceeded, thermal Lorentz have an initiated in I. minor the same only conceivable interpretation is planets the that of small thermal part Anderson, Jovian would within The element no always more a of field. the at to of and 0 too superadiabaticity. the The hypothesis, conduction discussed temperature magnetic the strongly assumption colder. force the first-order of have for also conclusion hydrogen 1 for apply each the of the ' fluid Saturnian adiabatic 2 profile constituents radio force Saturn (or melting is rotation, unlikely effect detail adiabatic Of K, the transition. the adiabat and be once Data higher convection be this Jupiter assured be of an other (Hide depth. course, the state for This evolution inhibition transition, transition only possible, fluid inhibiting "anomalously" very will is emissions is Hubbard, this field which of homogeneous, by (Gierasch (Pollack the System opacity liquidus Saturn. it not temperatures phase gravitational can was is temperature that valid as could temperature rotation ensures if adiabat to slightly by for can Gulkis 1974). be remains gradient is (Graboske difficult To the a is No adiabatic it inhibited to found reached. showing dynamo merit matches requires at become Jupiter, exceeds Steven­ not change Vol. is transi­ a region region and moves super­ of for break but block effect spec­ et Even most from then very of that Ap­ and and and less has has the ex­ (1) al. 35 to to to is if a a a a 1977ApJS...35..239S generate the change No.2, internal the tive mation, where absence specific assumed effective on where unity adiabatic effective the models with while where 1968), changes 1't as elapsed where magnitude. at where and Xm Xm--+ of For oc the the © a constant = Stefan-Boltzmann degenerate gradual temperature, a L time T is in optical 00, r.,;;re,i American L P 1977 g beginning •. ~ t same T. of both = 0 heat 1 insensitive of the since of is the is cooling The to pressure is temperature. little, is transmission a 0.25. than 47TR internal r index. to the a the an = - = a the be is Graboske It atmosphere) adiabat: loss luminosity per = = Jupiter during characteristic depth solution 7 3 [ 1 2 it internal the follows acceleration The where convective, excess a(Te cooling phase, Cv "age" (present (1 even (a) phase. unit first - - 1 of T of From or (i.e., - thermal 0 present Astronomical T 4 value to (present unity the is 1 primordial q4) R. - and most as - 4q volume, luminosity, Te,t became ~ of constant, the that of Xm opacity heat et To Since the To degenerate Furthermore, 4 excess is J the T. T. equation the of 1 and - corresponds Saturn 4 is for xm effective a!. T ) then energy: ( effective internal P of due an pressure 1 a p: source, changes 12q heat p.)n ~- the 1 X virial 77 planet is is T 4 ' 3 n and Xm degenerate), (1975) and the adequate the HYDROGEN-HELIUM luminosity) of 1 dx related to 8 - substantially Te ~ changes equal effectiv.e heat ~ content) + ): , at R the q gravity Pe (2) 0.25 T temperature evolution, is theorem coolmg. entire pressure, (rrrR 3. Society 0( is (the Cv 1 temperature, present, at the by is can the indicate is during atmosphere. to the q12) In to to is is some optical then first time 3 more an actual atmospheric temperature Cvi't), be the T. the the and the ] radius, interior ' q The . (Clayton by approxi­ order regarded Pe less q =T that average average average the • limit so K rate 4 rapidly that in depth is being effec­ is ~ Provided value 0 than /Te.r that a and de­ the the the has (3) (2) (4) 0.5 (5) (7) (6) of In of as is is K FLUID geneous unity precise goes analysis. are direct et luminosity, The The in they with the with planets, Stevenson for either geneous, in Jupiter. have has cluding 1976) inhomogeneity may uncertainties 1977) 10 tion. valid hydrogen, this order. III. two the near the perature conductivity flux enough) investigated from interface, ing very transition range stratified densities change order Stevenson away by 9 the a!. 1979. We that for greater assumed transition. assumptions molecular-to-metallic uncertainties an can calculation THE phases the be is differ no overestimated DISTRIBUTION back the about because deep 1975; and if the from small of Furthermore, that at do consider evidence Jupiter At evolutionary mainly resolved at planet "anomalously" In§§ first-order In but conditions, inhomogeneity also the the models or MOLECULAR-METALLIC interface NASA across to and The the transition. the Saturn into not 1977). latent interior, greatly "natural" Saturn for in and the at the the (1976) is Hubbard age neither ensure temperatures molecular-metallic superadiabaticity. for 1 is IV affect are transition least time. the end heated then and Saturn, suggests carried In differ x always major transmission interface is now two adiabatic it. and of suggests both with transporting relating The heat of still were (especially Astrophysics 10 (Podolak reaching luminosity is in is t the about of phase The may the 0 L/T, and the consider convection is it 9 calculations Jupiter V, For inhomogeneous a thin, even phases age ~ large the In the from two (i.e., greatly is 1977) L there years' but low excess possible uncertainties by pure needed: explanation. we evolutionary entropy that have the that found x 2 are where a in large 4.5 flyby transitions between "typical" from that next to turbulent most with the hydrogen examine of assumption enough gravitational) recent in if this average within. 1974), any is and hydrogen Saturn Jupiter considered. in (Stevenson 10 a opacity, luminosity. interest, x of the increases the uncertainty. a small from not the the that a different section of 9 the excess the Tis a conclusion, HYDROGEN likely (i) 10 everywhere difference signs homogeneous years reason for Different self-gravitating Saturn appreciably transition Saturn small homogeneous enormously paper, ages 9 Data the heat one the (or planets present a terrestrial It the adiabatic years. may the Jupiter value convection adiabatic, the is We transition either, fractional time or are the explanations is but for transition phases, the luminosity we planet for consistent finds hypothesis evolution rapidly flux change temperature would to of that (Pollack assumed not Under This by System and above units, are Saturn, Salpeter specific of in the (There opacity scale. discuss believe between luminosity) TRANSITION the modes The (Graboske the were the Pioneer were across well be t t which observers since gradient. 0 0 planets.) dilemma in not Salpeter different less the derived present Jupiter not ~ in is homo­ as Saturn follow­ density homo­ a phase­ with evolu­ major region crude which more eta!. fluid, from away is each with first­ first­ heat x 4 that, pure wide tem­ that high 243 how than that heat pre­ one (see and the are the the are the no II be at a 1977ApJS...35..239S 244 unity; (in determined is is then viscosity magnitude mined scopic not number between planet convective crucial. in Boltzmann's cussion incompatible Schubert, continuity, the defined analysis, dominates. is must by superadiabaticity state (because adiabaticity irrelevant, the the them the buoyant. phases superadiabaticity satisfied speeds envisaged consider and in greatly moving time steady situation, speed, essentially difference of difference not This Of A The indeed This apparently the the fluid © two the finite Busse superadiabaticity affected rate linear these deduce all be in to so American scales. high Earth's by on second levels; have state, (ii) and elsewhere, conclusion phases of interface inhibits hydrogen-helium case If separate (defined does parcel enormously which considered: these the and for a and amplitude of For a of first-order conditions the Turcotte, in between the in and conditions simple, flows stability where time viscosities To a of but "isothermal" by no separation its different of by fluid phases, constant; which of K Prandtl L the since a the olivine-spinel with not substantial To satisfied molecular-metallic (iii) conditions, aspect are is mantle. understand conditions order of the Schubert order ::::: two-phase the "two-phase" central the Jupiter rather scale are as becomes under the nature latent must becomes phases fluid k one-dimensional intermingled molecular prove so the summarize, it Astronomical is a ours, a Pr flow. analysis phases 8 dynamics flow and greater inhibited thermal T, at the elsewhere, (see two-phase numbers. would much has densities, substantial k are in of prevailing a unity. Earth's = (iv) They 8 the this in a is can both be T and temperature); dynamic of the predictions contrast Oxburgh Turcotte heat), two planets vjK, in a probably in (1971). that unstable "two-phase" the no in (ii) the positive why (rather action per large. a the are requires less solid-state very so region change was than the separate Jupiter problem This diffusion the heat propose microscopic Saturn, where greater aspects and discussion mantle by viscosity of region. large Earth, finite our than there stratified their particle, and region and and the in carried solid small is surface unity the This viscosity. of latent They hydrogen steady flux and to instability (1970) (v) to the the than satisfied. model is surface phase, a neither conclusion gravity. vis Society typical or of whose that conclusion amplitude is most that yet phase mixing fractional almost the and phases, than is phase no of are where Schubert coefficient) average factor which Earth's that conceivable criterion Since persists. linear a (v) Saturn, STEVENSON out the found region the is heat layers "adiabatic") where the state tendency for with in entropy in still the heat an and particularly for reached cool large-scale convective convective phase essentially a average important kinematic for transition transition transition However, energy nature Paper their when of""' at criterion viscosity upward­ problem is interiors the the stability L, and Prandtl • mantle, macro­ a remain value. can flux that L super­ In flow. of (1971) where deter­ where sound of is is down k Provided well­ with flow that pre­ 8 10 > dis­ two dis­ not our the the the not by I), an be of in is is is 24 0 a a AND 6..p interface. interface the phases position) (1976). amount consider density other communication) turbance those where velocity sibly scale stable experiments, average 1968b; direction) tfle Turner is, time) where substance 0?/ (Hubbard flux. compared case, large ing theless proportional (1975) entrained they the tion, tially have as provided defined, kinematic Stevenson I0- ::s; by SALPETER Experiments An 10- however, follows: ::::: is (the wave 14 molecular a the probably Even the the a - 10 the both distances different during characterizing given macroscopic modes 2 em (Salpeter g "isothermal" have Linden was (1968b) thermal 0?/ Experiments ergs ejected smaller is interfaces is em fluid although metallic for density of all is NASA in major) velocity 0?/ s- The ejected can and fluid, the is if (1976) viscosity. the and with In correct. not by been the interface. s the em- a « 1 possible Long's the • that to eddy the a small be acceleration density. are the 1 phase. the latent The from O?iw ejection 1973; and volume from by for large effect considered and Smoluchowski characteristic by 0?/ although 2 recoil source contrast interpreted the has expressed O?iw phase can at Astrophysics on two not), molecular-metallic s - 3 net were the (between Long and Linden , Jupiter amount but size latent To experiments high I the Stevenson 1 The but on the the interface the Ljj Long heat The Reynold's pointed = be sensible modes comparable effect fluids of convection in per The the finite velocity is prove Re of considered ( turbulent p0liw "normal" applied phase interface: the I, speed on turbulence conclusions This a gl at about sign and magnitude, small heat (1973). = and due interface unit mass 1975) in as large and of of the 6..p)1/2 amount p shear-induced as 2 latent fluids probability for are 0?/ljv that, an out Salpeter < has sensible appears of heat ejection the turbulent implying to boundary.) Saturn from 1976). interface is flux interface, number 1973), 30% to are in transfer n eddy finite-amplitude indicate 1. entrainment L phases gravity, Data Neglecting » to that entrainment comparable not would "wrong" = Jupiter (n heat we ' total of is were In would applicable flux. one OlieL (i.e., the ejected 1, hydrogen more 3 of not Turner = that and been would and and different heat is so a to this where n according System limit flux of fluid 3) area Salpeter sensible volume, of found between zero. major that (convective) and is be known, I Unlike = eddy and be I still that encounter­ dense has turbulence is case. Stevenson done, fluxes negligible ::::: the case, therefore rotation, is 2. phase (in substan­ a per unit (Turner (private into velocity need ensured a v Vol. p to be 10 Saturn (which (There hit transi­ In across length never­ O?ie is to small is (pos­ com­ size), same each than 9 heat O?iw, well dis­ and this but can but the the the the the the the are em (8) (9) be 35 to to ::::: at 1977ApJS...35..239S ever, initiate superheating tion to superheating fluid a interface tion hydrogen No.2, atom, T evident 1970), em. great change the thermal boundary and gravity inhibited thickness heat phase simple small other and and This Figure temperature temperature to between positive almost phase ILifkBTis surface Suppose, ~ FIG. the rapidly © boundary This is no the were region, Salpeter between 10 conduction it boundary, as z American same temperature probably boundary Salpeter 1977 2.- adiabatic is according mixing-length 4 significant is 2 latent nucleation from B waves inability boundary phase K 10 (D.T mainly energy C still possible. relative by of Temperature layer and probably somehow change profile profile in 3 specific labeled is now, order em Figure heat heat or Jupiter C is not 1976). or layer, E. while supercooled and on except itself the and instantaneously. enlarged supercooling for represents comparable is dominates for would of and The to to nucleation. possible supercooling leakage -fj,T-\ of layer that 10 entropy. by the Astronomical phase, inhibited only Stevenson find drop We AB Hydrage Metal slightly 2 a vacuum D the pure possible, Consider the with one or being analysis em, for no 11T, that actual macroscopic is versus "-#.1,.•,····· be interface, and Saturn, nucleation nucleation between predict longest nucleation. I \ superheated a conduction 'D D.T theory for phase ic ABEF, across is \ the \ since the for \ n thin \ or EFare I In I appreciably a and less (small-scale temperature exaggerated (about by clarity). If ~ to moving vertical might a yields molecular fluid a then actual HYDROGEN-HELIUM the region heterogeneous fully I0- within than large homogeneous is the that the with molecular, wavelengths which the of the adiabats with Instead, that BCDE case between volume mechanism. ( never 0.1 2 only is Society density be · Lang a superheating adiabatic a phases, unity of · coordinate K, near formation thickness up and amount Flow ·) boundary two-phase the possible. the eV needed. amplitudes there for profile less. pure is convection where and as viscosity). infinitesimal corresponding is metallic. the also enough per bulk and (Stevenson amount clarity. of across there part B difference while shown T in is down interface. metallic case, ,\ of • and fluid nuclea­ nuclea­ (---) surface shown. L a of Using Kohn which How­ ~ of region z, of Provided layer fluid It of very > The the the (or are the the the 10 the the At for to of to as of in A is is is 0 a 9 FLUID interfacial itself below interface Magnetic" the heat. the probably supercooling) flux a entire drogen interface. lic. ag discontinuity large interface tributes rapid process profile by effects and actually ary inhibit interface. This son mixing would within regions flux, adiabaticity a a approaches tions than rotation yields where becomes pressure Simple similar son dicts Coriolis or is by small wave distance The The In at the Saturn, molecular fluid If and Salpeter layer the 1974) superheating must most in a is DISTRIBUTION can the so Consider, enough nucleation is diffusion crest an v(z) effect be fractional effect the excessive length E crest are to will amount its satisfied mixing-length on cools to just arise, a force, scale ~ is NASA smaller small. itself more case is order be According in The is many that about z absence, buoyancy" is the be if temperature interface an waves, 10-4, unimportant, no difficult the the has is rearrange from the or neither much of below of the the z, phase to is L height. phase much constant, the lies upward for of trough Lorentz and of longer magnetic so planetary and hydrodynamically interface, of absence for cooled heat Far wavy change in D.T < generally superadiabaticity than orders seeds convective mixing 1976; Astrophysics magnitude. heat the in L it fluid no on metallic thinner the 0, Jupiter in but but begin Q superheated example, to change to the larger ~ seems may > flow. from similar interface, Allowance is no buoyancy the might of electrical onto theory longer heat important crest interface. itself most are w- assess, Gierasch it 0 to has the force inversion its of of since the length it fields thermal rotation supercooled if is to if a phase enhance This the much in 3 available, structure. or nucleation. than planetary the the follows flux. there likely point still changed magnitude T is hydrogen of grow metallic rotation velocity dynamo enhance so Since the circumstances. (without exists. a is superheated ~ Saturn fluid especially local on bubble. magnetic given temperature is interface, much force). properties crest that boundary. nor for 10 no and this, Since less presence Since Data are at (i.e., is the on that E of boundary may the quiescent the K. at that a reached ~ fluid phase greater rotation phase appropriate supercooled, theories the angular This the Stevenson then these is no the by rate than by structure of thermal less or rotation the and 10-a However, magnetic This the Coriolis rotation) less be the System amplitude unimportant. waves upward metallic order immediately the if z superheated interface pressure determined interface than of bubbles and the nucleation boundary, considera­ before needed across thus the ~ total However, there in inversion occurs than interface than at velocity. (Steven­ rotation (Steven­ layer thermal As than 10 bound­ Jupiter metal­ super­ of is which of at latent unity. 1977) 5 force con­ only heat heat field 245 to pre­ (10) em, is one hy­ the not the the the the the the the of to in of at is is it is a a 1977ApJS...35..239S tion Stevenson from and consideration. hydrostatic where Hubbard's where have tion. sense. field, essentially is likely according consequences as accurate only 246 known hydrogen models Salpeter at for assume or modifying transition we present specific entropy are that boundary primordial structure evolution is ture-dependence present by noting in contemporary content where homogeneous. per geneous the To The We evolving the The a a this the assume the © atmospheric two factor proton is planet for 10% It qualitative instead the summarize: that transition full consider This Sc, first-order, 11S American fi.S existence primordial follows at core normal first-order of entropy that heat in is heat an ways planet [1976]-implicitly interior large-scale at resulted that Satm lower or to of on is (1976) = the Tc is Jupiter of the adiabaticity as in hypothesis, the is is pressure, (Stevenson that latent adiabatic, p the differs equation L/T 20%, the the much 2 determined the content therefore an the the ;:); the uncertainty content for compositional in latent are that (Stevenson metallic now by In modes are densities in entropy or feature 0.1 regions extreme the planet). entropy cooling are planet of is which Sc age and latent model. If metallic and character a during the the heat the but Jupiter, Astronomical absence a independent from less the multiplicative g the the the + actual convective applicable, heat this central of (eq. em- (11) the homogeneous specific atmosphere even multipolarity Saturn--except which is and f1S modified of core entropy the cools, effect than may effect heat transition of the change which molecular-to-metallic of by Second, and that upper should change the in conclusions effect. different (The and could 3 [6]), the the planet, core, assume = , planet Salpeter most of the temperature this equation since the cooling of temperature is effect and states at the Satm formation result. for of entropies because a Salpeter 1976) (i.e., planet, no the provided planet. may bound, uncertainty is pattern adiabat at of metallic well-defined change then be this and not by the and of critical thermal the is an factor not longer temperature. There , the is pressure. (eq. 11S transition, wrong since that proportional roughly Society 1976). the rate planet. the readily be first-order of have adiabats adiabatic (11) In the large phase respectively, of negligible volume transition of in = Stevenson For [12]). and STEVENSON for an of (Busse at the exp or densities. nonadiabatic Salpeter at the the present Tc contrast 0.) equal temperature differs temperature is the addition structure. by the observable The the deviations important simplicity, molecular very the evaluated it wrong boundary evaluated First, in magnetic ( This the tempera­ interface planet This in as -fi.S/2), are is • and specific central change transi­ transi­ homo­ age to 1976), 11S either much phase phase more in Provided from same early in to (11) heat well (12) and and is is All the the we we by an to its kB to of is is a a AND 0.5 As energy The multiplicative the of or being (Flasar equation, The where where boundary, (dTJdt) Salpeter at geneous T at mantle where then proton fact, either ergs 10 Jupiter tinuity molecular because (see, degrees of gradients larger metallic thermal here helium omena. QL QL by SALPETER The These ~ Thermal -H in the the the helium. or is the is effect other. additional the 10 for both very s the positive to at K the as contemporary case, - boundary, effect fi.S total L planet changes the 4 is (Stevenson of grows 1 and There 1973). IV. phase of of planets models example, • s K, 1976) convection, most phase those large NASA is the = calculations metallic - on dense is purely In helium planets the freedom 1 This mantle. not convection derivative The QL and CONVECTION heat QL , heat TI1S Saturn, on the in Saturn rate which cools, factor. regardless Saturn, is COMPOSITIONAL smallness at 10'7 is as According a ~ luminosity kB boundary. that transition, ~ conditions an of molecular Gibbs simple is leads the when 11v the content at 2.0, QL, Astrophysics is x 6 thermal _ Spiegel1972), 0 fraction contain per core Jupiter and released.) extensive cooling is ( can one which of 4rrR almost is If which the latent dT volume dP) ~ This because expense is appropriate but smaller, the 10 g proton. is the are in to fi.S generalization phase Salpeter 3a the phase admit grows 2 of IN of 23 L ph= latent evaluated the resides is 0 the factor to (see (11S) from gravitational one and a the total inequality heat 3 hydrogen. at the Assuming (dP) which THE < of are compensate, rate /proton probably dT helium substantial literature change the presence 0, the temperature a of GRADIENT grows rule metallic qualitatively since 11S Since limited §II), sign 11v at of not this 2 effect PRESENCE first-order heat (see latent heat but ph 1976), ergs the could then in Clausius-Clapeyron Data additional to the arise latent (dT) enforces pure of can a dt the we one along at adiabatic, at §VI). metallic is I11SI (Stevenson flux is s- smaller of is per closer 11S. expense on core. dTJdt heat of the the be the usefulness it much limit 1 nonuniformly mass homogeneous metallic in the System finds have and hydrogen. even , heat , this OF follows composition of (If as is gram, < expense molecular­ transition. the new hydrogen­ generated molecular Note net a changing A to ourselves x 5 ~ available small 11S I fraction smaller. internal problem core. fraction that a Vol. discon­ greater release homo­ kB of -2 unity. phase effect phen­ much core > (13) that (1 that (15) 10 and and per the for for 35 as of 4 In In x 0, 24 ) 1977ApJS...35..239S the ture thermal and phases always, distributed. The the steady efficiencies cient almost homogeneous stable. can the to mode upward, by same ambient where which, No.2, ambient is If where scale-height, arguments then where With Consider, the we the x :; © "salt consider presence parcel exactly gradient in and derive unstable composition. define condition American is I sis = z 1977 states The (see certainly after well-known X is where e the these diffusivity. is medium, fluid, expanding qualitatively finger" (~;t,s dp dp - _ - the _ p, the a (Shirtcliffe purely Paper transport of must a ! solvable. first, respectively. -~ vertical p In some are: then a mixing is of velocity assumptions, ( p entropy, thermal i.e., D mode parcel simple ar (~;L + a destabilizing. (ap) valid for particular, P) modes is two the + ox purely diffusive, then I, diffusive with (:;t.J~;- elementary Astronomical The We the ( x.v adiabatically instability. simple §§VII coordinate (!;)s,p(~;) length, ap) ap unstable q,T of .. It for diffusive of is 1967; v: new [dT mixing-length convection, composition pis helium also condition dp have + (Turner heat E need dz dx x,s fluid a The both >X (!;t.T(~;) and solution _ and H the mixing-length it < simple assume v. we Turner or v lower ( not mode. The parcel is in dp dp' manipulation, molecular = ' diffusivity HYDROGEN-HELIUM ar) Generalizing ap VIII). pressure, processes and solute. owes 1967). assume possible equilibrium and (gHv) for + concern (~~L.J overstable, first x,s and is generalization Society that Hv and 1968a). density (~:t.P(~)' In is instability J well maintaining its theory, The The dp then 112 dz is direct assumption is density the and and that to existence and the is of H understood theory, highly us overstable remaining v the eliminate displaced tempera­ the than different with becomes pressure ' metallic analogy and further. • K we D is sound is given Provided usual that (16) effi­ < (19) (17) (18) un­ can the the the the we to of is K FLUID heat speed, in where and consider also distributing The lected the properties small fluid on obvious changes then the the tion (Gierasch force. exceeds a is In the medium. unstable small (diffusive) for and of characteristic overstability oscillation oscillations 8(Hv) E of Molecular by displacement > Consider The heat Jupiter the given (e, the overstability. instability is Cv evaluate 0, the the rate ambient same flux fluid the DISTRIBUTION is for compositional a x)-space, (as ::::; we displaced not In and X> convection diffusion overstable is stably the = correct density the consequence situations NASA I0- If at FT it region for molecular the have the and if negligible, would - is regime. is element multiplicative the viscosity g now 0). density can which FT is right the ( x 6 possible, is Fx above). the , is constant absence the medium 8 a time of stratified Stevenson = density of used ::::; criterion, so In bounded contrast grow In result the the position, ::::; (K and relative _v_ CT solute solute mass 0, Astrophysics order be an The ypv.3e(e sides T tX work properties. situation p) mode in that In pv.x(e + decrease for overstable and the acceleration diffusion element modified. into v Thus, gradient of on q,x which because Ke = v)e driven increase Figure of incorporating is regime of pressure is of course, heat fact e is (e is y-::::; between to the > ' the always these > - 1977).] on heat is viscosity, and ::::; p - factor the equations p enhanced, done if < helium of Dx. (D x) x from that most effect 10- x of of x)li2(1/Hp)2' by other one flux diffusion x), 3, effects y 112 of For can ::::; of mode. and In from ;): order roughly yv.z fluid surrounding but results + the = specific interest (1/Hv) against the 4 important, the 1. the but due 10- overstability the Data the side Fx: 8(1). v)x for diffusion. have example, efficient ( of solute The side it thermal the this a a that stability fluid vis greater to (20) In 8 small-scale next does v. and solute rotation ln ln to , 2 (e x is the 2 across be • this effect speaking [For condition heat heat. System Fx(lfHv). a gravity P) by T (Walin is = gravity. (K > diffuse element and that large section, not convection significant. a sufficiently 0, s,x' the mode, when x) buoyancy > diffusion, e efficiency Consider however, diffusion ·growing diagram ambient We of Jupiter, ::::; (22) I change is D is a a by stable in = effect 1964) rota­ from w-s fluid fluid very neg­ slice (20) The ::::; (22) (21) 247 just (24) (23) and can An the the the for Hp re­ we by v, 1977ApJS...35..239S they element greater is than than ratio by plitude "I· situations sections.) same ("wavelength") effect-see adiabatic analysis, accurately laboratory-sized as conclusion. is defined, 248 than the matically the the the heat heat provided stable E a of origin. in assuming tion, many unstability = contour accompanied only true In FIG. a the x the © amplitude reduction overstable E and an a would flux, Figure mechanically = of 1 as orders K. Usually to American X 3.- convective total (a but 0 than thermal it order situations is cannot represents but The unstable intercept solute v excess. to would constant The but of comparable to of occurs represented Paper is of Overstability for be temperature 3, experiments overstability the constant il?fK;:::: E1 interest .. magnitude of not in characteristic stability of because the modes from .. > by fluxes (the to magnitude. experiments, be be ..\ in wavelength, ~ .. the ( This 0, mixing a I, E behavior many dashed a solute is enhanced a Astronomical is that constant if = for a value § x gradual region estimated (HP/vs)(x pure oscillations since typically only VII). never Eo) > are (x diagram relative heat criterion of in larger all (at 0, orders we is should of gradient, (Caldwell1974) process. - fluxes the not the line Figure diffusion diffusion. of K shown thermal x a The heat E is flux. .. very The > reduction E for consider than given horizontal diffusion if oscillation ;:::: complicated, a and very inherent that for - schematically to of D. flux from should of physically we onset pure therefore is 1). transition In .. well-displaced is great, Ea. This The normal )- thermosolutal 3. order is is heat not always much magnitude diffusion contour. never were neglect the The not Society 112 The satisfied heat Once This dashed linear of in just means in be process flux) , inefficiency greatly STEVENSON stable probably transition time: 10 acting. length conduction) indicate vertical overstability greater E enormously be convection, roughly means subsequent satisfied from for reasonable unstability and the em For the is indicates regarded is line in stability that from less • not a region, greater convec­ driven super­ in rather clarity, is many Soret given (This over­ Provided scale than sche­ from that that (25) than am­ less well not the the the the of in is AND ilpr, the to intermittent limits the to ment transported the velocities. which units. predominates, feature from exceeded of and away as evolving convection. changes density distances To These likely general layer The Provided thermal that to there thermal ments until plate by plate z (see, equation Let Now Stommel1964) (Kt) and 1976) by SALPETER = In Consider this initially t "'10% be a conclude: 112 1 subsequent elapsed introduced the 0 let Fr ' then equation following the rigid, for 2 thermosolutal suppose the is near is and show by grows, surface, For to the positive, at has and core * and and equations ilpr features a no units. of plume solute caused diffusion a example, be hydrogen-helium plate, convection and is NASA constant, for ilpr a over of theory work the relevant constant x the perfectly conducting their Under deviation localized stable that there If exact: (Dt) intermittent. a theory and time boundary away the = a that and property It Fx semi-infinite If Fx (19) > unstable effect and forms and situation which 0, equation layer and diffusive but by done ratio are thermal solute 112 a * follows ilpx Fx* * ilpx, (Linden is Astrophysics and layer where > diffusive solute until these by Jeffreys the then values are be given heat mass the for a time-dependent the all convection, that (Howard (D/ perturbations from is is be of .::;; is which new convection. in the a layer the K amount that whole at stabilizing.) must the constant; can heat (D/K) K)l thermal the approximate, regime thickness predicts ( thermal the and that energy is (a) circumstances, redistributing solution rate is is E upward D)l/2 (20) of K 1974; most intermittent neutral density also 12 planets. ;:::: and respective pure "core" the Fx of introduced both removes local Fr is subsequent be D solute. form and Fx. 112 Eo process * formed 1964) *,then particular and shows introduced Jeffreys Fr* thermal flux. in (D/K) ilpr plate. flpx. of transported Linden nonlocal + = fluid, effect Assume very ilpr can heat need (b) Rayleigh stability solute, Data changes However, "-'(Kt) near which (D/K) of x/3. solute forms. K The effects at (Both This the the indicate (see 112 and still the is a z slow Incident which flux that bounded boundary z is not helium but stable An z Fr* then and solute 1950), = z 1 exact 112 convection ifsvt. A diffusivity. buoyant characteristic System = ' that destabilizing .. = Paper helium ila 2 relevance (Turner limit respectively. form = transported At in at Fr* , 0 are Fr. may concern extend - 0, convective interesting number in E where 0. the Shirtcliffe x repeated. 0 grows the away if the fluxes diffusive the - at increase x then that form Experi­ Experi­ locally. but defined density upward density applies Vol. can on ilpx « occur. at I) below t x plane fluid, layer frac- ratio edge fluid (26) (27) = and this « this .. the the are tis all by an by be 35 us of in in > is is is 0 .. 1977ApJS...35..239S is drogen. convective layering invoked insolubility gravitational thicken (Smoluchowski face corresponds of of desirable described entrainment criterion chemically available, diffusion denser in not situation subsequent solute superadiabatic tion simple theless, where whether solute theory and plicated density be an assume by diffusive This Flasar metallic regions). which possible No.2, x 3 (Linden It the Salpeter The In In Finally, this convection, satisfied thin, © fully analogous order oceanographic (e.g., might 10 between is addition this formation uniformly American 1977 concentration paper. (Linden 42 concentration ll.pr, helium steady thermal 1973). negligible drops with time with throughout 1974). global hydrogen layers steady the may to diffusive is ergs or for in times developed section, at here, of in we be homogeneous originally times. not in explain experiments in overstable sections. to tl.px an Saturn (i.e., layering discussion nevertheless magnitude principle, convective supposed to at to across a V. states temperature Helium resides could This instabilities should the and some state are if interface satisfied, 1967), there energy of cooling mixed the the HELIUM present. (both if and layers transition helium we "hot" Salpeter tl.pr tl.px Astronomical a situations wave-breaking Shirtcliffe enormously criterion convective the (Pollack this planetary form provided steplike change already turbulent diffusive-convective of the difference, proposed in are is Furthermore, consider consider but the convective differentiation state. < that content positive) excess . hydrogen-helium the if (Turner a between direct section of then where IMMISCffiiLITY and fluid case structure smaller impossible (~)112 at take An becomes system central are difference appears D Jupiter (1973) rapidly. all. chemical is is situations this et distribution in considered. Equation 1976) of diffusive the evolutions HYDROGEN-HELIUM favored even satisfied not for speeds luminosity a!. place which the mixing that large the convection, The whether of Sector that respectively. and ' layers phases, are section than reverts diffusive core first-order. 1977). at the to pointed show Society effects to changed Jupiter insoluble temperature larger usual Experiment this it and in in compared Stommel separation layers the one the now was are be relative (destabilizing) in considered (Kiefer of (28) or I are is wave the the that of we in (stabilizing) A equilibrium or more insolubility to most planet. even molecular­ more laboratory helium essentially of where overstable interface). of not interfaces energy originally separated solute, the Figure out is form common may presence presence an consider a because from this • Jupiter Never­ helium in If speeds purely to about cases. 1967; inter­ more 1964) com­ Provided clear total than with that (28) and and and this We hy­ not is the are the by in in 1. is a a FLUID P region transition, becomes about fraction immiscibility I would the molecular-metallic the homogeneous, helium. homogeneous planet of immiscibility. helium planet by corroborates =Po. FIG. a region actual small the DISTRIBUTION is is 8000 4.- occur In as cooler still P,T insoluble NASA (b) a of rocky The temperature) function helium is K H in immiscibility The (middle), predominantly layer H(He) still, shaded. first but (see which inhomogeneous this core) dashed and Astrophysics phase in forms, Fig. when transition. of mixed the guess. The the + is the now I the within P 3, only line separation planet center has the but metallic actual the = Paper hydrogen. in At Pc. first-order is inner expanded temperature the the evolution has Our of the solar In the pressure I). the helium-enriched region cooled planet. (a) phase, actual is Data discussion The molecular-metallic planet In proportions about (top), transition somewhat. (or, (c) is of critical down helium The System predominantly but (or drops (bottom), a the equivalently, to the hydrogen­ more, region in near is planet begin number helium surface central An helium below Paper first 249 and the the in­ of at is 1977ApJS...35..239S everywhere. concentration is 250 time the tially reduction phases. planet, separate). solar IJ..p::::: droplets is transition, rich below phase some of of convective cooling Stevenson We must rounding of Can gravitational difference where and b teristic and This roughly enough convective are where droplet droplet consider 1 Vb the the would by Vbfv em, derivable Suppose First, The the primordial supercooled be © planetary helium transported fluid additional the hydrodynamic consider Lifshitz droplets will time be abundance Vb): of p. ): separate Re its phase American pressure which preferentially diffusivity Cn diffusion x(P) is from to we surface. to Thus the 10 D hydrodynamic are begin be whether terminal ::::: in needed? is At fluid. between speeds 1977]). have 3 T(P) as 10 from is ::::: number overcome time , consider reached planet, temperature much Eventually, 10\ 1959). boundary needed is we and of a an three em 10- composition the the Xc shown rotation surface to sphere metastable the into this P a by can early The x mixing-length is time confirming A substantially 0 s- nucleate ("' scale, drag terminal 3 droplets Let 0 of , less velocity, fluid drag Astronomical It pressure the cm convection T(P) helium the ::::: 1 measure size important fraction. close Vb at approximate 10 dissolve. for how hydrogen-rich helium is in velocity for b is stage convective provided 2 0.1 energy than 2 forces: actual temperature which em s - also incorporated be helium be so the pressure ::::: coefficient. as would Figure efficient leads > the large (Cameron from 1 velocity to of fluid. s - the 20bg, grown? would abundance, the and droplets in 10 (Paper Tph(x ratio in our adequate of in 1 this droplet A T(P created. temperature However, 8 the ). exceeds the to is radius to theory distorting questions: the s, planet a droplet This preferentially let the metallic choice b): supercooled 4a. Droplets 0 of a a motion found helium Cn helium-rich a 0 size differences therefore , degenerate ) in molecular-metallic mixture P) macroscopic How region IJ..p I), typical Society the (the = distortion Assuming and A convective 1973) 1 can excess is ::::: of (with to are and [Gierasch This the cools so em. be Tph(x of STEVENSON and slight work b 0.05 a and we approximate much by 2 temperature hydrogen a separation? helium-rich / before the grow fragmented droplet, the Cn. where primordial of x(P) D What where within large-scale and droplet the ratio must of For Tph(x, preferen­ 0 equating (Landau down, the , ::::: done mixture helium­ • cooling charac­ droplet further on density P typical phase­ super­ effects of grow. Re speed 0 10 Provided layer = large ) they they is b (29) sur­ (30) also and xis (31) size the for the the 3 P) let x by = to ::::: 8, is s. a 0 AND inefficient. is the droplets gravitational cooling) The a Jupiter, the where where 0.5k sound surface fluid. a calculations could of cooling. an eta!. Jovian nucleated helium-rich where required It supercooling enough decomposition If immiscibility supercooling According regions neglect (i) fall evolve because The quired times the x(P) of ::::: by 0 cascade SALPETER the Since Once eventually the the is T(P); upper heterogeneous and typical downward life 8 toward intermediate the homogeneous T a (1966) = be surface slightly This velocity. are A exact € J energy heat at area typical 0 !J..T along to of the (iii) is x of 3 are are much helium can Eo For is NASA superheating ' bound 1 of (iii) 2 and T::::: the ensure much the distance a to < ::::: relative is the efficiency flux acts droplets near fluid the for as values J are energy helium-rich be formed region. grows becomes a at energy (Stevenson x supercooled 10-s latent P Figure the larger more output. the region = 10 occurs interparticle flow 0 Forb rough the nucleation found just center to ; to separation > homogeneous the as Hv. 4 Astrophysics a region (ii) that less to ratio 0 K, nucleation the P '1l right-hand-side nucleation ., pb of 4 a entire to release and to energy heat a of by per because 1 maximum P a At the = exp (see droplet of is 3 4b, ::::: nucleation so (see gHJ: estimate 0 In than the and larger, nucleation would by T. about servomechanism, droplet, of successive helium not P ( of immiscibility. < is 1 surface right ln -- we droplet Hba) per our ao4 it [ It supercooled ignoring em, 1975) = parameters, the mixture necessarily P Fig. Paper 2A is the is rate x(P) YJ much has has convective separation, is -a finally each Ph 2 < at atom falls. ::::: subsequent 1 released, (k be then we rate level planet. € is almost heat stable em droplets least P of 3 ' o and and 8 4b): 100. particle, composition of k when been -3/2 !J..T)2 = 1 I). rather even which indicate Data droplet find rate fragmentations, 8 possible, less !J..T, where at The T for is size this more get droplets x boundary to !J..T flux fully of 2 P (i) The size. ] given inhomogeneous (and Consider, initiated, 8 than the Since layer, the certainly smaller. > it heating we supply order energy IJ..TfT::::: = and complication. ' times, than nucleates System a ::::: is is P and Tph[x(P), is can keeping droplets to Xo. analysis, P convective. droplet addition ::::: Regardless equate theoretical the 1, not that the < 0 clear unity, begins by then pV , and diffusion per spinodal 10- produce pb so Regions v. x x P Vol. 8 release the of up it super­ super­ 3 3 highly 0 Feder If is = = • I0- these gJH, small three now, 2 His that unit A (33) (32) For and will but has out the the the the the the the the are Ry x1. Xs. re­ we P] 35 of to ::::: 2 • 1977ApJS...35..239S inhomogeneous evaluate The it the there thickness relative helium upward expansibility appear find down helium where because be "convective Salpeter P from given The other the homogeneous Choice drop given that evaporating, core this, helium begin evaporates, critical In hydrogen. being evolve During the composition homogeneous No.2, does Consider, Consider, Later = found Figure © phase right critical thermal uppermost !:1TJT P is equation !:iT, a parameters is helium-rich the by enriched as American to 1 along 1977 homogeneously !:ix region not • that of phase this the no droplets well-defined with to point The in 1973) the form. it phase approximately side d. 4b diagram and x ~ = take nucleation. 1 point fluid loses This the the for unstable phase and enriching the overshoot" the x (say) now, penetration 10/:ix and negative (i.e., velocity x reaches diagram ensures pressure (17). 2 region. intermediate = or with parts This is x example, diagram E layer most into solute evolution, is - phase phase as could buoyancy (x can ~ ~ x a the then droplets of degenerate itself. Astronomical reached 4 x the j(o and In a , predominantly 0.05 3/:ix( helium, = 1 is of the account function modes the • from density likely the Vc. the interaction that In nucleate slope the gradients boundary However, indicated Xc) (Stevenson Its gives mixed. be drop this thermal For boundary e by and ( droplet ( pjo a evolution critical This lJ) of Gierasch « limit ~) penetration misleading, or fluid inner below, the case equating fall the eddy and first. never but of X· In a the the M, the layer. the such and of discontinuity unique ( ' eddy where HYDROGEN-HELIUM predominantly T)x,pl This the ~P) eddy is innermost and all can penetrates composition is d, merges total structure in until eddy are region center it across metallic of has shown, and The the exist still nonlocal 1975) composition, the phase 1971 helium Society Figure The must the then impinges inequality its then ' convection grow. of « d helium a solution it layer however, distance begins assuming predominantly inner « small initial with for either size ; of droplet 1). the way with be we temperature in helium-rich diagram Shaviv continue HP, of the hydrogen­ 4c. hydrogen­ core the thickness, evaluated any It which effects We l layer exists typically the region Xc. content. thermal moving without • to we helium. the reaches kinetic on for helium Notice planet. would layer. arises h Provided After since layer inner must slow with that first then find and and (35) can the the are the in­ on as at to is FLUID !:ix. effect fluid it eliminated eddy fluid efficient the most more where roughly relative (o of droplets. is theoretical not helium. parable particle, phase inhomogeneous immediately h heat function before limited verge 10 overshoot thickness 10- able that between Regardless Thus, For where energy the jdxfdzj by !:lp In ~ p The 3 Suppose, follows ln inefficient, an surrounding in s. 3 Suppose lower the 10 vc "waves" § pfo (i.e., ~ from cm likely. eddy probably diffusion DISTRIBUTION might eddies of Since helium-rich general uppermost the II, separation gerr the 5 !:ixo[( to ~ to ~ by We 2 separation to em, To!:ix. In the nucleation. of NASA homogeneous the Since s-1, the the is 10cm the density second that 10- can eddy was one is calculations T)x,P the by supercooling, D, of it shall now, become less fluid the a these high-speed in since as work latent surrounding a occur, a 9 takes droplets of nucleation, transport the discussed phase fraction occurs nucleation, convective !:lp In Consequently, cm- is In comes the layer. fluid s-1, effective ~ than Jupiter, this the h~ Astrophysics now portions term inhomogeneous is is T p) that than much gerr helium process. > done -0.05, the heat droplets 1 latent hotter Since ,..., accompanied since R,p buoyant l ; amplitude into by 0 eddy ( here, about 10 Droplets we show to ~ gjdxfdzj = reach arises (Stevenson droplet 8 the provided the !:lp, + heat for and against is 4 ejection nucleation v/1 deceleration rest. 10 -gh of get more it the s fluid say), of whereas nucleation it velocity. diffusion heat "positive"), (I than growth, if 9 for pure began requires ambient where ambient this that follows the em 10 a h because through begin the inhomogeneous _ other if The dz. dx it )1/3 by size 5 velocity ~ the would exist gravity is efficient). g a)(a the eddy it em, (the excess is of hydrogen. eddy this Data o layer: 10 at ~ 1975) by (o droplet of eddy loses to . more of Nevertheless, < small would during 5 to and coefficient surroundings most an that at ox 10 is In fluid then In largest medium. em. instability the are the of heating order 0.97, 1 grow is penetrate 3 in so such unreasonably is the System pfox)r,p separate is p) em indicate the em to much some removal helium-rich fluid if dense volumes This layer. hjdxjdzj penetrating completely part T,p then separation only convective For is the transition A penetrate radius the which s- at eddy: knT a conceiv­ J , ( similar on of layers. strong is means 2 larger of a recoil (since of Since begin com­ layer D than does ~ that rate still (38) (37) and (36) 251 per out 39 the the the the the by its of of = in 2, ~ is ) 1977ApJS...35..239S thermal that than would entrainment. equation inherently would predicts separation elsewhere. 0.1x. ably The ture that the will (D schematically satisfied, decreasing 252 [24]) function region separation, to hydrogen-helium temperature (He) of X We FIG. © form, increasing + inhomogeneous in predicted be criterion gravity stable the In American and conclude v) decreases have 5.-Temperature the innermost be larger energy Paper and of ;:::; D 1:1TfT;:::; (9) but thickness the very the is pressure temperature 0.005 decreases. possible). A no with the indicates time. even waves accompanied in for central Since condition for I, within consequence significant because inefficient, temperature that or planet. Figure we cm respect 10/:1x, layers Note Astronomical overstability. helium. if P of the 2 found layer at it Much temperature (or the regions s-I, that the The the the T However, is, that inhomogeneous the 5. externally. (eq. radial and for equation to the total effect phase so will so inhomogeneous curves inhomogeneous later the gradient (K by in now HELIUM (K convective the overstability helium of of [28]) temperature + the coordinate amount internal not an (D), + the Overstable on increases v) A, temperature this (schematic) diagram early the diffuse increasing v)E in may differ (35) ;:::; Society the B, a planet, This composition SEPARATION the helium 0.5 > is heat C, application of stages distribution be STEVENSON layer inhomogeneous (D predicts overshoot r) as D is greatly that cm ejected "interface" is flux gradient despite marginally in layer are region modes the (Paper illustrated core + 2 tempera­ satisfied. -P of -r • gradient a is s- is v)x in helium external cooling Provided x helium prob­ 1 begins lower. from than fluid order E and (eq. as the are (so or of of in ;:::; I) a ' AND flux from for inhomogeneous it the tional release with radius. nor proceeds prolonging scale significant formed ever, evolutionary approximated case, dominantly luminosity The Fig. ward, rate homogeneous we substantially x (1967) (even ofthe differentiation consider x this heating. moved where where of 10 approximation The mass related the However, by 2 2 becomes SALPETER The To Assuming 8 masses -10 = assume to inhomogeneous the the becomes very is discussion inhomogeneous of gravitational gradually 5). His the 0.12 of convective an as conclude, x but overstable and 9 (dMjdt)He Me effect (The from the and down 2 which to early em, the NASA Suppose The the what + near with examination atmosphere, roughly T is this for that x fraction internal) and thermodynamically planet or the 2 1:1x 2 inhomogeneous precise Flasar helium-rich , the differentiation overstable a outer of less, M the sequence according stages a correct is different heating the as decreasing is 2 an 100'J increases the of§ planet. fraction thermal significant • distance He during x QGrav eventually helium layer mass Astrophysics helium always 1 modes.) temperature The flows, is QGrav• and the energy the metallic efficiency can "' "' inner = layers IV of the value 0 (1973) energies layer of • (1 0.06, layer vertical can core mass It of the of this procedure be indicates ;:::; and of the rate differentiation, is An differentiation modes, evolution from to - and and of given counteracted H leads the the separation the region of ( release effective large. As transports be of planetary 41:1x the to d:)HlH' layer and X stable the fraction composition a in respectively, early indicates QGrav and that d of at 2 that inhomogeneous layer outer core. for the estimated. the heat planet )(1 which thermal depends separation compare at a 2 by that which calculations to favored, miscibility helium molecular Mc We the that gravity becomes the is a + planet the has is planet is Data with stages for depletion is temperatures. is content We of helium sequence available therefore 3x neither thickness can boundary shall can mass. negligible of convection of has required some the metallic formed a that 2 structure constructing respect on by assume ' ) required on helium where cools, total the field System For the cool), an be of changes encompass of we the planet. even the estimate nucleation. fractions fluids. curve. the of helium the adequately evolution). the very separation of of adiabatic, of for Jupiter, planetary find efficiency g. effect (gravita­ to the (case layer the d between that for thicker. core, no (a we cooling models mass Vol. centers helium energy In that Kiefer excess of above large­ How­ to small Since Once Thus good from heat core d pre­ first (40) (41) (42) this this our up­ the are the do an B, 35 of ;:::; is is is is if a 1977ApJS...35..239S effects considered changes index in P tion with which Nevertheless, (eq. (especially decreases during gravitational in According temperature that the monatomic through total the where be T for (adiabatic) adiabatic assumed assuming corresponding differentiation. Jupiter. dT./dt, must region where ratio No.2, Cv T 2 1 1 We Te helium must if and ~ temperature atmosphere changes © x (47) during most a [5]) heat of n 2 be 2 n proceed American Cv 1977 P. are systematic where and the ~ were Te convective is the (Similar Pe x change the radiated. then to index. is as is 0.1. is affected significantly affected content lie content secondary 10 x in of to core, constant by early the average the differentiation be 2 the inhomogeneous whereas much the T approximately 8 constant. (~)Ph= on numerical the the acceleration the Te ~ implies 2 changing ergsg-I, now The • constant. is specific (Cooling EGrav to As helium at molecular to The Eth figures 0.1, Eth is the Eth• !:::..T2 pressure stages also --~---· din H-He outermost error energy the of The (Trafton the the T Astronomical less highly to dt transport core ~: in thermal 2 ~ same ·= ratio and is the hydrogen ,..., "' only T y boundary evaluate + helium effective the (A yC,Ta!:::..x content ~ heat 1 ratio therefore apply calculations of the rapidly phase To ~ of yCvTo (~~)Pht::..x2. !:::..T temperature rate The 3gH planet and that release (;:r. at adiabat, is change slightly of 1.5, typically envelope. first also differentiation.) inefficient ~ level 2 dinT. per and actually layers) constant. , layer. above Te energy Eth is to differentiation transmission dt where 3 P diagram ceases HYDROGEN-HELIUM the decreases, "' ,..., H approximation. by temperature, 1 g 2 of here unit x may changes is than and Saturn.) , Mc, in Stone to from 0 Pe, ~ affected in ~ the the 10 cooling · so diatomic. the 2 is indicate because 20-30%. Society EGrav 3 less mass 4 4 n n We actually defined ' increase to and inhomogeneous that T an K heat helium to x The is also x atmosphere 1 (Paper differentiation 1974) the than We 10 little, assume and the upper 10 the n and by is rate dominate.) opacity 3 Let can 9 that boundary indirectly proceeds, helium transport adiabatic therefore conclude since a cm decrease to it cm Since and increase average content • can y change 1), of during would Equa­ so bound all T 2 mean is these Provided 1 (48) that s-r, (47) (46) (45) (43) (44) the the the the for be Pe be be of at is n FLUID evolution than helium because prefer sider exceeded evolution, evolutionary apparent drogen-helium designation about limited gravitational is For Both metallic recently wllere rule phases sidered. entiation, equation envelope entiating distribution 0.09 ation, complications dramatically T discussion In dT./dt that molecular-metallic model tion Conservation and of by adiabatic order. Equating 10 -1.5 by 1 indeed This In Consider The Let 9 the conjunction = 20-30% yr example, requirement increases the requires Jupiter to more DISTRIBUTION "hot" in K/10 10 the x for Menv to molecular-metallic whereas the In ~ must above section 1 planet. outlined solubility x A 4 Q this hydrogen be (within of 1 NASA be (and 2 -1.3 ( for a K the in Grav the model 49) in QGrav once qualitatively general ~ 9 critical actual when few of the x is "hot" a) the of and mixed when yr. is have a section changing implies the a" energy 0.07). which 4 calculations present next ,..., the of with calculations 10 atmosphere) the _ VI. Paper that The not K/10 corresponds composition its hydrogen-helium or Saturn megabars. region In 2 with initiated, cold" 24 "cold" large that = the helium for (1 above helium Astrophysics of the mass different x central this MORE primordial Kelvin 5 cases, contrast, section, ergs or" with transition, differentiating "Cold" 5 temperature QGrav - hydrogen 9 arise helium times the 10 how release Jupiter coexisting last the era yr, (from x I pressure x the starting restriction uncertainty 9 cold" of may still s 2 new 4Menv indicates molecular 10 )(1 GENERAL starting suggests implies years. -l whereas actual both (Hubbard distribution results that hydrogen the luminosity-time 10 = temperature in time therefore (§ 24 more helium we are have at + Starting in of is to feature 9 Hubbard's already has x 2 II), eq. ergs requires determining need molecular helium years) present, the when therefore 3x the point preference consider transition hydrogen Sector If applicable molecular by in temperature rapid already points Mc!:::..X 2 of these· not [47]) for 10 homogeneous s- )T is CASES first-order planet Data that molecular that Saturn's or mass hydrogen. the not a in is 24 1977). 0 transition 1 relaxed. the corresponds Point the has have factor begun at dT1 (i.e., reached dt the for dT.jdt ergs III in the cooling. 2 then Tc(H-H imply innermost r~sults that are the helium of homogeneous, the fractions. least envelope. homogeneous ~ been System pressure must the a g in Gibbs is of and makes Jupiter relationship been For H the s the only - from considered molecular­ luminosity hydrogen­ the additional - of molecular differenti­ envelope. character Figure not dTefdt Menvt::..xh its ~ anything 1 · was has effect We depleted metallic and 2 be Saturn, suggest implies 4 planet. several helium ). Differ­ -7 simple would evolu­ differ­ if phase x to early or itself first­ con­ con­ only 253 first The The ( The 1 and less hy­ the the 49 K/ an _ ~ ~ of 1. 5. ) 1977ApJS...35..239S 254 P in increases temperature.) evolves occupied excluded innermost molecular megabars. As most hydrogen ward point point content comes and assumption diagrams, the helium-rich molecular two relevant covering molecular fined, amount metallic metallic FIG. = Consider, this © helium-poor the Cameron very P B, composition along hydrogen-helium 0 6.-A and American possible, "cold" and ~ along pressure. planet of layer layer (x using layer hydrogen-helium phase different the by and region We 3 lies this fluid helium-rich = from the The sequence Mbar a first, at rocky is the any 1974), x and the case, assume small exactly phase 1 A formed is continues dashed and phase ) becomes metallic phase. metallic (shaded (b) P- cosmogonic (a) If becomes is cases, more form given lower (see Astronomical the the T p inhomogeneous, no a core. metallic of but rocky relationship p Pc excluded is time droplets phase on on that heat phase molecular Paper line "stable" fluid fluid formed, equal H(Hel depending element not dense. H and top phase in the 2 The (He) to is core. flux the slightly in is is diagrams Fig. crucial droplets of reached considerations form evolves phase to region 1). just grow. contract, ~ less interface Figure center a is I I I I I actually boundary. Once ~ x I I it FIG. homogeneous This Nucleation of 0 transported 6a). beginning case fluid , or while at settles and upon Society is boundary to 6a,b fluid helium-rich C. for more Meanwhile, of when into shaded. of is 6a. This a our existing STEVENSON is in the An a lower the the between a the macroscopic homogeneous whether into evolves sharply Eventually, reasonable to dense inhomogeneous which entire argument. There the the "cold" occurs (Podolak planet be then pressure metallic The through • within for a helium inner­ compressed phase Provided layer molecular than actual and this up­ the de­ the the, the be­ are A stable X X at is the layer. helium-rich AND (d) helium (c) planet. case. metallic layer into Subsequently, this Pc ture. the position simple: content. drogen is tion reached more metallic molecular happens helium since shows outer and dense, molecular-phase molecular thermodynamic phases, by H SALPETER As These discussed the composition (The phase begins interface, is the compression core the layer X The dense the phase molecular or then the into pressure diagrams the content. at as phase, simultaneously new NASA becomes forms We helium to planet begins boundary which metallic layer FIG. phase excluded in shown situation molecular than reached be later. metallic fluid (c), then is 6c, have, tn uniformly even coordinate represented composition. are to principle is d X X the layer insolubility the Only continues the the Astrophysics still added In effectively form almost the region. in inhomogeneous. slices phase at in molecular toward innermost molecular fluid hydrogen when of the fluid Figure P remains subsequent retains which in the at depleted to of P course, = Later, A. latter and the remains already the by can the this continues to P rather the a Even occurs. A 1 fluid the region. process subsequent molecular 6c. more equally three-dimensional not in cool, its phase steady-state same in phase without case, less of metallic-phase dashed (b), Data later, Figure evolves assumed (This He primordial He evolution present. helium. a uniformly thin helium-rich The an a Notice dense ceases well as is consequence to the molecular at time inhomogeneous line. always System changing evolve is intermediate and the into 6c. (d), former be contraction insolubility The configura­ Figure a In to labeled than that is that must the the metallic original Vol. general metallic helium mixed, P, (a), be a rather along com­ triple triple little mix­ case T, less hy­ the the the the the 35 be 6b by of x 1977ApJS...35..239S just coexisting in toward the reason remains presence separation. separation (see for ciently for from convective assume example, at content evolution. This droplets is is is Jupiter many molecular homogeneous mixing helium helium helium heat which while helium begins by plicated Notice fraction amount are of at of our degrees convecting evolution The ward molecular mixing upward. analogous of No.2, fully This Even Return, ultimately Band heat helium-rich applicable opacity (P, C © the the 10 the atmosphere phase Paper triple from being is the atmosphere. the and 10 the American in T, orders to 1977 solute slowly, is will rather upward separation, separated depends is molecular that is a through an metallic helium later remains the of as up when fully years of then the In x)-space. constant, that the be Jupiter probably of core than the that phases. diagram region considerations now, the may are layer point freedom unstable to 1), formed, be fluid. work heat Figure depleted into primordial an our present in metallic a derived proceeds the mixed is rocky of immiscibility to molecular could, the simple rather we we D/K fluid more becomes the have and it be inhomogeneous the fixed to begins the on primordial is added layer, continues context, evolves engine magnitude work hydrogen phase. Jupiter, required is Provided found is consider the The situation Astronomical at given Figure The 6d beginning :::; or nearest model.) innermost the evolution, are thermal nucleate molecular-metallic now core. situation, internal occurred and not at inefficient. in in dense most picture fluid from I0- latent heat shows growth atmosphere required exactly to value final the helium principle, but available at helium that progressively If, layer heat in (Graboske prohibitive into can 2 the The be 6a the , then to (as (DjK)1 to the to at region Notice the as above the still. triple flux daai contraction, adiabatic than so we diffusivities, the heat, heat it from where (zero the larger depleted flux can of but Figure to redistribute A. helium evolve a is and question do the seems we so criterion molecular as would HYDROGEN-HELIUM the atmosphere, relative lower The solute triple to discussed content, the layer Fr, eliminated be These atmospheric flux 12 point Fr, for there we either the which become supply must) point expands or triple helium. work the Fr, separation completely that temperature) upward Society during a increased. which during actual et current likely, 6c: discussed even heat to of even all metallic this molecular point. content boundary more begins molecular where in is is boundary. to al. is beyond the droplets Jupiter is for lower point the composition, be of some transporting no energy respectively, helium. added and whether region fluid determined conduction helium more radioactive in If 1975). to is was amount innermost now. convected mixing by but electronic the the the depletion transport enriched. complete Droplets this D fill state §IV, to • of directly mix layer evolves the helium helium helium helium of in§ above. In acting surely which which and Provided other more com­ these early suffi­ suffi- early layer form form state fluid of now case The The For up­ We the the the the in­ V. or of of of in is a K FLUID since for comparable difference in in in likely. the the metallic that than the velocity time where helium become phase previously by boundary 6a speed follow core). of cient diffusive-convective amplitude envelope is diagram, layer the negative the mined heat happens homogeneous s-\ since outward a diffusion out by by positive It We A small. energy and fact, the virtue inefficient, Jupiter thickness stable metallic actual inhomogeneous the pure at that seems of Rayleigh- flux. and DISTRIBUTION the to the (Fig. only at now vis diagrams However, phase the We Figure will time instability. the quite equal. double which fluid NASA from is (in disturbances characterizing of at temperature case, of is units. coexisting the An relative between Djvb amount molecular-metallic slope! phase discuss might "'A the then that suppose 6b) likely, determined fluid. the t to the its then the metallic boundary later, kinematic additional 6b, Vevt, immediately Taylor the likely) overstable is but layer the layer, the will The :::; helium the be planet the of molecular-metallic sense boundary the (Chandrasekhar beD This Astrophysics the Clearly, a to center the 10 therefore, close of that Equating molecular-metallic the Paper velocity there dashed form which 'TRT that solution. few helium 5 planet layer of the small metallic densities work that instability case em. of phase can the gradient :::; discussed overdense (at somewhere excess, 0 to thickness viscosity :::; by wavelength,\ center the complication times B of the exists even w-s modes), I the (with the where occur gt\f1p the The done in but value is line thickness beneath 4vp continues indicate the of content the that t is early evolution upper Figure Helium hydrogen phase. of expense more ceases cm self-consistently the in to instead a Djvb, of in planet. must upward then the is redistributing ' if the slope in and the thin Data of molecular 2 the 1961) Vevf the the Figure it rRr between the molecular-metallic now s-\ evolution interface. limit § coexisting it. that D/K in heat molecular is 6b dense l1p be III). to take where The inhomogeneous interface presence molecular planet. transport to of can maintained can gives forms of latent Typical possible the of Here vb System no possible. rate, is be very transport transition of(D/K) ensures contract, this 6b flux the which to :::; What the then participate theoretical the longer the less molecular layer than ceases vb the attain w-s a heat metallic is large Vev is The density carried and helium phases Figure purely phase, moves values of deter­ is phase dense arise: in (and, as quite layer 112 time then that The (51) (50) is was and 255 has the the the em for in­ FT by an or so to of in in is is a 1977ApJS...35..239S for P fluids from droplets choice in terized the small stratified sively direction ary droplets forbidden a convection or for layer For For to merge motion helium-poor helium-poor upward. upward Subsequent evolution of continues the not at 256 the A 2 chemical ~ The ), P FIG. Figure fall the 2. two molecular planet breakup © arrow much as but 0.1 = Dev V the (about The at near 6 more a American 7.-The P1 in v type by of subsequent reasons. they into em. substantial ~ of P of remain in rather ~ (with helium-rich of have 7, a to the cools, phase region different helium potential to 10- = size in 10- P the helium-rich the but discussed of the fluid dense Thus metallic 10 merge fall fluid P the breakup = the unstable convect 9 3 more 2 the lead 4 1 First, arrow it helium-poor em interface so s) P are diagram ems em, at in entire under same 2 may diffusion relative above. the equilibrium layer: t~ helium-poor , Astronomical of evolution than with P fraction from s-l Figure difference to not helium in most region to fluid -I, this the = "cold" in in of density. instability actually a ( and helium-poor P within the in (a inner --2 gvev gravity. the vHP diffusive-convective helium-rich a 1 Figure figure the §IV. in , t These in to and very phase of present-day and 7 ~ is helium-rich time of fluid. Figure transport the metallic case. is )1/3 stable Jupiter), in Droplets shown region tending 10 the the a is with an evolve Except rise tend actually thin droplets 7, scale 5 time this similar effects coexistence: They for . is s order innermost The Second, helium-poor becoming to case. the Society layer 7, typically to layer and and as ~ such in case at they height, phase. merge coexisting break to : I core for these become t to evolve homogeneous the value phase He(H) some STEVENSON of expanding will P may ~ drive helium-poor The of Fig. (P A would droplets is the magnitude a > direction 10 at away fluid X ~ 1 regions), not diffusive with nucleate progres­ droplets P solution For so actually 6. 9 charac­ there for • bound­ P shaded shaded < helium helium 2 10 in s stably phases • these layer = Provided P take from stop into The (52) and em. the the the the P2. as < of is is AND P important, times ~z paratively so case, found f3 interface interface In the wavelengths the to whence tude phase Stevenson I situation situation (b) (a) core and density dial the metallic velocity the waves have region penetrates properties somewhat small stable, for difference densities For values by 1 ~ SALPETER ~ The Despite We will h break be other a the the actual metallic 10 planet. simple helium mixture, Stevenson 0.55. will ): the gravity the the predominantly their and transitions occurs 2 the now is to then equality the in metallic I interface. helium-poor of em. we interface (i.e., interface, Vc have limited same is will leads away NASA in are words, have Figure distance then (homogeneous) the between stable of like Consider a other own (197 a so mixing-length find "isothermal." consider PVc eventually and At phase Figure b) which distance waves admixture, the then and have and wavelength equal, larger a it 2 Pmet density. Figure to 6; small this from expands The" f3 (1976), and length of lower is pure metastable) only (hjl size ~ The the will 7 Astrophysics another ~ (c) also relative possible f3 be densities because measured excited the ~ ~ the size, will larger molecular Sa distortions helium metallic pg(fJ- a an the are however, Hot" < h amplitude similar still the considerations hydrogen helium I densities by Po(l 10-4( into 6c. evolve case interfacial given phases. shows see hs-xldd region phase-excluded incident 1). until much expand molecular viscosity eddy interface and the exceeds theory, coexisting not determined The helium to novel the - by amplitude for Starting § of in ~p at core a)(h/Hp)h at the III) to content into lower by upward be it layer a!:lzjHP)' fluids, later convection. are molecular opposite one the which of drops r/9 of the the In this comes phases, of Figure completely at and feature: molecular on De has wave distribution the still the Data gravity metallic the the and substantial roughly molecular-metallic in above a ' particularly compressibility position amplitude ~ Point phases respectively. the· than form triple of of pure already two the from the 2 interface mainly in apply, case 1 10- but 6c, amplitude) proceed rather phase. 2 size System Salpeter fluid contact • interface. large-ampli­ evolution influence phases waves this the 6 In except molecular­ a point. phase. fluid v.(ljHp) destroyed. the at where are a is are ,..., diagram Salpeter formed, and primor­ at P than ~ at Longer density Vol. not helium by 10 of begins equal­ in likely a some com­ = 0.45, were P 2 with with (one that The (53) (55) (54) The and this few Let the the the the em yet for 113 P by 35 = of of of It 1 , 1977ApJS...35..239S pressure are equilibrium like metallic No.2, (bottom), subsequent equivalent In helium molecular have at drogen, arrow. metallic FIG. (a) A © always Figure and lower (top), concentration, American P,T 8.- 1977 These P, P,T then layer helium-poor fluid T rise P envelope, to The helium-poor evolution density less = evaporate Fig. with forms. droplets 8a, along P H( droplets "hot" dense 2 6d. • He) the so while than From the The and Astronomical metallic they phase eventually dilutes case. at for droplets phase the final helium-rich nucleate a P. the any helium-rich Xo The phase begin < boundary, droplets state the diagram P fluid plausible (B) dashed become < HYDROGEN-HELIUM excluded helium at is to P nucleate molecular 0 at not • at B. region rise, as Society In line P as A, pure shown These 1 indicated phase I I I region (b) content = He(H) shown X represents from helium-poor maintaining P forms. (middle), metallic 1 fluid, since no is diagram X droplets the • shaded. by of by longer In Provided so fluid it the hy­ the the the the (c) is a FLUID P fragmenting. in in change the never the they rise pure exists. either face larger arrow indicates Presumably, the steady-state medium, surface, that molecular to stable, separating primordial, forward. away presumably but and at actual drogen considerations. line homogeneous mately the metallic helium now because phase state so as Figure Figure molecular molecular but origin. by The As The In = this P as Figure the helium. the way density region segment, same to have continue at we P convection the slightly = is DISTRIBUTION to Figure metallic and case, excluded 1 x, the This dynamic section homogeneous grow in pressures molecular paths since 8b. have subsequent We P 8c. P planet in similar a phase, ensure of to but assume the 1 unless that hydrogen NASA equal Figure constant-density as = Eventually Fig. 6d. VII. become fluid no near layer pure convective There two phase phase the layer first because of also P a exists, Above by pieces in region The the much displaced metastable 1, to microscopic 3 the The on very layer, longer the can 8b) DISCUSSION hydrogen • rate mainly for greater We density. which region steady they the molecular P the evolve hydrogen summarize at The maintains extension lower has forms. by waves droplets 8a. that Astrophysics phase instability are exists boundary excluded droplet = helium "hot" buoyant be long shall P at of evolution P of "cold" perhaps larger droplets. entire nucleation extending are molecular this P a As interface rise, this ~ which characterized an transport. now the the > 2 evolves state stability. by the diluted along than from Helium-poor at at lifetime, is is transported P not P At usual at metallic evolve and 0 state continuous inflection AND because evaporation a is layer, metallic 2 of distribution which and being regions. determined interface. compositional evolution droplets than two resulting is in remains by constant-density presumably evolution. Pe, discuss is the P even the region which the to in the is "cold" composition toward This sufficiently not qualitative is CONCLUSION = up at ( losing layer begin within The the cf. interfaces, droplet quite phase to same but Figure the reached continually then 1 P an qa density equal Data P lower to some catastrophic. phase em 0 hydrogen passes become § This the pressure • as discussion = fully coexisting by is expands even cannot develops V), of metallic the It addition dilution to of inhomogeneous They their a starting as the low quite P boundary. would Thus a by at radius details convection. illustrated converts diagram is has level System 1 form metallic a 8c, are the semi-infinite pressures, atmosphere, is the could through mixed they the fashion uniformity. is delocalized latent droplet. rather hydrogen­ at essentially relative helium pressures, shown evolve the can . different. at interface enriched approxi­ the straight­ and ambient droplets droplets (labeled then P "mist" droplet of toward a of points, cannot of phases in before earlier which break inter­ level, layer since = back pure such final this, now heat The 257 un­ hy­ the the the the the all P be In to as in in A a 1 1977ApJS...35..239S the ated geneous ceeds interface initially impact is inner layer inhomogeneous than the temperature region to helium-poor mixed helium poor premature separating separate. and depleted atmosphere and dominantly except after, a evolutionary case. depending Tc(H-He). coincidental complexities I 258 molecular molecular metallic the mixture helium begins begins begins six of of of Tc(H-He). Later, Unstable: and more helium It The Sector Sector Sector Sector Sector Sector essentially characterize a helium possible © planet. metallic eventually stable a is the by hydrogen-helium metallic Figure and the III. immediately, region predominantly American evident a complications to molecular when in to exist, dense composition. abundance. of and an an predominantly coexisting metallic region II III I helium-enriched II from has III Subsequently, I core separate a A separate the and layer. (cold).-During homogeneous our is If at inhomogeneous (hot).-As (hot).- (cold).-Similar case, evolutions inhomogeneous helium on helium-enriched then rise (hot).-As phase. 9 starting typical similarity ( arise primordial sequences no transported slowly Tc(H-H than separated cold).-(a) the droplets this metallic shows metallic that (or the metallic lead considerations since the to and density regions Much begins layer except envelope. metallic stage. the because helium-enriched dilute out. homogeneous molecular This detailed core. Astronomical Subsequent out At usually to sequence evolves it points relative a 2 one of ) helium coexisting nucleate corresponding the the planet, combines of also first, layer the somewhat is later phases. At droplets helium also intermediate there and by composition. very discontinuity. Sector Depletion the To molecular Stable: possible that and inner the fortuitously upward dilution planet planet phase Sector first, the molecular in numerical by an complexities begins Later, layer. molecular give have a inner toward still, (probably) and phase, early core can values of densities Figure on four helium-poor the early we depleting region core from inhomogeneous evolution II molecular a events If the an molecular cools nucleate cools becomes be helium helium further the the II depleted somewhat shall are into begins region of Later, in to molecular-metallic of and inner the helium then interfaces. begins evolution, Society corresponds indication effects regime as the of a to I. the the A helium thermal begins form, layer is calculations inhomogeneous very not envelope. predominantly down, metallic metallic would many the down, Some Tc(H-H STEVENSON possible consider of small forms mixture, to formation helium distribution. atmospheric region) begins evolution is alternatives, to exist, an phase, of to separation less the from of uniformly similar molecular coexisting molecular is difficult similar envelope, while form. to from enriched inhomo­ concern formed, as form. amount helium­ of • Sectors be history helium helium phase­ "hot" a 2 of separ­ These layers phase dense layer. ) to three Soon high­ these Provided from then how pro­ to that rise, pre­ and the the are the the (b) be to to of of a AND is the character countered affected The lasts luminosity). continues most helium-depleted we mediately high-luminosity phase rich. "age" central the tions. tive, models helium understood planetary since (Bodenheimer about considerations using Saturn, 8c. (Graboske starting on Tc(H-He) by SALPETER Consider, In P,T FIG. probably the have helium homogeneous, the gravitational only we about The and the ~ ~ p3 IQ diagram pc several We 10 9.-An (i.e., planetary redistribution point). pressure). for but since consider 6 a NASA a begins = of homogeneous years center to as molecular shall hydrodynamic "cold" et the core the and 50,000 short first, 12,000 appropriate, we the the Nevertheless, This expand intermediate a!. the possible (Paper 1974), time very may better phase also old. becomes choose to cooling center metallic molecular 1975; only becoming Astrophysics time, is The Tc(H-H phase energy K, starting form K essentially taken evolutionary early . neglect generally envelope actually I) but throughout and constrained the and of central choices it first Hubbard suggests excluded Jupiter rate. and cooling Jupiter, case would 2 this to progressively evolution evolution so release hydrogen effects ) the Tc(H point, the metallic = degenerate becomes reach latent the grow a (Sector planet is temperature has 60,000 be retains of 2 helium-rich that not sum -He) age of because but 1977) Data models may would models the by the the region and Tc(H-H more at heat subsequent no are greatly the of region is transition current II the degenerate. is since dominant K. observed = evolution, its the Fig. probably be concern of more are (P very System unstable effects, 6000 He(H} substantially expense prolong In relevant in it of 2 Fig. primordial of ), important this first-order 4c is ~ modified, affect core is this (Fig. Figure observa­ specula­ Jupiter, helium­ but and K. 1 then Jupiter Vol. 1, excess to better is phase Mbar effect He here, since only case case Our The of cold and Fig. im­ the the en­ the for 7). 35 to at 7 a 1977ApJS...35..239S temperature helium deep indicates transport helium slightly. amount calculation continuing central latent than since 60,000 "hot" state tinuity years. words, an clear above. This have would rise No.2, a occurs. gravitational and 10 adiabatic, is later Jupiter the homogeneous (see correct of explanation lesser made pressures (These adiabatic, likely) luminosity Podolak transition within Saturn's ably account about the with Consider, 9 helium-poor Consider, In Similar inhomogeneous consistent x 5 © Saturnian cooling central to years already § interior the that a the that in is it of degree heat for American V). 10 1977 K helium-rich be would temperature starting In if the by oe pressures lower abundance would 10 Helium-poor conclusions until necessarily A alternative and the essentially preceding Jupiter 9 of interior various and that upward unless through in all 9 this for if upward depleted years. rough is In elapse effects comments last homogeneous helium years. (Pollack temperature homogeneous. in the helium Jupiter for evolution-about been finally, the rate the now, is first-order are of immiscibility Cameron exist energy Jupiter, a with the case, excess only model, § point molecular 10 decreased, the certainty. homogeneous various Jupiter V, excluded uncertainties. is decreased calculation, transitions are would encountered, are 9 convective Nevertheless, before convective the rate discussion anywhere in molecular differentiation. core T Astronomical and be of not observed depends years, as differentiation by observations, the present of a assume Tc(H-H 0 et is (H-H luminosity metallic lower released, both in inhomogeneous consistent precise helium, 50,000 and immiscibility apply perhaps of illustrated al. 1974). is compatible which in to is that would latent Sector be the Present-day cooling even region or much model. envelope. 2 negligible, by Saturn, 1977). dilute evolutionary begins then ) the 2 than and that ) much on anomalously = transport. analogous layer transport is we K droplets to a situation the Immiscibility = joins calculation but planet, 10 HYDROGEN-HELIUM and all integrated 20% heat the state slightly about comparable present 20,000 III Tc(H-H2) then the are have not older layer 0 with temperatures those 10 in current Saturn, A the is be when in in and is K. (Unlike Once no but slower this molecular-metallic planet of continuously with years Society Figure efficiency reliable.) possible (see, effects may substantially in of The corresponds larger the then Figure molecular In an temperatures to formations. layer.) density Figure then than because ultimately K. not at of is If progress, enriched Jupiter to luminosity in this allowance age begin for tried an T helium again, estimates models but the have (until from atmosphere comparable :( helium large luminosity, is that the The than 0 the in Figure than 7 (H-H2) form, associated 4.5 fractional has attractive 20,000 example, in case, (but difficult, In expands effective 8b. 1. • Jupiter. present of discon­ to with in§ simple in would of within excess excess begun initial x now). much prob­ other for Provided in Over layer it keep then with with with fluid heat The to and and less less the We the 10 the are an V, 8a = o of in K is is a a a 9 · FLUID reasons. temperature probably predict phase" partitioning (i) essentially extensions ation among cussed, evaluating distribution differentiation), droplets Paper modynamic stantially ture slowly require (i.e., or in one Except ment. unlikely slowly). equilibrium faces. considered diffuse thermodynamic to rapidly with engine cient limited will this helium helium phases. siderations especially distribution atmosphere and Earth the atmospheric lacks However, of ,...., by These We 10- Nevertheless, We In more cases such Paper present diffusion (and the CH paper the also transitions DISTRIBUTION homogeneous redistribution, quantitative 3 seed, conclude proceed, I Nonequilibrium Any masses more than cm the but diffusive in than regions could a by (on An data The to 4 (§ probably are First, molecular in where NASA different not applies. rules diffusivity probably I). redistribute 2 adiabatic. special thereof) dynamic VI) the Saturn. encounter energy would purely for of various observational s - analysis. the of the equilibrium of if any gradient solute second 1 of than helium, is the is helium time. If, em compositions Paper 1 in temperature in minor the the minor , of likely exist helium: Jupiter by the a the any now, provide are fluid basis layer entropy) equilibrium. as the Astrophysics species and principle (iii) droplet in predictive about thermodynamic cases considerations from be droplet material noting partitioning For achieves phase that Unfortunately, layers, that hydrogen-helium and prefer process "abrupt"). considerations solute is just less difficulty radius near (ii) then I growth constituents would First appropriate a to to constituents state At accurate (see§ of (§ minor likely, larger substantially typical in the a partitioning more In (such upward be one may than as VI) a diffuses the each (a a which is would test thermodynamic first-order it some less the and molecular moves would inhomogeneous brief everywhere. ambiguous. and they following continuous. transport which (the likely tends power IV) unique from order not of a indicate not of discussion important of Unlike the constituents. ,...., temperature as that parameter interface, molecules than foremost, droplets determinations of distribution of move 10- which did have the consideration diffuses convective helium achieve (the is be to the at Data to situation, environment more redistributes to prefer solute the minor of (such the because may the not we 6 different for overstable 4.5 the considerations phase be achieved prescription a would magnitude that rules em and phases to helium, at convective several inadequacies giant forms region fundamentally center trapped (iv) interpretation from helium. have System cause preclude x 2 rapidly as the beginning of move diffusivity ,...., to values(§ diffuses constituents much our equilibrium s- view. helium-rich H would will transitions 10 10 gradient the layers, water). § No according The 2 be transport 1 therefore tempera­ close 0, environ­ was III, 9 • at a analysis planets, em for already tens of of the This helium critical to apply: 10 during modes years). of of nucle­ "two­ mixed in Ther­ inter­ NH more more non­ than 8 con­ sub­ heat s - 259 and two dis­ effi­ less not the the the the V), for the em re­ In of to of of in is is is 1 3 . , 1977ApJS...35..239S latent temperature evolutionary entiation of following fluids instability. may knew helium parameters properties of are predictions constraints order Latent 260 metallic tinuity to of ous, --. --. Busse, Busse, Bodenheimer, Aumann, Caldwell, Clayton, Jeffreys, Cameron, Chandrasekhar, Gierasch, Ingersoll, Howard, Flasar, Feder, Anderson, Gierasch, Gulkis, Graboske, Hubbard, Hubbard, Hide, Hubbard, Kiefer, Notwithstanding 1. Numerous 2. Phys., Ap. Ap. 6, Stability Physics Nucleosynthesis chap. 14, R. Mechanics Gehrels Arizona Jupiter 1976, have immiscibility magnitude. © likely 36. The J. eim differentiation Helium R. adiabatic the 599. have J. J. transition F. heats F. J., must American F. heat H. S., 1975, in 1977, 1973, H., 20. latent miscibility (Letters), (Letters), in 15, 1974, hydrogen H. H., D. A. D. H. L. Russell, P., P. W. A. W. M. W. hydrogen-helium and (2d to H. J. major (Tucson: H. conclusions Press), or (New and helium of Jupiter, and could P., are 111. effects on 1971, 1976, R. D., of H., and G. N. (Berlin: and B., be be overlooked D. B. C., Icarus, Ap. 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