Dynamic fission instability of dissipative protoplanets

I( A t . u s

63. IZ;4-152 (1985)

Dynamic Fission Instability of Dissipative Protoplanets A. P. BOSS AND H. MIZUNO I Dclmrtme,t (!f "ferrestrial Ma~ncti.wn. ('arm'v.ic Instil,lion q f Washi..~..ton. 5241 Broad Bram'h Road. N~', Wa,~hin~.,t.n, I).('. 20015 Received January 2, 1985: revised April 30. 1985 All theories of fission require a catastrophic, dynamic phase in order to produce two separate bodies. We have used nonlinear numerical and linear analytical calculations to show that the d y n a m i c fission instability probably does m~l occur in dissipative protoplanets. The numerical calculations were performed with a three-spatial-dimension hydrodynamical code, with the protoplanet represented by a fluid with a M u r n a g h a n equation of state. The kinetic energy in the protoplanet (other than rigid body rotation) is dissipated throughout the evolution in order to simulate the effects of viscous dissipation. Protoplanets rotating above the limit for d y n a m i c instability were given initial a s y m m e t r i c density perturbations; in each case the a s y m m e t r y did not grow during a time on the order of the rotational period. This dynamical stability has been verified by including the dissipative terms in the tensor-virial equation analysis for the stability of a Ma~ claurin spheroid: the d y n a m i c instability vanishes when the dissipative terms are included, while the secular instability (with a growth time m u c h larger than the rotational period) remains. Fhe result applies to bodies of radius R with a kinematic viscosity u > 4 x l0 t`` (R/6400 km) 2 cm: s e c t and hence m a y be applicable to any terrestrial protoplanet ,ahich is not totally molten. Current thermal histories for the Earth predict a partially molten mantle with a viscosity greater than this critical value. Depending on the detailed rheoh)gy of the early Earth. our results appear to rule out the possibility of forming the E a r t h - M o o n s y s t e m through a dynamic fission instability. , L,~s> &cademic Press, [no

I. I N T R O D U C T I O N

One fundamental problem in astrophysical fluid dynamics is that of the equilibrium of a rotating, self-gravitating, fluid body as the amount of angular momentum is increased. The history of work on this problem is a long one (Chandrasekhar, 1969). In 1687 Newton considered the equilibrium of homogeneous, uniformly rotating bodies in the limit of small rotation rates. Maclaurin extended the theory to include the equilibrium of rapidly rotating, spheroidal bodies, while Jacobi showed that equilibrium figures exist also tbr ellipsoidal bodies. Meyer demonstrated that the Jacobi sequence bifurcates from the Maclaurin sequence above a critical amount of angular momenPresently at M a x Planck lnstitut ffir K e r n p h y s i k . 6900 Heidelberg 1, Postfach 103 980, Federal Republic of G e r m a n y .

tum, and Poincare discovered the existence of pear-shaped equilibria which bifurcate from the Jacobi sequence at the point where the Jacobi ellipsoids become dynamically unstable. Poincare then conjectured that the pear-shaped sequence would lead continuously to the equilibrium of two bodies, one larger than the other, whereupon Darwin hypothesized that such a fission mechanism might account tbr the origin of binary stars. When Liapounoff found the pear-shaped sequence to be dynamically unstable, Jeans (1919) proposed thal the dynamical evolution of an unstable pear-shaped perturbation would quickly result in fission into two equilibrium bodies, whose form is described by Darwin's equilibria for tidally distorted, binary objects. Lyttleton (1953) argued against the possibility of fission, by insisting that dynamical instability implies reversibility through the complete absence of dissipation, whereas 134

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DYNAMIC FISSION INSTABILITY the reversal of time for a binary system simply results in orbital evolution in the opposite sense, not merging together into a single body. Dynamical instability in the real world undoubtedly involves some dissipation, however, and Darwin's hypothesis has continued to intrigue cosmogonists. A dynamical phase of evolution is necessary in order to produce two bodies through any fission sequence. Because of the limited usefulness of analytical methods for dynamical problems, the classical workers could do little but hypothesize about the outcome of a dynamic fission instability. As stated by Darwin (1880), " F o r although we know the limit of stability of a homogeneous mass of rotating liquid, yet it surpasses the power of mathematical analysis to follow the manner of rupture." The advent of digital computers has meant that previously insoluble problems such as the fully three-dimensional evolution of a dynamic fission instability can now be rigorously treated. The fission of a rapidly rotating body into a binary system is clearly an attractive means for accounting for the formation of close binary stellar systems (reviewed by Tassoul, 1978; Durisen and Tohline, 1985). Numerical techniques have been used to construct equilibrium sequences of rotating, polytropic (compressible) bodies (Hachisu and Eriguchi, 1982; Eriguchi and Hachisu, 1984) in much the same spirit that the classical workers studied the equilibrium of uniform density, incompressible bodies. More importantly, numerical techniques have also been used to study the dynamical evolution of unstable, rapidly rotating polytropes (Lucy, 1977; Gingold and Monaghan, 1979; Durisen and Tohline, 1980; Boss, 1984; Durisen et al. 1985). These dynamical studies have shown that, at least for the case of a polytrope with pressure depending on density as p -- Kp~ and 3' = 5/3, the outcome of the dynamic growth of a bar-shaped instability is not fission into a binary system, contrary to the classical expectation (Durisen and Tohline,

135

1985). It is convenient to quantify the critical amount of angular momentum through the ratio/3 = T/I WI of the rotational energy to the absolute value of the gravitational energy. The classical workers found the Maclaurin spheroid to be secularly unstable at/3 = 0.138, where the Jacobi ellipsoids bifurcate, and dynamically unstable at/3 = 0.274. A rapidly rotating y = 5/3 polytrope with/3 > 0.274 will indeed begin to form a binary system, but at the same time it generates trailing spiral arms, which gravitationally transfer angular momentum outward (Boss, 1984). The loss of orbital angular momentum causes the nascent binary system to merge into a single central body, rotating with less than the critical amount of angular momentum necessary for dynamic instability. In the meantime, the spiral arms which receive the angular momentum are ejected outward, carrying away a small fraction of the total mass and a large fraction of the total angular momentum. Thus these numerical studies strongly imply that close binary stellar systems do not form by the classically envisioned fission mechanism; the dynamical, nonlinear evolution of differentially rotating, compressible bodies is quite different from what had been expected based on the consideration of idealized, equilibrium sequences. Fission has also long been considered an attractive mechanism for producing satellites of planets. The fission of a rapidly rotating protoearth was first proposed by Darwin (1879) as a means of forming the Moon from the outer layers of the Earth. Darwin originally suggested that solar tides might resonantly excite a rapidly rotating protoearth into fissioning, but Jeffreys (1930) showed that the frictional damping due to motion of a fluid mantle over a core alone is sufficient to damp any tidally induced instability. Darwin (1880) also hypothesized a purely rotationally driven fission origin of the Earth-Moon system, and analyzed the evolution of the newly formed Moon's orbit. Moulton (1909) argued against a fission origin of the Moon by

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BOSS AND MIZUNO

pointing out that the amount of angular momentum in the Earth-Moon system is insufficient to have resulted in a fission instability in a combined Earth-Moon protoplanet. Ringwood (1960) renewed interest in the fission origin of the Moon when he proposed that the protoearth formed very close to the limit for dynamical instability, and that the decrease in the moment of inertia accompanying rapid core formation pushed the protoearth into the unstable regime. Wise (1963, 1969) argued that a fission origin for the Moon would explain several key features, such as the low eccentricity of the lunar orbit and the mean lunar density, and sought to avoid Moulton's criticism of insufficient angular momentum by proposing the loss of mass and angular momentum from an atmosphere tbilowing fission. O'Keefe (1969) similarly proposed that the tidal dissipation produced by asynchronous rotation immediately following fission would volatilize sufficient matter to carry off the missing angular momentum. Interest in the fission origin of the Moon has been sustained by the results of the Apollo sampling missions (e.g,, Binder, 1974, 1977). Forming the Moon from the outer layers of the Earth has the great benefit of providing a ready explanation for many of the geochemical characteristics of the Moon (reviewed by Ringwood, 1979). Three dynamical questions about a fission origin of the Moon have been discussed since Moulton's time, namely the insufficiency of angular momentum, the inclination of the lunar orbit, and mechanisms for the initial spin up. These threc questions have been interpreted as evidence against a fission origin (Kaula, 1971 : Kaula and Harris, 1975), though uncertainties still remain (e.g., Rubincam, 1975). To these three concerns, we wish to add a fourth dynamical question of even more fundamental importance. The dynamical evolution of the fission instability for a rapidly rotating, homogenous, inviscid fluid body has never been determined, much less

for a compressible, differentiated, viscoelastic protoplanet. The classical fission hypothesis assumes that a rapidly rotating body will dynamically evolve from the unstable equilibrium of a single body to a nearby stable equilibrium involving two bodies in mutual orbit. This single hypothesis underlies all of the controversy about a fission origin for the Moon. If this hypothesis is shown to be correct, then the controversy will continue; but if the hypothesis is shown to be incorrect, then the fission origin of the Moon can be definitively ruled out. We have seen that the classical fission hypothesis for the origin of close binary stellar systems has apparently been disproven by recent numerical calculations of the evolution of rapidly rotating, inviscid polytropic bodies. The rotational instability leads to the ejection of a ring of matter containing a small fraction of the mass, not to the fissioning into nearly equal masses required to explain close binary systems. The former outcome, however, seems perfectly reasonable for forming a satellite like the Moon (Durisen and Scott, 1984), and it might explain the loss of substantial angular momentum through inefficient formation of the Moon from the circumterrestrial ring. The question remains whether the numerical calculations, originally performed in order to simulate stellar bodies, have any application to protoplanetary fission. Durisen and Scott (1984) noted that the numerical calculations need to be extended to polytropic bodies with values o f y which produce a variation of density with depth in the protoplanet similar to that in the Earth (i.e., y 3, instead of the value of y --: 5/3 used in Durisen et al., 1985). Our results (Boss and Mizuno, 1984) indicate instead that the most important omission of the previous modeling has been the neglect of viscosity, which is reasonable for stellar models, but not for planetary models. This paper is thus concerned with investigating the dynamical evolution of a rapidly rotating, v i s c o u s protoearth, in order to

DYNAMIC FISSION INSTABILITY

137

The classical results on the equilibrium configurations of inviscid fluid bodies were extended to include viscosity by Roberts and Stewartson (1963) and Rosenkilde (1967), who showed that Maclaurin spheroids continue to be secularly unstable for/3 > 0.138. However, when the effects of both viscosity and gravitational radiation reaction are considered (Lindblom and Detweiler, 1977; Comins, 1979), the two effects may cancel each other and can even suppress the secular instability until the sequence reaches dynamic instability at/3 = 0.274. Gravitational radiation reaction is of course negligible for planetary bodies. It is important to note that all of these results hold only in the limit of very small viscosity. Planetary bodies will not be nearly inviscid unless they are completely molten, which formally prohibits applying the clasII. VISCOUS FLUID MODELS AND THE sical or contemporary work on equilibria to PROTOEARTH solid or even partially molten protoplanets. Previous Assumptions Wise (1963) was perceptive of the neglect in The classical work on the possible fission the classical studies of fission of the possiorigin of the Moon implicitly assumed that bly overwhelming importance of viscosity the protoearth could be represented by a for the fluid dynamics of the protoearth, low-viscosity (inviscid) fluid. Darwin (1879) stating, " F o r an Earth of 1022 poises it is assumed the protoearth to be a molten, vis- inconceivable that separation could be accous body, and in the 1880 paper consid- complished without viscous frictions." ered it to be "partly solid, partly fluid, and Wise refers to the understanding that the partly gaseous," with no further elabora- present Earth's mantle responds over geotion. A fluid protoearth has also been as- logical time scale like a fluid with a viscossumed in contemporary work (e.g., Binder, ity of 1022 P, whereas the viscosity of water 1974). O'Keefe and Sullivan (1978) pro- (a relatively inviscid fluid) is about 10-2 P. posed that the protoearth was initially a to- Of course the viscosity of the early Earth tally molten Maclaurin spheroid, rotating was probably much lower than 1022 P, bewith 0.274 > fl > 0.138 (secularly, but not cause of the strong dependence of viscosity dynamically unstable). O'Keefe and Sul- on temperature. For example, the viscosilivan pointed out that the presence of a ties of terrestrial magmas are in the range of small amount of viscosity would transform 10 2 t o 10 6 P (Clark, 1966). such a protoearth into a Riemann S-type ellipsoid, which could eventually be unsta- Viscoelastic Models ble to a pear-shaped instability and then We do not have a rigorous understanding catastrophically fission. Considering that of the large-scale physical processes that the Riemann S-type ellipsoids are inviscid occurred in the early Earth, though a rough fluid bodies which contain significant inter- outline exists. As the Earth accumulated nal motions, the protoearth must be nearly through the impact of planetesimais (e.g., inviscid throughout this scenario. Wetherill, 1980), the heat of impact was learn whether the hypothesis of Darwinian fission is a possible means of producing a satellite like the Moon. It should be emphasized that we are only concerned with pure Darwinian fission, where the instability arises because of rapid rotation, and not with other scenarios involving formation of the Moon from the mantle of the Earth that are sometimes termed "fission" (see Ringwood, 1979). In Section II we discuss how the Earth may be modeled as a dissipative fluid, and in Section III briefly describe the numerical techniques necessary for following the three-dimensional, dynamic evolution of a rapidly rotating protoplanet. The results of the numerical models are presented in Section IV, and a linear instability analysis for dissipative bodies is given in Section V. The conclusions are discussed in the final section.

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BOSS AND M1ZUNO

partially trapped in the outer layers (Kaula, 1980) or in the primordial atmosphere (Hayashi et al., 1979), producing locally molten regions where iron-silicate differentiation initiated the processes that led to core formation (Stevenson, 1980). The convective processes in such a situation are very speculative (e.g., Boss et al., 1984), but models of global heat flow should be reasonably accurate. These models of the early thermal history of the Earth do not predict or require a totally molten phase in response to the deposition of the energy of accretion. which on energetic grounds alone is quite capable of melting the entire protoearth. Rather, convection in large, partially molten bodies is very efficient at cooling the planet (Turcotte et al., 1979; Kaula, 1980: Schubert et al., 1980; Cook and Turcotte, 198I). A viscosity on the order of 10 ~ P or greater is thought to be sufficiently low to transport the accretional energy out of the mantle (Schubert et al., 1980). The rheological behavior of the early Earth is equally uncertain. Indeed, there is considerable debate about how best to model the rheology of the present Earth (Peltier et al., 1981). The possibilities include (Christensen, 1971) modeling the Earth as an elastic solid, a Newtonian or non-Newtonian viscous fluid, a viscoelastic fluid (Maxwell model), or as a viscoelastic solid (Kelvin-Voigt model). Viscoelastic models may be linear or nonlinear as well. For p h e n o m e n a with time scales less than the Maxwell time (about 200 years for the present Earth; Peltier et al., 1981), a transient, viscoelastic rheology is necessary (Yuen et al., 1982). The Maxwell time measures the time needed for a viscoelastic body to relax following an applied stress. The time scale for dynamic fission instability is the rotational period P, which must be on the order of a few hours if fission is to occur. The effective viscosity for short time-scale p h e n o m e n a (about 10 ~7 P) like seismic waves is considerably less than for long time-scale p h e n o m e n a (about 102: P) like mantle convection (Peltier et al., 1981).

Poirier et al. (1983) give formulas which allow the effective viscosity ~, of viscoelastic models to be calculated from the specific dissipation function Q, the rigidity /~, the density p, and the forcing frequency n. For the K e l v i n - V o i g t model, u = ~ / Q p n , whereas for the Maxwell model, u -- IxQ/ pn. The seismically inferred Q of the Earth's mantle is around 100, while estimates of Q for the present-day terrestrial planets vary from 10 to 500 (Goldreich and Sorer, 1966). For a rigidity of 10 ~2dyn cm 2, a density of 3 g c m ~, a Q of 100, and a frequency n = 2~r/P = 6 × 10 4 sec ~,the Kelvin-Voigt viscosity becomes about 5 10 j: cm 2 sec 1, and the Maxwell viscosity about 5 × l0 f6 cmz sec ~. Peltier e t a [ . (1981) favor a Maxwell model for the Earth, while VanArsdale (1982) used a K e l v i n Voigt rheology to model tidal detbrmation. The effective Q for fission of a partially molten protoearth would have been much less than 100, because Q is dependent on temperature and on the frequency and amplitude of the motions being dissipated. B e r c k h e m e r et al. (1982) have experimentally determined the frequency dependence of Q for two possible mantle materials, When their results are extrapolated to frequencies of 10 4, Q decreases to values of 1 or even less. Berckhemer et al. also found Q to decrease by a factor of 10 as the temperature was raised from 1200 to 1400°C. A value of Q = 1 yields a viscosity of about 5 × 1 0 j4 c m 2 s e c -i for both the Maxwell and Kelvin-Voigt models. Most of the experience with rheological models is limited to the small strains (of order 10 --~ to 10 6) that occur in the present Earth. Very little is known about the theological response to the global strains of order unity that must occur during a fission instability. The stress that occurs during the fission instability must be larger than the tidal stress in the Earth that would exist if the Moon was at several Earth radii, which can be estimated to be about l0 n dyn cm 2 (100 bar) for the Earth (Rubincam, 1975; Kaula and Harris, 1975). Stresses

DYNAMIC FISSION INSTABILITY larger than this certainly involve nonlinear rheoiogy, such as dislocation creep (Stocker and Ashby, 1973). Stocker and Ashby give the strain rate for olivine as a function of the applied stress and the temperature. If we define an effective viscosity to be the stress divided by the strain rate, then the viscosity due to dislocation creep for a stress of 103 bar ranges from 10 ~s to 1029 P for temperatures of 85 to 45% of the melting temperature, respectively. These estimates of an effective viscosity apply only to a coherent body. Because the stresses involved in a fission instability must exceed the tensile strength of the material, the fission process at some point must involve shattering the protoearth into a number of coherent bodies. The effective viscosity for relative motion of such bodies will be dependent not only on the parameters already considered, but also on unknown quantities such as shape and the number distribution as a function of size. Apparently we do not know within fairly large bounds what the effective viscosity should be for a dynamic fission instability of the protoearth, or even whether an effective viscosity is adequate to model the response of what must be a viscoelastic body. Estimates based on thermal histories predict effective viscosities appropriate for convective processes of at least 10 ~5 P. Estimates based on rheological models of viscoelastic materials range from 10)3 to l017 P or more, depending on the degree of partial melting. Ill. N U M E R I C A L M O D E L

Equations o f Motion We model the partially molten protoearth as a fully three-dimensional, self-gravitating, rapidly rotating, fluid body. The dynamical evolution of the protoearth is then determined by the equations of hydrodynamics, written here in spherical coordinates (r, 0, 4)): 0£ Ot + V • (pv) = 0,

_

139

( O~

_

O(pv,)~t

+

v , (pv~v)

-

=

p ~

Op) + 7r

+ P- W~ + u~),

(2)

r

O(pvo) + v.(p o t

ot

o(pa_)ot+

-7

=

V-(0Av)

1(

p o--~

+

-~

P--(v~vo- v~ cot 0), r

;

-

( 0 ~o.

+

,

(3) (41

where p is the density, v is the fluid velocity with components (vr, Vo, v~), p is the pressure, A is the specific angular momentum (A = r sin0v~), and G is the gravitational constant. The gravitational potential q) is determined by the Poisson equation V2~ = 4rrGp.

(5)

These equations are solved by a numerical code which has been extensively described previously (Boss, 1980). The basic code is the same as the one used to study the dynamic fission instability in inviscid polytropes (Boss, 1984; Durisen et al., 1985). The equations of continuity (1) and momentum (2)-(4) are solved in conservation law form using a modified form of donor-cell finite differences. The Poisson Eq. (5) is solved by a spherical harmonic expansion including terms up to (, Iml = 8. The protoplanet is able to deform arbitrarily in three dimensions, subject to remaining symmetric above and below its equatorial plane. The numerical grid contains 20 points in r, 17 points in 0 (with the assumed symmetry), and from 32 to 64 points in q~. Any implicit (artificial) viscosity arising from this coarse mesh is much less than that associated with the fluid viscosity. The numerical grid remains fixed (Eulerian) while the evolution proceeds, and any matter which flows to the boundary of the grid is assumed to have been ejected and is removed from the calculation. Pressure Relation

(1)

The interior of the protoearth is repre-

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BOSS AND MIZUNO

4.0 ix 1011

% u

;

,2

DENSITY

I 3.0 F" "\

-110 %

\

t~

20i \ z

8 \

1.0l I

PRESSURE

!4

\

iI

OL f i O

3

6

9

12

i5x

10 8

RADIUS {cm)

FIG. I, Density and pressure as a function of radius for a spherically s y m m e t r i c model of the protoearth b a s e d on the M u r n a g h a n equation of state.

sented as a slightly compressible fluid, whose d e p e n d e n c e of pressure on density was derived by Murnaghan (1951). The matter exterior to the protoearth (=atmosphere), as well as any matter which is expelled from the interior, is represented as an inviscid, pressureless fluid, which should simulate material on ballistic trajectories. The pressure equation of state is then p

=

(Ko/3`)((p/poP'

-

I)

p = 0

P

:>

pll

P <- pn,

where K0 is the bulk modulus at zero pressure. We take K0 = 1.25 × 10 r2 dyn cm 2, 3' = 5, and p0 = 3.0 g cm 3. The Murnaghan relation is consistent with seismic determinations of the variation o f the bulk modulus with depth in the Earth (Stacey, 1977), and yields a sound speed o f about 6 km sec J which is also appropriate for the Earth. This equation o f state allows a similarity of flows to hold for the numerical calculations, which means that the results for the protoearth can be scaled to protoplanets o f arbitrary size, provided that the scaled pres-

sure law still makes physical sense (Mizuno and Boss, 1985). The pressure relation was used to construct a nonrotating, spherically symmetrical equilibrium protoearth, by using the hydrodynamical code to relax an arbitrary initial density distribution to equilibrium. The resulting equilibrium is shown in Fig. 1, where it can be seen that the central density (4.20 g cm -3) is only slightly larger than the mean density of 3.48 g c m ~. This spherical protoearth model has a mass of 6.15 × 1027 g and a radius of 7500 km. The numerical code is capable of maintaining this equilibrium configuration for a period of time much longer than the rotational period. The Murnaghan pressure relation results in a less centrally condensed protoearth than the present Earth. Previous work on the dynamic fission instability has been limited to polytropes which are considerably more centrally condensed than the Earth (Durisen et al., 1985). Thus the present models explore not only the effects of viscous dissipation, but also greatly increased incompressibility compared to previous work. The pressure is assumed to vanish in the protoearth atmosphere, as can be seen in Fig, 1. The original numerical code (Boss, 1980) has been altered in order to follow thc boundary between the interior and atmosphere of the protoearth, Details of the alterations are described by Mizuno and Boss (1985), who used the same numerical code to study the tidal disruption of dissipative planetesimals. The key change is to modify the donor-cell flux at the interface between the interior and atmosphere in order to avoid unphysical situations which may arise because of the original formulation of the donor-cell flux (i.e., matter being forced out of the interior when the interior is trying to contract instead). Simulated Viscous Dissipation

We assume that the protoearth can be modeled as a fluid with an effective viscos-

DYNAMIC FISSION INSTABILITY ity on the order of 10 ~4 P or greater. The importance of viscosity for such a body can be estimated on the basis of time scales. A conservative estimate of the time scale for viscous diffusion and dissipation rdiss is approximately RZ/v, where R is the protoplanet radius and v is the kinematic viscosity. This estimate should be an upper bound, because significant velocity gradients will occur over distances less than R. The time scale for the dynamic fission instability to occur is the rotational period P 104 sec. Viscous dissipation of the velocity field is important when ~dis~ < P, which occurs when the viscosity exceeds a critical value given by

141

redistribution of momentum that would occur if shear stresses were properly represented. The simplified viscous dissipation is further discussed in Mizuno and Boss (1985).

E s t i m a t e d Effective Viscosity We can estimate the effective viscosity associated with our simplified viscous dissipation scheme. Consider a generalized momentum equation for inviscid flow: OVi

at -

OVi vj ~ + ai,

(6)

where v~ and x~ are velocities and coordinates, respectively, and ai is the body acv > vcr = 4 × 1013(R/6400 km) 2 cm 2 sec -I. celeration (pressure and self-gravity). The simulated viscous dissipation consists of reThe estimates of the effective viscosity of placing the prediction of the inviscid equathe protoearth considered in the previous tion for the next time level v~+j with fv~ +j, section generally exceed this critical vis- where f is less than I. When this substitucosity, often by large factors. When v > vc~, tion is made in the finite difference replacewe cannot rigorously model the Navier- ment for (6) using explicit time differencing, Stokes viscous stresses with this numerical the finite difference equation then reprecode, because of the severely small time sents the partial differential equation step required. Considering the uncertainOvi ties both in the proper viscoelastic model Ov~ c3t vi Oxj + a~ for the protoearh, and in the magnitude of the effective viscosity, it appears to be ade[ Ov~ vi 1 + (I - f ) Oi~a i - ~ , (7) quate to model the effects of viscosity in a simplified fashion. We have therefore simulated the effects where the terms multiplied by (1 - f ) are of a large viscosity by strongly dissipating the effective viscous stresses, and At is the the fluid velocities that develop in the inte- time step. The time rate of change of the rior in each time step of the evolution. No specific kinetic energy e = 002/2 is Oe/Ot = dissipation was applied in the protoearth at- OV~avi/Ot, assuming constant density. From mosphere. The Vr and Vo components of the (7) we have velocity in the interior of the protoearth Oe Ovi - p f v i a i - pfvivj ~iaj were reduced by a factor of f from their predicted values for that time step. The v6 (l - f ) component was handled differently: only At pvivi. (8) the difference between v6 and the solid body rotational velocity was dissipated, by Using ai = - p lOp/Oxi - O~/Oxi, and Ovi/Ox~ a factor of f each time step. This results in = 0 for incompressible flows, the first term conservation of the total angular momen- on the right of (8) becomes equal to tum of the protoplanet, while still allowing -f[O(pvi)/axi + O(pCbvi)/Oxi]. The second advection to occur in the ~b direction. Note term on the right of (8) is equal to -fO/ that this dissispation scheme results only in Oxi[vj(pv2/2)]. The total kinetic energy of the the damping of fluid motions, and not in the protoplanet E is defined as f e d V , where the

142

BOSS AND MIZUNO

integration is over the volume V. From (8) and the previous results we can obtain the time rate of change of the total kinetic energy:

+ ~ v~ v

~

E.

{9)

The volume integral in (9) can be replaced with the surface integral fp(v2/2 + p/p + O)v - dS. This surface integral vanishes when it is evaluated at a surface outside the protoplanet, where p = p = 0. Thus dE/dt = - 2(1 - .OE/At. From the N a v i e r - S t o k e s equations for viscous fluids, the time variation of the total kinetic energy is

dt -

3Or ,-

\Or + .

[LVi

'

dV.

(10)

Replacing Ov/Ox with (I - 3)v/R, where R is the protoplanet radius, we have dE/dt - ( 4 - 40)vE/R 2. Comparing this with the result of (9), we see that p -- (0.1 - 1)R2( I - .f)/(2At).

(II)

For a model of the protoearth with.l' - ._',, At 1 sec, a n d R = 6400 kin, we have u ~- 10 ~ to 10 j7 cm 2 sec J as the effective value of the viscosity we are simulating in the protoearth. F o r f = 0.995 and At --~ 0. I sec, the effective viscosity drops to about 10 ~ to 1016 c m 2 s e c 1, These estimated viscosities are several orders of magnitude larger than the critical viscosity given in the previous subsection. IV. N U M E R I C A L RESUI~TS

Initial Conditions The appropriate initial conditions for a dynamic fission instability depend on the time scale for secular instability 7.... compared to the time scale for spin up ~-~pm(Tassoul, 1978). If 'Tse c is much less than ~-~pi,, then the secular instability at /3 = 0.138 should distort an axisymmetric Maclaurin spheroid into a nonaxisymmetric Jacobi el-

lipsoid, and the Jacobi ellipsoid should then become dynamically unstable when the pear-shaped bodies bifurcate at/3 = (:).163. If ~-~ is much longer than ~-~pi,, then a Maclaurin spheroid will not undergo appreciable secular deformation prior to reaching the dynamic instability at/3 - 0.274. Usually ~-,~ is estimated to be equal to the time scale for viscous dissipation, ~'~Ji~. Because a partially molten protoplanet will have an extremely short time scale for viscous dissipation (Sect. 11), on this basis we might expect a Jacobi ellipsoid to be the appropriate initial model for protoplanetary fission. However, this estimate means that the time scale for secular instability (~ T,,i~0 is actually shorter than the time scale for d y n a m i cal instability (~'dy., -~- P, where P is the rotational period), which reverses the temporal distinction between the two types of instability, and thus implies that the estimate of 7,~ is too small. We see in Section V that the secular time scale is much larger than the dissipation time scale when very large viscosities are involved. A lower limit on r~.~ would seem to be T,,y, P. Thus an axisymmetric initial model should be appropriate unless 7~pi, is much greater than P. For example, if the protoplanet is considered to have been forced into rotational instability as a result of a relatively sudden increase in angular momentum |ollowing a giant tangential impact, or through a catastrophic core-formation process (Stevenson, 1980), then an initially axisymmetric model, rotating with/3 > 0.274, seems appropriate. However, if the protop[anet reaches higher /3 through a more gradual process of core formation, then an initially nonaxisymmetric model would seem appropriate. If we consider an initially axisymmetric model with/3 > 0.274, we should be able to study either the dynamic instability of an axisymmetric body, or else its rapid deformation into a nonaxisymmetric body and the subsequent dynamic instability of the latter body, because in this case all possible limits for dynamic instability will have been

DYNAMIC FISSION INSTABILITY

143

l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

llltllFIG. 2. Density contours for an a x i s y m m e t r i c , rapidly rotating model of the protoearth. The density in this plot as well as the others is c o n t o u r e d with the innermost c o n t o u r corresponding to densities of 3.40 g cm 3, and s u c c e s s i v e contours indicating d e c r e a s e s in density by 0.40 g cm 3 with the last c o n t o u r being at a density o f ! .00 g cm 3. In this case the protoearth is constrained to being s y m m e t r i c about an axis which is the left-hand border o f the plot. The protoearth is also s y m m e t r i c above and below a plane which forms the bottom b o u n d a r y of the plot. B e c a u s e o f the a s s u m e d s y m m e t r y , then, only one-quarter o f a c r o s s section through the center of the protoplanet is shown. The protoearth is flattened b e c a u s e of its rapid rotation (fl = 0.30).

exceeded. In addition, if we apply a large enough nonaxisymmetric perturbation to an axisymmetric, unperturbed model, some components of the perturbation should mimic the nonaxisymmetric model. A natural means of constructing a dynamically unstable initial model is to spin up a model protoplanet while forcing it to remain axisymmetric (i.e., two dimensional). When such a model is then treated three dimensionally, nonaxisymmetric instability will immediately be able to grow. The same type of initially axisymmetric models have been used in the previous studies of fission (Durisen et al., 1985), which allows a direct comparison with the previous results to be made. We have therefore used axisymmetric initial models for the rapidly rotating protoearth. These models were constructed by starting with the spherically symmetric protoearth model of Fig. 1, and then adding a small amount of solid body rotation. The configuration was then allowed to hydrodynamically relax to a flattened equilibrium with solid body rotation. This procedure was repeated until the desired amount of rotation was reached. Because of the nearly incompressible na-

ture of these models, the sequence that results is very similar to the Maclaurin spheroid sequence, both qualitatively and quantitatively (Mizuno and Boss, 1985). Figure 2 shows a cross section through the rotation axis of one of the initial models with/3 = 0.30, showing the high degree of flattening obtained. Results

We have calculated the outcome of the dynamic fission instability for the four different models listed in Table I. Each unperturbed model was initially axisymmetric, with a mass roughly equal to that of the combined Earth-Moon system, and rotating with a value of/3 greater than the limit TABLE 1 INITIAL CONDITIONS FOR MODELS OF THE DYNAMIC FISSION INSTABILITY OF THE PROTOEARTH

Model

/3

a

N+

m = Nsymm

f

A B C D

0.30 0.30 0.33 0.33

0.10 0.50 0.50 0.50

32 64 64 64

2 2 2 2

0.5 0.5 0.5 0.995

T

i,

\

*

~' ~' i

r

'

!o "7.

_- -c .r_ - ~ ,~

I

0 ....

i

=~

...-..;..\ i ...":.....

)

',~ -~.

c_ >,

"".:/i -".""

i

L)

I-

=.

r_ r-.-

>~-~

=_,.>i ¸ '

iI'

,, /

/

• !

I /

III

144

~,~,

> >

.-

DYNAMIC FISSION INSTABILITY

FIG. 4. Time evolution of the amplitude of the m = 2 (binary) mode in model A. The amplitude is shown also as a function of radius for the interior of the protoearth. The amplitude monotonically decreases from its initial amplitude of 0.10; the protoearth model is completely stable with respect to binary fission because of the presence of strong viscous dissipation.

for dynamical instability (0.274) for bar modes. The protoearth models were perturbed with a variation in the density of the form p = po(1 + a cos(m~b)), where pu is the density of the unperturbed (axisymmetric) spheroid, and the amplitudes a of the perturbations of mode m (= Nsymm) are listed in Table I. The models were constrained to be symmetric through the rotational axis; Nsymrn = 2 means that only modes with m = 2, 4, 6, 8 were allowed to grow (binary instability). For Nsymm = 2, this artifice allows the effective spatial resolution of the numerical grid to be N6 = 64 with 32 active computational cells in ~b, while still allowing the binary mode to grow unimpeded. In three models viscous dissipation was simulated using a value of f = ½ (leading to an effective v ~ 1016 to 1017 cm 2 s e c - I ) , while the fourth model used f = 0.995 (v ~ 10 '5 to 1016 c m 2 s e c - I ) .

The time evolution of the dynamic fission instability in model A is shown in Fig. 3. Initially the model is mildly deformed into a binary protoplanet (Figs. 3A). The velocities (in a frame rotating with the mean angular velocity of the protoearth) that result from the perturbation are initially on the order of 103 cm sec -1. The apparent rotation of the outer layers in Figs. 3A is due to the very small amount of differential rota-

145

tion in the initial model, which is greatly magnified in this plot. The binary perturbation, however, dies out. By the time of Figs. 3D (1.1P, where P = 1.15 × 104 sec), the protoearth has become nearly axisymmetric again, and the fluid velocities have degenerated into radial oscillations in the rotating reference frame. The inability of the binary perturbation to grow in this dissipative protoplanet is also depicted in Fig. 4, where the m = 2 mode is shown to decay from 0.10 to essentially 0 by 1.1P. Model A did lose about 0.6% of its total mass during this period of time, but this loss can be traced solely to the application of the initial density perturbation, and not to any dynamic instability, because essentially all the mass loss occurred within the first 0.25P of the evolution; thereafter the protoearth was quite stable. In any numerical calculation of the dynamic fission instability, the growth or decay of a perturbation is dependent on the competition between the strength of the instability and the numerical resistance to the instability that results from finite grid spacing (e.g., Durisen et al., 1985; Tohline et at., 1985). The latter effect means that there is an effective numerical viscosity which acts to diffuse perturbations; this implicit viscosity decreases as the number of grid points increases. For growth of nonaxisymmetric perturbations, the numerical damping depends primarily on the number of 4) grid points (N~). Tohline et al. (1985) found that for inviscid polytropic (3' = ]) models of binary fission with N+ = 64, the initial m = 2 perturbation did not grow for/3i = 0.28, but did grow exponentially with time for/3i = 0.30, 0.33, and 0.35. Considering that the analytic linear theory predicts growth whenever/3 > 0.274, the absence of growth for/3, = 0.28 found by Tohline et al. (1985) implies that for this case the numerical diffusion associated with N~ = 64 roughly balanced the tendency for dynamic instability. Models with higher/3i have greater tendencies toward dynamical instability, and in these models (/3i = 0.30, 0.33, 0.35) the nu-

146

BOSS AND MIZUNO

merical diffusion was unable to prevent the instability. The numerical results reported by Tohline et al. (1985) were obtained using a numerical code with roughly the same numerical diffusion characteristics as the code we are using (discussed in detail by Durisen et al., 1985). H e n c e we can expect that for /3i = 0.30 or larger, N~ = 64 or larger should assure us that numerical diffusion is not significantly affecting the results. Considering that model A has/~i = 0.30 and only N~ = 32, it is clear that models B and C are necessary in order to be certain that numerical diffusion is not responsible for the failure of the initial perturbation to grow in model A. Based on the foregoing arguments, model B (with/3i = 0.30 and N~ = 64) should not be dominated by numerical diffusion, yet its evolution was qualitatively similar to that of model A. Similarly, on the basis of the inviscid theory, model C (with /3i = 0.33 and N~ = 64) should have been considerably more dynamically unstable than models A or B, yet once again the initial perturbation decayed. The combination of all three models clearly demonstrates that our dissipative protoplanet models are dynamically stable to m -- 2 perturbations in a regime where dynamical instability is predicted by the inviscid analytical theory, and furthermore that this stability is not caused by problems related to numerical resolution and diffusion. Model D extends the range of effective viscosity where this conclusion is valid. Because the mode in model D was observed to decay rapidly (decreasing in amplitude by a factor of about 5 within 0.15 of a rotation period), this model shows that for effective viscosities of order 10 j5 to 10 ~6 cm 2 sec ~. dynamical instability is prevented by dissipative effects. Presumably this dynamical stability will continue to occur for even lower viscosities, perhaps an order of magnitude lower (10 ~4 to 1015 cm 2 sec ~), considering the rapid decay encountered in model D. While we have not shown how low the viscosity must be before an instability is recovered which occurs on the rota-

tional period time scale, on this basis, it appears that our criterion for the importance of viscous effects (Sect. III) u > v~ ~ 4 x 1013 cm 2 sec may be the appropriate criterion for determining when a fission instability does not occur on a rotational period time scale. In both models A and B, which differ only in N~ and in the amplitude of the initial perturbation, the maximum amplitude of the binary perturbation drops by a factor of about 100 in model A and by a factor of about 50 in model B within about IP. On the basis of the results of Tohline et al. (1985) and Durisen et al. (1985), model B should be relatively unaffected by numerical damping, so the question arises of what is the source of this damping. We answer this question in Section VI, after we derive further information necessary for understanding the damping in the next section. The next section demonstrates that the dynamical stability of these models is indeed caused by the inclusion of strong dissipation, not by the greater degree of incompressibility in these models (y = 5) compared to previous calculations where the instability grew (y = ~; Durisen et al., 1985). V. ANALYTICAL MODEL In this section we extend the tensor-virial equation formalism developed by Chandrasekhar (1969) for studying the linear stability of rotating fluids to include our simulation of viscous dissipation. Previous workers (Roberts and Stewartson, 1963; Rosenkilde, 1967) have considered the effects of viscosity on linear stability, but only in the limit of very small viscosity. The equations of motion in the presence of our viscous dissipation are equivalent to the inviscid equations with the addition of terms which simulate the viscous stresses (i.e., Eq. (7)). The tensor-virial approach begins by taking moments of the hydrodynamical equations. When the moments of

DYNAMIC FISSION INSTABILITY Eq. (7) are taken, the resulting virial equations of the second order are

fp

xgv = f f + (1 - f ) [

pa~rjdV 0 f ff-~xk(pvivk)xjdV

147

where V~;j, Vij, 817, and 8Wij are defined by Chandrasekhar. We take the protoplanet to be rotating about the 3 axis, i.e., l~,- = 8,3~q, and assume that the perturbations have a time d e p e n d e n c e of the form exp(;~t). Then Eq. (14) b e c o m e s

~k2Wi;j - ( 2 f ~eie3 Ve;j

l fpv&dV].

At

(12)

It is convenient to work in a rotating frame of reference, in which case the body accelerations ai must also include the centrifugal and coriolis accelerations. The first two integrals in Eq. (12) are the same as in the inviscid case and are given by Chandrasekhar (1969, Chap. 2). The third integral can be rewritten to yield

d

fpv~flV=f[2Ti~

- FV~j)X = f[SWij + f/2(V~j -- 8,3V3j ) q-

[),2 + F~k - 2f(l~ 2 - 2BII)]VI2 + f ~ ' ~ X ( W l l -- V22) = 0

-4f~)tVt2 + [h 2 + Fa - 2f(D 2 - 2Bj0](Vjl - V22) = 0,

[fo

~Xk (pViVkXj)dV

l fpv ,dV],

At

(13)

where Tii, Wii, ]ii, and 17 are defined by Chandrasekhar, and f~/is the c o m p o n e n t of the rotation vector. Equation (13) describes the equilibrium states. In order to learn about the stability o f the equilibria, the response to small perturbations ~:(x, t) must be determined. This response is found by considering the first variation of the virial Eq. (13). All the terms in the resulting equation are the same as in Chandrasekhar, except for the 6fO/Oxdpvi vkxi)dV term, which can be shown to vanish in the steady, unperturbed state (vi = 0). The resulting equations for the steady, unperturbed state are

d2

dt---2 Wi;j ~- f

(15)

where F = (1 - f)/At. In the inviscid case, the stability analysis of a Maclaurin spheroid (Chandrasekhar, 1969, Chap. 5) involves the toroidal mode described by Eq. (15) even in the index 3. The relevant equations are then

"~ Wij + ~2]ij - ~'~i~~klkj

+ (1 -- f )

8ij~l-[],

[ ~Wij

q- ~'~2Vij - ~-~i~'~kVkj "-k 81"ISij d ] (1 - f ) d + 2t~itm~m -~ VczJ] At dt Vi;j'

(14)

(16)

where Bit is defined by Chandrasekhar. Notice that in Eq. (16) and hereafter, the unit o f l l and h is (lrGp) v2 and therefore F = (I --f)/(At('trGp)l/2). In order to have a solution for the two linear Eqs. (16), the determinant o f the matrix of coefficients must vanish, leading to the characteristic equation [?2 + F~. -

2f(1~ 2 -

2Bj0] 2

q- 4f2~2X 2 = 0. Letting

h = io-, t h e c h a r a c t e r i s t i c

(1 7)

roots be-

come

iF o- = f f ~ + T +

~ + i

+ 2f(2Bij - 1)2).

(18)

Chandrasekhar (1969, p. 95) defines dynamical instability as being an instability (Irn(cr) < 0) which occurs in the absence of dissipation, whereas secular instability describes an instability which only occurs if dissipation is operative. In the absence of viscous dissipation, a neutral point (or point o f bifurcation, where o- = 0) occurs at

148

BOSS AND MIZUNO

~2 = 2Bll (or/3 = 0.138), whereas dynamical instability occurs for ~2 > 4Bj~ (or/3 > 0.274). These results may be obtained from Eq. (18) in the inviscid limit ( f = 1, F = 0). The neutral point implies that if there is a lower energy configuration nearby, it can be reached through dissipation of energy, so that the secular instability in this limit is implied. In the limit of small viscosity. Roberts and Stewartson (1963) and Rosenkiide (1967) have shown that the dynamical instability remains for/3 > 0.274, and that for 0.274 > /3 > 0.138 the Maclaurin spheroids are explicitly unstable, with the

instability arising from the terms introduced by viscosity. Thus slightly inviscid Maclaurin spheroids are secularly unstable for 0.274 > / 3 > 0.138 (the secular instability disappears for /3 > 0.274). The same results may also be obtained from Eq. (18) in the limit of small viscosity ( f - : 1 ~:, ~: I). This shows that our simulated viscosity behaves correctly (i.e., like a true viscosity), at least in the limit of small viscosity. The most remarkable feature of Eq. (18) is its behavior in the limit of large viscosity. the correct limit for viscous protoplanets. Because At ~ P and B~ ~ O(I), we have I: ~ = B~ in the adopted units. Then the characteristic roots can be simplified to become (to first order in lI/k" or BL~/F, which are both ~ I)

2f

(r, = 2 / ' ~ + iF[I - ~ _ (2Bll - ll~)]

2f

o-_ = i ~ (2Bll --

f~2).

(19)

The criterion for instability is lm(o-) < O. We see first of all that there is n o l o n g e r a n y d y n a m i c a l i n s t a b i l i t y fi~r fl > 0.274; the terms responsible for the dynamical instability in the limit of small viscosity have disappeared. Furthermore, because F ~> and B l l , Ira(or+) > 0 and the + mode is always stable. H o w e v e r , the - mode will be unstable w h e n e v e r 2B~ < 1)2, which corresponds to the secular instability limit for Maclaurin spheroids,/3 > 0. i 38. This instability is secular because it is produced by

the terms introduced by our simulated viscous dissipation. Moreover, the time scale for growth of this secular instability is quite long. The ratio of the time scale for the growth of the - mode to the rotational period is ( - l/lm(~r )) (1/~)

-k'~ 2l'(2Bll

-

-

~2) F =f~

I.

(20)

Because this ratio kT[~ - P / A t .~ 104 for the models w i t h . / ' = ~ and -103 for the model with f = 0.995, it is not surprising that the initial perturbations given did not grow appreciably in - P . In the limit of infinitely large viscosity (At --* 0 or f--~ 0), F --> ~, and hence the time scale for growth also becomes infinite. For fixed viscosity, the growth time also becomes infinite as/3 0.138+, because of the denominator in the second term in Eq. (20). In the limit of small viscosity, it can be shown from Eqs. (18) and (11) that the time scale for secular instability is equal to the time scale for viscous dissipation (R2/u). H o w e v e r , in the case of extremely large viscosity considered in Eqs. (19)-(20), it can be shown that "rse~/"cdi~ "= (0.1 -

1)/f((l - .DP/(47rAt)) 2.

For our rapidly rotating protoearth models, r ~ ~> ~'di~,~, contrary to the result for small viscosity. The ratio of the secular instability time to the dissipation time is about l04 to 105 for models A, B, and C, and 102 to 103 for model D. We have seen that the dynamical instability of the inviscid or slightly viscous Maclaurin spheroid disappears when a strong simulated viscous dissipation is operative, while a secular instability occurs f o r / 3 > 0.138. This result is completely consistent with the dynamical stability found by numerical means for highly dissipative protoplanets. Because of this analytical work, we can say without qualification that the inclusion of our simulated viscous dissipation alone is responsible for eliminating the

DYNAMIC FISSION INSTABILITY dynamical instability of highly viscous bodies. VI. CONCLUSIONS We have shown that the dynamic fission instability does not occur in rapidly rotating protoplanets that have an effective viscosity much greater than about 4 x 1013 (R/ 6400 k m ) 2 c m 2 s e c - l . The failure of the fission instability in our models differs dramatically from the results of previous studies of the instability in inviscid, stellar models (Durisen and Tohline, 1980; Boss, 1984; Durisen et al., 1985). Because the basic code used in this study was also used in the stellar fission studies, where the instability was found to grow, we can rule out numerical techniques as a cause for this difference. While the protoplanets we have studied are considerably more incompressible (3' = 5) than the stellar models (3' -- ~), it is clear that the dynamic fission instability failed in our models because of the presence of a strong dissipation, the effects of which have heretofore never been considered. This assertion is based on our analytic study of the linear growth of perturbations using the tensor-virial equation formalism pioneered by Chandrasekhar (1969): when our simulated viscous dissipation terms are included in models of incompressible Maclaurin spheroids, dynamic instability for/3 > 0.274 still occurs in the inviscid and slightly viscous limit, in agreement with previous workers who used the correct formulation of viscosity. However, in the limit of strong dissipation, the dynamical instability disappears. Highly viscous protoplanets are still subject to a secular instability for/3 > 0.138, however. Hence once a nonaxisymmetric perturbation is introduced into a numerical protoplanet model with/3 > 0.138, the perturbation should grow on the secular time scale, which is much longer than the rotational period for the models we have considered (as the viscosity increases, this secular time scale increases). The fact that the initial perturbations in our numerical

149

models damped and did not experience slow secular growth (i.e., did not remain essentially unchanged for one rotational period) implies that while dissipation prevented dynamical instability, numerical dissipation must have been responsible for the damping of the secularly unstable mode. This is understandable, because while the numerical dissipation associated with the spatial resolution used in our models is insufficient to prevent a robust dynamical instability from occurring (Durisen et al., 1985; Tohline et al., 1985), this amount of numerical dissipation is quite capable of obliterating the extremely weaker secular instability that remains for highly viscous protoplanets. In other words, our spatial resolution was sufficient to detect any rapidly growing, dynamical instability, but not to follow a secular instability with a growth time much longer than the rotational period. The analytical work of Section V is crucial for proving that it is indeed the strong dissipation that eliminates the dynamical instability. Depending on the detailed theology of the protoearth, it appears that Darwin's hypothesis (1879, 1880) of the formation of the Moon following dynamic fission from the protoearth can be ruled out, unless the protoearth was nearly inviscid, i.e., completely molten. Thermal histories, while implying a " h o t " origin for the Earth and other terrestrial bodies, do not favor a completely molten initial state for the Earth. Hence the controversy over the fission origin of the Moon may finally be resolved, because the most fundamental part of the fission scenario, the dynamic fission instability itself, simply could not have occurred for a partially molten, viscous protoearth. Fission of a rapidly rotating planetary body has also been discussed as an explanation for the origin of other bodies in the Solar System. Weidenschilling (1980) suggested that the asteroid 624 Hektor is a binary asteroid which would be unstable to fission if it was a single body, but preferred an impact origin for Hektor. Lin (1981) pro-

150

BOSS AND MIZUNO

posed fission as a means of producing the Pluto-Charon system. Because of their relatively small sizes, however, the asteroids and Pluto could not have been completely molten early in their history, and hence binary fission cannot account for the origin of either binary asteroids or the Pluto-Charon system. What then is the fate of a protoplanet which is impulsively spun faster and faster'? Weidenschilling (1981) advanced rotational fission as a limit on the rotation rates for asteroids, resulting in a maximum rotation rate about one-half that allowed by equating gravitational and centrifugal forces at the equator. Our results imply that the maximum rotation rate for viscous asteroids or protoplanets should be limited by equatorial breakup. A protoplanet could conceivably be spun up above the centrifugal limit by the off-center impact of another protoplanet with a large relative velocity. Great impacts appear to be a natural consequence of the accumulation of the terrestrial planets from a swarm of planetesimals (Wetherili, 1985). If such a great impact gives a protoplanet sufficient angular momentum to raise it above the centrifugal limit, then enough mass and angular momentum must be lost from the equatorial regions to lower the protoplanet back below the limit. Because of the high specific angular momentum of this material, a small amount of mass can carry off a large amount of angular momentum. The fate of the matter which is ejected is uncertain. If ejected ballistically, the matter will return to the protoplanet's surface, which does not solve the protoplanet's excess angular momentum problem. Hence the matter must be put into orbit about the protoplanet. The details of this ejection process, and of the evolution of the matter in circumprotoplanet orbit, remain to be elucidated. Similar processes involving giant impacts have been suggested for the origin of the Moon (Hartmann and Davis, 1975; Cameron, 1976); certainly the giant impact scenario demands further investigation.

ACKNOWLEDGMENTS We thank Joel Tohline for a thoughtful review of the manuscript. The calculations were performed on the VAX computers of the Carnegie Institution of Washington. This research has been supported by the Innovative Research Program of the National Aeronautics and Space Administration, under Grant NAGW-398. REFERENCES BERCKHEMER, H., W. KAMPFMANN, E. AUI_BACH, AND H. SCHMEHN6 (1982), Shear modulus and Q of forsterite and dunite near partial melting from forced-oscillation experiments. Phys. Earth Planet. Inter. 29, 30-41. BINDER, A. B. (1974). On the origin of the Moon by rotational fission. Moon 11, 53-76. BINDER, A. B. (1977). Fission origin for the Moon: Accumulating evidence. Lunar Planet. Sci. Con/~ VIII, 118-120. Boss, A. P. (1980). Protostellar formation in rotating interstellar clouds. 1. Numerical methods and tests. Astrophys. J. 236, 619-627. BOSS, A. P. (1984). Angular momentum transfer by gravitational torques and the evolution of binary protostars. Mon. Not. R. Astron. Soe. 209,543-567. Boss, A. P., C. L. ANGEVINE, AND I. S. SACKS (1984). Finite amplitude models of convection in the early mantle. Phys. Earth Planet. Inter. 36, 328336. Boss, A. P., AND H. MIZUNO (1984). The dynamic fission instability and the origin of the Moon. in Papers Presented to the Conference on the Origin ~/" the Moon, p. 36. Lunar and Planetary Institute. CAMERON, A. G. W. (1976), The origin of the Moon. Lunar Sci. Conf. VII, 120-122. CHANDRASEKHAR, S. (1969). Ellipsoidal Figures of Equilibrium. Yale Univ. Press, New Haven. Conn. CHR1STENSEN,R. M. (1971). Theory t~/'Viscoelasticitv: An Introduction. Academic Press, New York. CLARK, S. P. (1966). Viscosity. Geol. Soc. Amer. Mere. 97, 293-300. COM1NS, N. (1979). On secular instabilities of rigidly rotating stars in general relativity. I1. Numerical results. Mon. Not. R. Astron. Soc. 189, 255-272. COOK, F. A., AND D. L. TURCOTTE (1981). Parameterized convection and the thermal evolution of the Earth. Tectonophys. 75, 1-17. DARWIN, G. H. (1879). On the precession of a viscous spheroid, and on the remote history of the Earth. Phil. Trans. R. Soe. 170, 447-530 (reprinted in Scientific Papers, Vol. 2, pp. 36-139. Cambridge Univ. Press, Cambridge). DARWIN, G. H. (1880). On the secular changes in the elements of the orbit of a satellite revolving about a tidally distorted planet. Phil. Trans. R. Soc. 171, 713-891 (reprinted in Scientific Papers, Vol. 2, pp. 208-382. Cambridge Univ. Press, Cambridge). DURISEN, R. H., R. A. G1NGOI_D, J. E. TOHLINE, AND

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