Calculat ions of the Evolution of the Giant Planets ~
PETER BODENHEIMER,*, t ALLEN S. GROSSMAN,t, $ WILLIAM M. DECAMPLI,§ GEOFFREY MARCY,* AND JAMES B. POLLACKt
* Lick Observatory, Board of Studies in Astronomy and Astrophysics, University o f California, Santa Cruz, California 95064; t Space Sciences Division, NASA-Ames Research Center, Moffett Field, California 94035;
:~ Department o f Physics and Astronomy, Iowa State University, Ames, Iowa 50010; and § Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California 91125
Received October 15, 1979; revised November 29, 1979
Evolut ionary calculations are presented for spherically symmetr ic protoplanetary configurations with a homogeneous solar composi t ion and with masses of l0 -:~, 1.5 x 10 -a, 2.85 × l0 ~, and 4.2 × l0 -4 MQ. Recent improvements in equation-of-state and opacity calculations are incorporated. Sequences start as subcondensa t ions in the solar nebula with densit ies of - 1 0 -'~ to 10-" g c m -'~, evolve through a hydrosta t ic phase lasting l0 ~ to 107 years , undergo dynamic collapse due to dissociation of molecular hydrogen, and regain hydrosta t ic equilibrium with densit ies - I g cm -3, The nature of the objects at the onset of the final phase of cooling and contract ion is d iscussed and compared with previous calculations.
I N T R O D U C T I O N
A number of recent theoretical calcula- tions have been made of the evolution of the giant planets Jupiter and Saturn, under the supposition that they form by a process of instability in the solar nebula at about the same time as the formation of the Sun. The evolution from the initial state to the final state of a condensed planet can be divided into three major phases: (1) an early cool phase, starting with protoplanetary mate- rial having a radius a few thousand times the present radius of the planet, during which the hydrogen is in molecular form and internal temperatures are less than 2000°K; (2) a hydrodynamic collapse in- duced by the dissociation of molecular hy- drogen, ending when temperatures reach 2 x 104°K; and (3) a final slow contraction and cooling, starting at a radius a few times that of the present planet, proceeding on a time scale of 109 years or more, and charac- terized by a fully convective structure.
Within this evolutionary framework and under the assumption of spherical symme-
1 Contr ibution from the Lick Observa tory No. 421.
293
try, two different major physical scenarios have been investigated: (A) a rocky core of a few Earth masses (Me) forms first by accretion of small particles, then gas of solar composition accumulates around the core and reaches a point where it becomes unstable to collapse; (B) the protoplanet forms as a chemically homogeneous sub- condensation in the solar nebula and subse- quently contracts and collapses under its self-gravity, either remaining chemically homogeneous during its entire evolution or developing a rocky core by precipitation or by gas-drag capture of small stray bodies (Pollack et al . , 1979) during the course of the evolution. Phase (3) under either sce- nario must be similar. Detailed calculations of phases (I) and (2) under scenario (A) have not been carried out. However, Perri and Cameron (1974), Mizuno et al. (1978), and Harris (1978) have studied the transi- tion between these two phases and have investigated conditions necessary for gravi- tational collapse of the gas onto the core. Grossman (1978) has calculated the evolu- tion of Jupiter and Saturn through phase (3) with a rocky core of a few Me. Recent (rotating) models of the present state of
Jupiter have been presented (Podolak, 1977) with cores between 16 and 18 Ms, of Saturn (Podolak, 1978) with cores between 20 and 25 Ms, and of Jupiter and Saturn (Slattery, 1977) with cores of 14-16 and 15- 17 Ms, respectively. Core masses in this range produce fits to the observed masses, radii, and gravitational moments of the present planets.
Spherically symmetric calculations based on scenario (B) can be summarized as fol- lows. Bodenheimer 's (1974) calculations for 1 Jovian mass (Mj) show that phase (1) is a quasi-static contraction proceeding on a time scale of 10 '~ years. The calculations continue through phase (2) and part of the transition to phase (3). Graboske et al. (1975a) present the phase (3) evolution for 1 Mj, starting with a fully convective struc- ture with about 16 times the present radius ( R j - - 7 × 10 -',cm) and ending at a point where the model can be compared with the present properties of Jupiter. A parallel calculation for Saturn was accomplished by Pollack et al. (1977). DeCampli et al. (1978) reexamine phase (1) for 1 Mj with improved opacities and equation of state and con- clude that (1) the models are largely con- vective in contrast to the radiative structure found by Bodenheimer (1974), and (2) the assumed ratio of ortho- to para -hydrogen plays an important role in the earliest phases of the evolution. The improved physics is also employed in a more exten- sive discussion of the phase (1) evolution of protoplanets in the mass range 0.3 to 5 Mj by DeCampli and Cameron (1979) who reach the important conclusion that in a certain mass range grains can grow and precipitate to the center of the protoplanet resulting in the formation of a rocky core prior to the onset of phase (2). Thus the assumption of a homogeneous initial state is not necessarily inconsistent with the presence of a rocky core in Jupiter and Saturn at the present time. If the grain fraction is 3 × 10 -3 by mass, then the process can produce a core of about one Earth mass, smaller than the present values
inferred for the giant planets. Mechanisms to enhance the heavy-element content of the protoplanets, such as mass transfer from the solar nebula and capture of chon- drules, are discussed by DeCampli and Cameron (1979) and will be further studied in the future.
In this paper we extend the work of DeCampli and Cameron (1979) through phase (2) and as far as possible into the transition to phase (3) for four masses: 1.0 Mj, 1.5 Mj, 0.285 Mj (Saturn), and 0.42 Mj. The chemical composit ion (hydrogen mass fraction X = 0.74, helium mass fraction Y = 0.24) and equation of state are the same as those used by Graboske et al. (1975a) and Pollack et al. (1977), so the present calcula- tions can be matched at the onset of phase (3) with those earlier ones. The combina- tion of these solutions provides a complete evolutionary history for both Jupiter and Saturn under the principal assumptions of (1) spherical symmetry, (2) chemical homo- geneity, and (3) isolation of the proto- planets from the influence of the surround- ing solar nebula. One of the important questions to be considered is: what is the maximum luminosity of Jupiter and Saturn during their final contraction phase, that is, at the onset of phase (3)? The temperature in the primitive nebulae surrounding the planets and hence the composit ions and mean densities of the major satellites are strongly influenced by this quantity (Pol- lack and Reynolds 1974; Pollack et al. 1976). The following sections describe the physics incorporated into the calculations and the resulting evolutionary sequences. The paper concludes with a comparison with earlier results and with a summary of the main conclusions.
DESCRIPTION OF CALCULATIONS
The computations are carried out with a computer program designed for stellar evo- lution (Henyey et al., 1964), modified to include hydrodynamic effects by Bo- denheimer (1968). The standard equations
EVOLUTION OF GIANT PLANETS 295
of motion, mass distribution, radiative or convect ive energy transport , and energy conservation are solved by a fully implicit technique on a Lagrangian grid. The source of energy is gravitational contraction alone, since nuclear effects are completely negligi- ble at the temperatures and densities under consideration. In convect ion zones the temperature gradient is set equal to the adiabatic gradient; more detailed mixing- length calculations by DeCampli and Ca- meron (1979) show that this assumption is justified except in the low-density outer layers during phase (1). During hydrody- namic phases of the evolution convect ion is not included since the time scale for evolu- tion is short compared to the time scale for convect ive transport. The surface bound- ary condition simply assumes that the sur- face pressure is zero (actually a small posi- tive value) and that the luminosity L and surface boundary temperature TB are re- lated by the Eddington approximation L = 87rR 2 o-TB 4, where R is the radius. A strong shock wave appears in the calculation dur- ing the transition between phases (2) and (3); it is t reated with the standard technique of artificial viscosity.
The main differences between this pro- gram and that used by Bodenheimer (1974) lie in the equation of state and opacity, for which data must be provided in the temper- ature range 60 < T < 60,000°K and density range 10-" < p < 10 g cm -3, skewed be- tween the limits. Improvements have been made in the equation of state both in the molecular hydrogen regime at T < 1000°K and in the high-density regime at p > 0.1 g cm -3 where nonideal effects are significant. While the earlier calculations simply took the internal energy per gram for molecular hydrogen to be En2 = 2.5RgT (where R~ is the gas constant), the present program takes into account the excitation of rota- tional levels (at T ~ 80°K) and vibration levels (at T ~> 1500°K). The ratio of ortho- to para-hydrogen is assumed to achieve the thermal ratio appropriate to the local tem- perature. The effects of varying this as-
sumption and the general effects upon the evolution of the ortho-para ratio are dis- cussed by DeCampli et al. (1978); the prin- cipal effects occur during the early part of phase (1).
In the nonideal regime at higher densities there are three major regions of interest. For p < 10 -t g cm -3, the low-density region, the equation-of-state data are deter- mined by a Helmholtz free-energy minimi- zation technique (Graboske et al., 1975b). The following elemental species are consid- ered, H2, H2 +, H, H +, H- , He, He +, He ++, M, M + (heavy elements), and e- . Solar abundance ratios are assumed for the heavy elements. The free-energy expression uti- lizes an internal partition function in which self-consistent t reatment of coulomb cor- rections, pressure ionization and dissocia- tion, and excluded volume corrections can be applied to the ideal, noninteracting, gas pressure. Within the low-density region three principal temperature subdivisions can be identified. From 60 to -2500°K the hydrogen will be in the molecular fluid phase, from -4000 to -6000°K the hydro- gen will undergo dissociation to the neutral atomic phase. At temperatures greater than -20,000°K at low densities (p < 10 -4 g cm -3) and greater than -100,000°K at higher densities (p - 10 -2 g c m -3) the hydrogen will be ionized and can be de- scribed in terms of a coulomb plasma model. The effects of pressure ionization and dissociation in hydrogen become strong in the density region p = 10 -3 to 10-' g c m -3.
For densities greater than 1 g cm -3, the high-density region, the equation of state is determined by a modified Thomas -Fe rmi model known as the T F K model (Graboske et al., 1975b) with additional contributions from the ion components (coulomb correc- tions) and additive volume effects. For tem- peratures less than approximately 3000°K, solid metallic hydrogen is the dominant hydrogen species. At the temperatures in the range 3000 < T < 20,000°K the hydro- gen is in the coulomb metallic fluid phase.
296 BODENHEIMER ET AL.
In the density range 10 -1 < p < 1 g cm -3 and temperature range 2000 < T < 20,000°K, the intermediate region, the equation-of-state relations were determined by numerical interpolation between the high-density and low-density relations. Er- rors in pressures and enthalpies in this region are probably less than 10%, while second-order quantities such as the adia- batic gradient could be in error by several tens of percent. The effects of changing the equation of state in this transition region and the resulting uncertainty in phase (3) evolution are discussed by Graboske et al. (1975a). A complete discussion of the equa- tion-of-state models used in this paper and comparison to all known experimental data and other theoretical approaches are given in Graboske et al. (1975b).
During the early epochs of the history of the Jovian and Saturnian systems under consideration, the nebular densities and temperatures are quite low. Hence, opaci- ties due to dust grains can be expected to be much more important than those due to gases. At later times, during and subse- quent to the hydrodynamical collapse, the grains will have evaporated in much of the protoplanet and gaseous opacity will be- come important because of the much higher temperatures and densities. A major im- provement is made here in the calculation of the grain opacity, which in the earlier calculation for Jupiter (Bodenheimer, 1974) was simply set to the constant value of 0.15 cm 2 g-i at temperatures below an evapora- tion temperature of 1400°K. As discussed by DeCampli and Cameron (1979) and in this paper (see below) the changed opaci- ties have a major effect on the structure of protoplanets during much of phase (1), causing the interiors to be largely convec- tive.
Pioneering work on the opacity of a vari- ety of particulate species has been done by Knacke (1968), Kellman and Gaustad (1969), and Cameron and Pine (1973). How- ever, these estimates of grain opacity use inaccurate values of optical constants, do
not include all relevant grain species, and employ wavelength averages of the opacity for only single species rather than for en- sembles of species. In an attempt to im- prove on this situation, Pollack (1979) has carried out a series of calculations of the grain opacity expected for a nebula having solar elemental abundances. Here we briefly summarize his procedures and results.
In order to determine the temperature structure of a protoplanet we require the Rosseland mean extinction coefficient fie, which is given by
~e -- Jo d T
1 (1)
where B, h,T, and K~(X) are the Planck function, wavelength, temperature, and monochromatic extinction coefficient, re- spectively. The factor Ke(~) equals the sum of the extinction coefficients of the various particle species, which, in turn, depend on the fractional abundance of each species, its size distribution function, and optical constants.
On the basis of "equilibrium condensa- tion" calculations by Lewis (1974) for solar elemental abundances and consideration of abundant opaque phases in meteorites and interstellar dust grains (Huffman, 1977), Pollack (1979) included the following grain species in his determination of ~: water ice, liquid water, low-temperature (i.e., hy- drated) silicates with and without iron, high-temperature silicates with and without iron, magnetite, and metallic iron. The lower and upper temperature boundaries of the stability fields for each of these species were obtained from the work of Lewis (1974) as a function of nebular density. The mass fraction of each particulate phase was found from its structural formula, solar elemental abundances (Cameron, 1973), and, in the case of magnetite, from esti- mates of its abundance in meteorites and
EVOLUTION OF GIANT PLANETS 297
interstellar grains (Huffman, 1977). In order to accurately evaluate the wavelength inte- gral in Eq. (1), it is necessary to know the real and imaginary index of refraction of each species over a very broad spectral interval that ranges from the near ultravio- let to the radio domain. Pollack (1979) assembled the best available data from the literature on the needed optical constants. Finally, he defined the particle size distri- bution function primarily on the basis of the distribution for interstellar dust obtained by Mathis et al. (1977) from an analysis of extinction data. Fortunately, for most cases of interest, the dominant particle sizes (0.1- 1 ~m) are small compared to the wave- lengths of greatest importance ( -30 -1000 /xm), and hence the values of ~e do not depend sensitively on this choice of size distribution (Huffman, 1977; Pollack, 1979).
Figure 1 illustrates the resulting Rosse- land mean extinction coefficient as a func- tion of temperature in the range T < 2000°K
for a density of 10 -6 g cm -3. Similar curves hold at other densities. For example, at 10 -l° g cm -3 the steep drop in opacity due to evaporat ion of iron and silicate grains occurs at l l00°K instead of 1700°K; other- wise the curves are identical. The names indicated at major discontinuities on the curve refer to the particulate species that ceases to exist at higher temperatures and hence is responsible for the discontinuity.
At temperatures above the grain evapo- ration point (1000-2000°K, depending on density), molecules constitute the principal remaining source of opacity. The Rosse- land mean drops by about two orders of magnitude at the evaporation point, then rises again at higher temperatures. In this region, up to T = 3500°K, the opacities of Alexander (1975) were used, extended to higher densities with values provided by Alexander (1977, private communication). A large number of molecular species were considered, of which H20 and TiO turned
I O . O i i ! WATER HIGH TEMP.
ICE Sl LICATE
LOW TEMP. / I SILICATE / I IRON
1.0
i:-. I - (.3
o
z < 0.1 uJ
D E N S I T Y = 1.0 x 10 - 6 g cm -3
0 . 0 1 I I I I0 I 0 0 I 0 0 0
T E M P E R A T U R E ( K )
FIG. 1. Rosseland mean grain opacities used in these calculations, plotted against temperature for a
representat ive density of 10 -6 g cm 3.
298 BODENHEIMER ET AL.
out to make the largest contributions. Above 3500°K opacities are obtained by a numerical fit to the tables of Cox and Ste- wart (1965, 1970). Figures 1 and 2 of De- Campli and Cameron (1979) illustrate the opacities used here at temperatures where grains no longer exist.
RESULTS
The four sequences, with masses of ap- proximately 1.0, 1.5, 0.285, and 0.42 Mj are initiated as adiabatic equilibrium con- figurations with conditions approximating those in Cameron 's (1978) model of the solar nebula (for details, see DeCampli and Cameron, 1979). At an approximate mean temperature of 45°K the radius of each initial model is chosen to be small enough so that the object is gravitationally bound, that is, the absolute value of its gravita- tional energy is larger than its thermal en- ergy. These initial models are tidally stable in the gravitational field of Cameron 's (1978) model solar nebula at the orbital radii corresponding to both Jupiter 's and Saturn's estimated point of formation. However , they may become tidally unsta- ble later in their evolution as the nebula transfers matter to the growing central Sun. Whether or not this tidal instability occurs depends on the unknown original distances of the protoplanets from the Sun, on the structure of the solar nebula, and on the time scale of evolution of the solar nebula compared with the time scale for contrac- tion of the protoplanets. Table I gives the properties of the initial models. The ratio of
central to mean density Oc/p is an index of the degree of mass concentration; for com- parison purposes the corresponding ratio in a polytrope of index 1.5 is 6. Other quanti- ties are the central temperature To, the radius in terms of Jupiter 's radius R j , the luminosity in terms of the solar luminosity L~, and the effective temperature Tell.
The main features of the evolution are similar for all four sequences; Fig. 2 com- pares them in the (log Pc, log T~) plane and Table II gives the evolutionary times cor- responding to the marked points in the figure. Here we describe in detail the evolu- tion of 1.5 M j, then make some brief re- marks on the other sequences.
The evolution of 1.5 Mj proceeds first through a sequence of quasi-equilibrium configurations on a Kelvin-Helmhol tz time scale (phase 1). The effective temperature increases slowly while the luminosity in- creases slightly at first, then decreases steadily, reaching log L / L o = - 5.8 at the point of dissociation of molecular hydrogen at the center. The structure is almost en- tirely convect ive except for a thin surface radiative zone that includes less than 0.005 of the mass. This zone decreases in mass as the evolution progresses, since the heating of the outer layers results in an increase in opacity, favoring instability to convection. The increase in opacity also results in the general decline of the luminosity. When the interior temperatures reach 170°K the opac- ity drops by a factor of about 5 due to the evaporat ion of the ices; however this rela- tively small drop does not result in stability against convection. The fully convect ive structure and the decrease in luminosity during the evolution resemble the corre- sponding behavior of pre-main-sequence stars of ~<1 Mo during the Hayashi phase; however the increase in Teff by a factor of 2.5 is a feature that differs from the stellar case, in which T~ remains relatively con- stant.
When Tc reaches about 1600°K the opac- ity drops sharply again by a factor of 1(30 because of the evaporation of the mineral
EVOLUTION OF GIANT PLANETS 299
I I I I I I I I I I
1.0 Md,/"~
~ 5 0.285 Mj
"~o 4x 4. I. 0
5 "0.285 Mj
2 I I I I -I0 -9 -8 -7 6 -5 -4 -5 -2 -I 0 I
log Pc (g cm-3)
FIG. 2. Evolut ion of the centers of the four calculated sequences (light solid lines) in the temperature-density diagram. Evolution times corresponding to the numbered points are given in Table II. The heavy solid line indicates the onset of dissociation of H~. The dashed lines show the evolution during the final contraction phase as calculated for Jupiter by Graboske et al. (1975a) and for Saturn by Pollack et al. (1977). The asterisks indicate the initial models of this final phase as obtained in the present calculations.
gra ins . T h e d e c r e a s e in o p a c i t y at this po in t is sufficient to s t ab i l i ze the ma te r i a l at the c e n t e r aga ins t c o n v e c t i o n ; thus a r ad i a t i ve c o r e fo rms tha t g r o w s in mass as the c e n t e r hea t s up. By the t ime Tc has i n c r e a s e d to 2000°K this r ad i a t i ve reg ion inc ludes a b o u t 0.4 o f the to ta l mass and is nea r ly i so the r - mal . T h e to ta l t ime r e q u i r e d for c o n t r a c t i o n
Time from dissociation point (years) 4 0.370 0.210 0.110 0.046 5 0.397 0.242 0:118 0.063 6 0.404 0.253 0.124 0.074
to the d i s soc i a t i on p o i n t at the c e n t e r is 2.1 × 105 y e a r s . Dur ing this t ime the mass c o n c e n t r a t i o n i n c r e a s e s to 22 at Tc = 1500°K (be fo re f o r m a t i o n o f the r ad i a t i ve core ) , then i n c r e a s e s to 80 at the d i s soc i a - t ion poin t . Thus the c o n t r a c t i o n is not ho- mo logous . The o u t e r rad ius d e c r e a s e s b y a f ac to r o f 26 dur ing this p h a s e o f the evo lu - t ion.
In s t ab i l i t y to g r a v i t a t i ona l co l l apse t a k e s p l a c e when Tc = 2100°K at Pc = 7 × 10 -6 g cm -a. The p h a s e (2) c o l l a p s e o f the cen t ra l r eg ion o c c u r s on a t ime sca le o f a b o u t 0.4 y e a r s and invo lves an i n c r e a s e in d e n s i t y o f a l m o s t five o r d e r s o f ma gn i t ude . The col- l apse is p r ac t i c a l l y ad i aba t i c s ince the o p a c - i ty i n c r e a s e s r ap id ly a b o v e T = 2000°K so tha t the r ad i a t i ve diffusion t ime f rom the i n t e r io r b e c o m e s much l onge r than the col- l apse t ime. A b o v e 0.2 g c m -a the inc reas ing st iffness o f the e q u a t i o n o f s ta te , c a u s e d by non idea l effects , e l e c t r o n d e g e n e r a c y , and c o m p l e t i o n o f d i s s o c i a t i o n , r e su l t s in a hal t
300 BODENHEIMER ET AL.
to the collapse and the format ion of a core in hydrostat ic equilibrium whose initial Pe = 0.3 g cm - 3 a n d Tc = 2.2 × 10*K. This core rapidly grows to include about half of the mass of the protoplanet ; an accretion shock then forms on its outer edge as the outer material continues to collapse onto the core. This shock is initially located at about 5.0 Rj (3.5 × 10 ~° cm) f rom the center; velocities of infall just outside the shock exceed 10 km sec -1. The pressure gradient in this infalling material amounts to about half o f the force of gravity, while typical tempera tures range from 1 to 3 × 10a°K. The tempera tures just behind the shock are about 3 to 4 × 103°K.
The remaining half of the mass collapses onto the core on a t ime scale of 0.1 year. The calculation was carried up to the point where 98.5% of the mass had been accreted onto the core. At this t ime Te = 3.2 × 104°K and p~ = 2.7 g cm -3. The inner 80% of the mass is practically isothermal at T~, while the density drops to 10-" g cm 3 at the edge of the isothermal region. The high opacities in this zone, in the range 103 to 105 cm ~ g-~, result in instability to convect ion, but on
the short t ime scale of the accret ion phase, which is determined by infall velocities at the shock, convect ive motions do not result in appreciable energy transport . At the end of the calculation the shock lies at a dis- tance of 20 Rj f rom the center; the region that is in complete hydrostat ic equilibrium has a radius of 12 Rj. At optical depth 0.7 in the outer layers the radius is still 150 Rj and the tempera ture 39°K. As this small remain- ing amount of material collects onto the core the convect ive motions are expected to result in mixing of the material in the outer layers down to high tempera tures where the grains will evaporate . The total radius of the protoplanet at the t ime all the material comes into hydrostat ic equilibrium has not been calculated; it is est imated to be considerably smaller than 12 Rj (see the calculation of this quantity for 1 M j, be- low). The variations of Te~, L/Lc, pe, Tc, and R with time for the 1.5 Mj evolution are shown in Fig. 3.
The evolution of 1.0 Mj proceeds in a similar manner. The initial conditions are given in Table I. The early contract ion (phase 1) is convect ive with a thin radiative
~2
2
f 1' 1 1 m
log pcF~ ~ 0
log T c ~ ~ J -3
-6
-9
1.5 Mj
~..~o~ c __~L~ i
0 .4 0 .8 1.2: 1.6 2 .0
TiME (10 5 YR)
log Tef f
I 0 2 4 6 8 /0 12 L4 16
T~ME (10 6 SEC)
FIG. 3. The behavior of central temperature, effective temperature, central density, surface luminosity, and total radius as a function of time for 1.5 M~. The left-hand portion represents the early equilibrium phase, with time measured from the beginning of the calculation. The right-hand portion represents the subsequent hydrodynamic phase, with time measured from the onset of collapse at the center.
EVOLUTION OF GIANT PLANETS 301
surface zone. The luminosity declines to log L/LQ = - 6 . 2 5 at the onset of dissocia- tion. The central radiative zone includes 35% of the mass at this t ime and is practi- cally isothermal at 1900-2100°K. The degree of mass concentrat ion is 22 at Te = 1500°K and 62 at Tc = 2100°K. The radius at the point o f dissociation is 168 R j, a reduction by a factor of 27. The evolution time up to this point is 4 × 105 years. Gravitat ional collapse sets in at Te = 2100°K, Pc = 2 × 10 -~ g cm -3, a higher density than the corresponding one for 1.5 Mj. The dissocia- tion collapse proceeds on a time scale of 0.25 years and results in the formation of an initial hydrostat ic core when Pc = 0.2 g cm -3, T~ = 2 × 104°K. The radius of the accret ion shock which develops shortly af- ter this t ime is 3.0 Rj. At the end of the calculation 96% of the mass has been ac- creted onto the core, p~ = 1.8 g cm -3, T~ = 2.9 × 10~K, the radius of the accretion shock is 8 Rj while the equilibrium region has a radius of 4 Rj. Parameters of the evolution as a function of time are given in Fig. 4, and the structure of the final model is shown in the (log p, log T) plane in Fig. 7. In this diagram are shown the various equa- t ion-of-state regions of interest and the ap- proximate locations of the H2 dissociation
zone and the H ionization zone. Note that, as expected, the molecular hydrogen heats up and undergoes dissociation as it passes through the accretion shock.
The final Jupiter model shown in Fig. 7 still does not represent the starting model for the evolution through phase (3), the final phase of cooling and contract ion to the present radius. A small fraction of its mate- rial is infalling toward the accret ion shock and is not in hydrostat ic equilibrium. The structure of the initial phase (3) model has been determined by matching a model at- mosphere boundary condition (see Gra- boske et al., 1975a) to the equilibrium core, under the assumptions that the infall has been completed and that grain effects in the a tmosphere have become negligible be- cause of mixing to deeper, hotter layers. The resulting model has p~ = 2.73, Tc = 34,670°K, only slightly different f rom those at the end of the hydrodynamic calcula- tions. The luminosity at this point is log L/LG = - 5 . 6 5 while the effective tempera- ture is 617°K and the radius of 9.08 × 109 cm is only 1.3 t imes the present value for Jupiter. This initial model falls on the evo- lutionary track of Graboske et al. (1975a) at a point corresponding to an age of about 107 years. An asterisk marks the position of
Y ~2
,.S
o~
5 T r T--T- ^_^
1.0 Mj I ~ 0
c
4
log ( ~ ) -3 log T c
3 2
-6
2
-9 log Tef f
I 2 3 4 I 2 3 4 5 6 7 8 9
TIME (10 5 YR) TIME (10 6 SEC)
FIG. 4. Evolution of 1.0 Mj. The same quantities are plotted as in Fig. 3.
log Pc
log R_ Rj
302 BODENHEIMER ETAL.
this model in the (log Pc, log TJ plane (Fig. 2). Note that a considerable portion of the earlier t rack is bypassed , that the tempera- ture max imum in Jupi ter ' s interior is only 35,000°K rather than the 50,000°K calcu- lated earlier, and that the luminosity maxi- mum during the final contract ion of Jupiter is more than two orders of magnitude lower than that calculated under the assumption of purely hydrostat ic evolution. The subse- quent evolution through phase (3) takes place as calculated by Graboske et al. (1975a).
The evolution of 0.42 Mj proceeds f rom the initial conditions given in Table I through the largely convect ive phase (1) on a t ime scale of 2 x 106 years , during which the luminosity declines by a factor of 50 to log L/LQ = - 7 . 2 5 and the radius by a factor o f 30 to 68 Rj. At the t ime of dissociation at the center the central radiative core con- tains 33% of the mass, slightly less than in the two previous cases at the corresponding time. The degree of mass concentrat ion is 23 at Tc = 1500°K and 63 at the onset of dissociation at Tc = 2200°K. The central density reaches 1.2 × 10 -4 g cm -a at this point, about a factor 6 higher than for the case of 1.0 Mj. The higher starting density results in a shorter collapse t ime of 0.12
o ~
5 - - T r r r
2
0.42 Mj
/
4 8 12 16 TIME (105 YR)
year. The hydrostat ic core forms and be- comes well developed at pc = 0.2 g cm -3, T c = 1.6 × 104°K, a somewhat lower tempera- ture but about the same density as in the case of the higher masses. Also the increase in Pc is not as large during the collapse. At the time when the accretion shock becomes well developed at a radius of 2 Rj the equilibrium core includes about 55% of the mass. At the end of the calculation, when the core mass includes 99% of the total, the shock radius has grown to 6 Rj while the region that is comple te ly hydrostat ic ex- tends to about 4.2 Rj. The radius that the final configuration is expec ted to have at the onset o f phase (3) has not been calcu- lated but it is expected to be slightly smaller than this value. As in the other cases, the infalling material outside the shock front attains velocities of over 10 km sec - ' . Core central conditions level off with t ime to pc = 0.4 g cm -3, Tc = 19,000°K. The evolution as a function of t ime is shown in Fig. 5.
The evolution of 0.285 Mj proceeds through the early convect ive phase on a t ime scale of 5 x l06 years, during which the radius decreases by a factor of 30 and the luminosity by a factor o f more than 100. The degree of mass concentrat ion is 24 at Tc -- 1500°K and 91 at the point of dissocia-
1 T - - T 1
log Pc - ~
log T c
L log 0 -I0 L~
log
bog Teff
20 i 2 3 4 TiME (10 6 SEC)
5
- 6 c~ o
- 9
FIG. 5. Evolution of 0.42 Mj. For details see Fig. 3.
EVOLUTION OF GIANT PLANETS 303
tion, which occurs at Tc = 2500°K, pe = 3.7 x 10 -4 g cm -3 with a radiative core of 40% of the mass. After a collapse time of 0.06 year, the hydrostatic core forms at Pc = 0.2 g cm -3, Tc = 1.5 × 104 K, values similar to those for 0.42 Mj. The accretion shock forms at about 2 Rj from the center when the core includes about 55% of the total mass. The final accretion phase lasts 0.01 year. At the end of the calculation the core has accumulated more than 99% of the mass and the shock front is at a radius of 7 Rj. The radius of the hydrostatic region at this time is about 4.4 Rj or 5.2 times the present radius of Saturn. Central conditions level offto Tc = 17,000°K, Pc = 0.32 g cm -3, central pressure 4 × 1011 dyn cm -2. The evolution is illustrated in Fig. 6, and the structure of the last hydrodynamic model obtained is shown in Fig. 7.
In order to obtain the luminosity and radius of Saturn at the onset of phase (3), after complete hydrostatic equilibrium has been obtained, we recomputed the final model with a static model atmosphere as outer boundary condition using the proce- dure described by Pollack et al. (1977). Note again that as in the case of Jupiter, this atmosphere includes molecular sources of opacity but no grain sources. The result-
ing model has Tc = 18,335°K, Pc = 0.28 g cm -~, log (L/Lo) = -4 .78 , R = 1.94 × 10 t° cm or 3.4 times the present radius of Sat- urn, and Teff= 660°K- The Saturn sequence calculated by Pollack et al. (1977) starts with a radius of 10.7 times the present observed value and a luminosity of 10 4 Lo. The present model, indicated by an asterisk in Fig. 2, for the onset of phase (3) lies on their evolutionary track at an age of about 10 '5 years. The subsequent evolution, start- ing from that point, follows the calculation of Pollack et al. (1977; see for example their Figs. 2 and 3). Note that the luminosity of Saturn at the onset of phase (3) is slightly larger than that for Jupiter at the corre- sponding point of its evolution and that the equilibrium track is reached before the inte- rior temperature maximum is attained.
DISCUSSION
The general results of these sequences may be compared with the earlier calcula- tions by Bodenheimer (1974) of the evolu- tion of 1 Mj. As mentioned above, in that calculation the opacity was set to a con- stant value of 0.15 cm 2 g ' at temperatures below 1400°K, while in the present calcula- tions the opacities in that temperature re- gime are in general considerably higher,
5 r r T ~ T 1 ~ ~ r o
0 . 2 8 5 Mj J
-
I ~ log Tel f
I0 2[0 30 40 0 0.5 1.0 1.5 2.0 2.5
TIME (~05 YR) TIME (106 SEC)
FIG. 6. Evolut ion of 0.285 Mj. For details see Fig. 3.
304 BODENHEIMER ET AL.
o
F-
0
5 I I I I I I / I I LOW DENSITY REGION
NON IDEAL GAS ~ • IDEAL
GAS HYDROGEN COULOMB METALLIC PLASMA
)OLATION REGION
DIS OClATIO[~ L
11% .1 II SOLID METALLIC
5 - I
I MJ DENSE MOLECULAR
FLUID
2 I 2185 MJI I I I I I l
-I0 -9 -8 -7 -6 -5 -4 -5 -2 -I 0 I 2 5 log p (gcm -3)
Flc. 7. Interior s t ructure of the models for Jupiter and Saturn at the end of the hydrodynamic calculations (heavy solid lines). The various regions defined by equation-of-state characteris t ics are indicated. Interior to the accret ion shock region (dashed lines) the models are in or near hydrostat ic equilibrium. Outside that region, they are nearly in free-fall.
I I I HIGH DENSITY REGION
METALLIC HYDROGEN COULOMB FLUID
reaching maxima of 7 and 6 cm '~ g 1 at T = 200 and 1500°K, respectively, for a density of 10-" g cm 3 (cf. Fig. 1). The 1974 models were almost completely radiative up to the point of dissociation, while the present ones are almost completely convective up to the point of grain evaporation, as a consequence of the higher opacities. While the 1974 sequence contracted through phase (1) with a practically constant L = 10 -5 Lo on a time scale of 7 × 10 ~ years, the new models undergo a decrease in L from about 10 -5 to 6 x 10 -7 Lo, more than a factor of 10, and the time scale is increased to 4.3 x 105 years, a factor of 6 longer. The temperature dependence of the opacity in the outer layers is the main reason for the difference. However , the evolution in the (log Pc, log T~) plane is practically identical in the two calculations, and the further
evolution through phase (2) and the accre- tion process leading to phase (3) exhibit only minor differences. For example, the radius at the point of dissociation is 168 R~ compared with 200 Rj in the earlier calcula- tion, while the values of Tc at the end of the calculation are 2.9 x 104 in the present case and 3.0 x l04 in the earlier case.
An important consequence of the con- vective structure during phase (1) results if rotation is considered, since the angular momentum distribution is modified by tur- bulent viscosity. Bodenheimer (1977) made an approximate calculation of the evolution of rotating Jovian protoplanets under the assumptions that (1) the structure is that of a homologously contracting polytrope of index 3 with radiative energy transport only, and (2) each element of mass con- serves its angular momentum during the
EVOLUTION OF GIANT PLANETS 305
contract ion. The present work shows that neither of these assumpt ions is correct . H o w e v e r the main point of that paper was that a significant d i lemma resulted f rom those assumptions: if the angular momen- tum distribution in a Jovian protoplanet was taken to be that of an initially uni- formly rotating uniform-densi ty sphere and if each e lement of material conserved its angular momen tum, then a model with sufficient angular m o m e n t u m in its outer layers to account for the orbital motions of the Galilean satellites would become unsta- ble before reaching the dissociation point. The result would probably be fission of the protoplanet into two orbiting bodies rather than formation of a single central object with a surrounding stable cloud out of which satellites could form. The latter sys- tem, it was pointed out, could result if the angular momen tu m distribution were modified so that it contained less angular momen tum in the center and more in the outer layers than in the distribution of a uniformly rotating uniform sphere (which, in the absence of bet ter information, may represent conditions at the time of forma- tion of the protoplanet) . Angular momen- tum transport outward during the contrac- tion is a natural way of modifying the distribution in the desired direction, and the present results show that such a mechanism exists in the form of convect ion. Thus angular momen tum transport by convec- tion during phase (1) is likely to modify the angular momen tum distribution sufficiently so as to avoid the fission instability and to favor the formation of a central planet plus satellite system. A more detailed discussion of the effects of rotation during phases (1) and (2) of the evolution will be presented in a future paper.
The present evolut ionary models for 1 Mj and 0.28 Mj during phase (1) are very similar to the " J u p i t e r " and " S a t u r n " pro- toplanets calculated through that phase by DeCampli and Cameron (1979) with " h i g h " opacit ies (similar but not identical to the ones here). There are small differences in
surface propert ies as a function of time; also the thin radiative zones that appear in those calculations at locations be tween the center and the surface at various times are absent in the present ones. The total evolu- t ionary time up to the point o f dissociation is 4.3 × 10 ~ years in the present calculation and 8 × 10 ~ years in the earlier calculation for the case of 1 Mj; the corresponding times for 0.28 M j a r e 4.56 × 106and 8.13 × 106 years, respectively. The shorter t imes calculated here are a consequence of the somewhat lower opacit ies in the outer pro- toplanetary layers (T < 100°K) which con- trol the rate of energy loss. However , the evolution of the centers in the (log p, log T) plane are practically identical. The latter point is important , since DeCampli and Cameron (1979) showed that the centers of protoplanets of 1 Mj or less pass through the region of the (log p, log T) diagram where liquid iron exists, jus t before the onset o f dissociation. As a result, grain growth and precipitation toward the center, coupled with convect ive mixing of the grains in the outer regions of the proto- planet, can result in the formation of a rocky core with mass about 1 M¢. The present results confirm the conclusions reached by DeCampli and Cameron in this regard. The internal structures of the models for 1 Mj and 0.285 Mj at the point where grain evaporat ion is taking place at the center are shown in Tables I I I and IV, respect ively.
In summary , these calculations of the evolution of protoplanets lead to the follow- ing main results. (1) A protoplanet of 1 Mj starting at R = 4500 Rj, log L/LG = - 4 . 1 , Tef~ = 25°K contracts in 4.3 × 103 years to R = 168 Rj, logL/Lo = -6 . 25 , Te~ = 40°K. Its structure is largely dominated by con- vection until T¢ reaches 1500°K, after which it develops a radiative core containing about one-third of the mass. It then col- lapses in 0.25 year , regaining equilibrium w i t h R = 1.3 Rj, log L/Lo= -5 .65 , To= 35000°K, and Pc = 2.7 g cm -3. The evolu- tion of 1.5 Ma is closely parallel. (2) A
306 BODENHEIMER ET AL.
TABLE III
STRUCTURE OF MODEL FOR 1 Mj DURING PHASE |
Age: 2.91 × 103 years
Central temperature: Tc = 1809°K Central density: Pc = 4.1 × 10 -6 g cm -3
protoplanet of Saturn's mass (0.28 Mj), starting at R = 1637 Rs (where Rs is Saturn's present radius), log L / L o = - 5.6, Teff= 20°K, contracts in 4.6 × 10 ° years to R = 57 Rs, log L / L o = -7 .7 , Teff= 32°K. As in the case of 1 Mj the structure is convective until a radiative core develops at temperatures above the grain evapora- tion point. The collapse induced by dissoci- ation of H2 takes 0.074 year, after which hydrostatic equilibrium is reestablished in the entire mass at R = 3.4 Rs, log L / L o = - 4 . 9 , Te= 1.8 × 104°K, p c = 0 . 2 8 g c m -3. The evolution of a protoplanet with 1.5 times the mass of Saturn is similar.
We also make the following comments: (1) In a rotating protoplanet the occurrence of convect ion during the early hydrostatic phase will lead to the transport of angular momentum outward, favoring the forma- tion o f a central object which is rotationally
stable surrounded by a low-density nebula. (2) Although protoplanets are initially tid- ally stable in a primordial solar nebula, their long contraction time, caused by high grain opacity in the outer layers, may lead to tidal instability due to formation of the Sun before the rapid decrease in radius caused by the dissociation collapse. Fur- ther detailed studies of the time depen- dence of the gravitational field in a " so l a r " nebula are required to clarify this point. (3) In the case of a Jovian protoplanet, the luminosity and radius at the onset of the final hydrostatic phase are considerably smaller than those used as initial conditions by Graboske et al. (1975a). In the case of Saturn the agreement is better: the initial radius used by Pollack et al. (1977) was a factor of 2 to 3 larger than that obtained here, but the luminosity at the beginning of
TABLE IV
STRUCTURE OF MODEL FOR 0.285 Mj DURING PHASE 1
Age: 2.59 × 103 years Central temperature: Tc = 1853°K Central density: Pc = 5.4 × 10 3 g cm-3 Radius: R, = 4.0 × 10 Hcm Luminosi ty: L, = 7.17 × 1023 ergs sec -~
the hydros ta t i c con t r ac t i on phase is compa- rable in the two cases . The s u b s e q u e n t con t r ac t i on h is tory to the p l a n e t ' s p resen t state is no t l ikely to be affected s ignif icant ly by the change in init ial cond i t ions . (4) Wi th in the con tex t of spher ical s y m m e t r y , the fol lowing impor t an t effects should be cons ide red in fu ture ca lcula t ions : (a) inter- change of mass (winds , accre t ion) b e t w e e n con t rac t ing p ro top lane t s and the su r round- ing nebu la , (b) the effect u p o n the evo lu t ion of a p reexis t ing rocky core of a few Ms, (c) a more deta i led s tudy of the process of core fo rma t ion by gra in p rec ip i ta t ion , and (d) deta i led cons ide ra t ion of the process of grain evapo ra t i on by convec t i ve mix ing at the onse t of the final con t r ac t i on phase . The speed of des t ruc t ion of the grains in the o u t e r m o s t layers and the resu l t ing change in opaci ty will s t rongly affect the luminos i ty at this po in t ; in the p re sen t ca lcu la t ion we have a s s u m e d that this p rocess occurs rap- idly c o m p a r e d with the evo lu t ion t ime. (5) A l though the o c c u r r e n c e of c o n v e c t i o n resul ts in s t ruc ture and evo lu t ion of proto- p lanets dur ing the init ial hydros ta t i c phase that is subs tan t ia l ly different f rom that found by B o d e n h e i m e r (1974), the dynami - cal col lapse phase and the t r ans i t ion to the final hydros ta t i c phase are unaf fec ted and proceed essent ia l ly as ca lcu la ted in that ear l ier work.
ACKNOWLEDGMENTS
The authors wish to thank Dr. D. Alexander for providing unpublished opacity data. P.B. and G.M. were supported in part by National Science Founda- tion Grant AST 76-17590 during the performance of this research and W.M.D. was supported by NSF Grant AST 76-80801.
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