Differentiation of Planetesimals and the Thermal Consequences of Melt Migration
DDifferentiation of Planetesimals and the Thermal Consequences of Melt Migration
Nicholas Moskovitz Department of Terrestrial Magnetism, Carnegie Institution of Washington [email protected] Eric Gaidos Department of Geology and Geophysics, University of Hawaii ABSTRACT Al and Fe to determine (i) the timescale on which melting will 19 occur; (ii) the minimum size of a body that will produce silicate melt and 20 differentiate; (iii) the migration rate of molten material within the interior; and (iv) 21 the thermal consequences of the transport of Al in partial melt. Our models 22 incorporate results from previous studies of planetary differentiation and are 23 constrained by petrologic (i.e. grain size distributions), isotopic (e.g.
Pb-‐
Pb and 24
Hf-‐
W ages) and mineralogical properties of differentiated achondrites. We 25 show that formation of a basaltic crust via melt percolation was limited by the 26 formation time of the body, matrix grain size and viscosity of the melt. We show that 27 low viscosity (< 1 Pa·s) silicate melt can buoyantly migrate on a timescale 28 comparable to the mean life of Al. The equilibrium partitioning of Al into silicate 29 partial melt and the migration of that melt acts to dampen internal temperatures. 30 However, subsequent heating from the decay of Fe generated melt fractions in 31 excess of 50%, thus completing differentiation for bodies that accreted within 2 Myr 32 of CAI formation (i.e. the onset of isotopic decay). Migration and concentration of 33 Al into a crust results in remelting of that crust for accretion times less than 2 Myr 34 and for bodies >100 km in size. Differentiation would be most likely for 35 planetesimals larger than 20 km in diameter that accreted within ~2.7 Myr of CAI 36 formation. 37 38
Keywords: planetary differentiation, planet formation, thermal histories, achondrites, iron meteorites < Al and Fe, were potentially significant sources of internal heat. 57 Most, but not all CAIs contained Al at a concentration close to the canonical ratio 58 Al/ Al ~ 4-‐5 × -‐5 (Jacobsen et al. 2008; MacPherson et al. 2010). The consistent 59 enhancement in Mg of planetary materials with respect to CAIs suggests that Al 60 was uniformly distributed in the solar system (Thrane et al. 2006; Villeneuve et al. 61 2009), although some refractory grains lack Al and there may have been Al-‐free 62 reservoirs. Near-‐homogenous Al supports an exogenous origin such as a 63 supernova or Wolf-‐Rayet star (Ouellette et al. 2005; Gaidos et al. 2009), as opposed 64 to the active young Sun (Gounelle et al. 2006). The initial abundance of Fe relative 65 to Fe is more difficult to infer and a reliable value has proven elusive, but was 66 perhaps no more than ~10 -‐6 (Tachbana et al. 2006; Dauphas et al. 2008; Mishra et 67 al. 2010). This isotope also seems to have been uniformly distributed throughout 68 the Solar System (Dauphas et al. 2008), and the initial value may be typical of star-‐69 forming regions (Gounelle et al. 2009). Table 1 gives the adopted initial 70 concentration and half-‐life of each SLR. 71 72 These concentrations correspond to an integrated energy input of 6.7 x 10 J/kg for 73 Al and 7.0 x 10 J/kg for Fe, assuming chondritic elemental abundances (0.0113 74 and 0.24 respectively; Ghosh et al. 1998). These energies were sufficient to melt the 75 interior of planetesimals: the Fe-‐S eutectic at 1200 K marks the onset of melting 76 (McCoy et al. 2006); 1400 and 1900 K are typical solidus and liquidus temperatures 77 for planetary silicates (McKenzie et al. 1988; Agee 1997); and the Fe-‐Ni metal that 78 dominates the melt compositions of iron meteorites at 1800 K (Benedix et al. 2000). 79 Partially molten metals and silicates can gravitationally segregate, forming 80 differentiated interiors. Subsequent collisional disruption of a differentiated 81 planetesimal would have produced a suite of fragments, some of which are 82 recovered today as meteorites, ranging from pieces of Fe-‐Ni core, to dunite-‐rich 83 chunks of mantle, to feldspathic crustal basalt (Mittlefehldt et al. 1998). 84 85 Both meteorites and asteroids provide evidence for widespread differentiation in 86 the early Solar System. The majority of the ~150 parent bodies represented in 87 meteorite collections experienced metal-‐silicate differentiation (Burbine et al. 88 2002). The canonical example of a differentiated asteroid is the large (~500 km) 89 asteroid 4 Vesta. The spectrum of Vesta closely matches the basaltic HED 90 meteorites, thereby suggesting that Vesta’s surface is an undisrupted differentiated 91 rust (McCord et al. 1970). This spectral similarity, the lack of other large basaltic 92 asteroids in the main belt, Vesta’s favorable location for the delivery of fragments to 93 Earth, and the presence of an extensive collisional family of Vestoids (Binzel et al. 94 1993) suggest that the HEDs are genetically related to Vesta (Consolmagno et al. 95 1977). 96 97 The thermal history and differentiation of Vesta-‐like bodies have been modeled 98 using spherically symmetric descriptions of heat conduction. Ghosh et al. (1998) 99 modeled Vesta’s thermal evolution through core and crust formation, assuming 100 instantaneous segregation of metal and silicate phases at the metal-‐sulfide and 101 silicate liquidi respectively. These calculations constrained the epochs of accretion 102 (2.85 Myr after CAIs), core formation (4.58 Myr), crust formation (6.58 Myr), and 103 geochemical closure of Vesta (~100 Myr), all of which are broadly consistent with 104 the measured chronologies of HED meteorites. In addition, Ghosh et al. (1998) 105 showed that concentration of Al into Vesta’s crust affected the body’s thermal 106 evolution, while the decay of Fe following its segregation into the core, did not. 107 However, the abundance of Fe adopted by these authors was more than an order 108 of magnitude less than current estimates (e.g. Mishra et al. 2010). Hevey et al. 109 (2006) presented a series of models for the heating of a planetesimal by the decay of 110 Al and traced these bodies’ evolution through sintering of the initially 111 unconsolidated silicates, and the formation and convection of magma oceans. They 112 found that the parent bodies of differentiated meteorites must have accreted within 113 ~2 Myr of CAI formation, but these authors did not take into account the 114 consequences of SLR redistribution. Sahijpal et al. (2007) coupled a planetesimal 115 accretion model with a thermal conduction model, and included volume loss due to 116 sintering, redistribution of SLRs due to melting, and the gradual formation of both a 117 core and crust. This study also found that differentiation occurred on bodies that 118 accreted within 2-‐3 Myr of CAI formation. In general, these works all suggest that a 119 time of accretion <3 Myr was necessary for differentiation. No general consensus 120 was reached regarding the importance of Fe, in part due to different values used 121 for its initial abundance. 122 123 Other work has described the dynamics of the melting and differentiation process. 124 Taylor (1992) described core formation on asteroids based on the analytical works 125 of McKenzie (1984, 1985) and Stevenson (1990). He concluded that high degrees of 126 partial melting (~50%) were required for metal to efficiently segregate into the 127 cores of iron meteorite parent bodies. Wilson et al. (1991) proposed that exsolution 128 of <1 wt% of H O and CO would increase melt buoyancy to an extent that basaltic 129 melt would pyroclastically erupt at speeds greater than the local escape velocity for 130 bodies smaller than approximately 100 km. Objects larger than this (e.g. asteroid 4 131 Vesta) had sufficient surface gravity to retain erupted melt in a crust. Such eruptions 132 could also occur following the pooling of basaltic liquid in large sills at the base of an 133 unaltered, less permeable layer in the upper ~5 km (Wilson et al. 2008). This model 134 explains the lack of a basaltic component associated with the aubrite, acapulcoite-‐135 lodranite and ureilite meteorites, each of which derived from parent bodies that 136 xperienced partial melting (Wilson et al. 1991; McCoy et al. 1997; Goodrich et al. 137 2007). 138 139 Here we address several aspects of differentiation not considered by these previous 140 studies. These include a realistic treatment of silicate melt migration rates and the 141 thermal consequences of such migration, constraints on the initial conditions (e.g. 142 time of accretion) responsible for the formation of putative magma oceans on 143 planetesimals like the asteroid 4 Vesta (Righter et al. 1997) and the parent bodies of 144 magmatic iron meteorites (Scott 1972), and quantifying the influence of Fe decay 145 on differentiation. Recently updated constraints on initial SLR abundances in the 146 Solar System (Jacobsen et al. 2008; Tachibana et al. 2006; Mishra et al. 2010) 147 provide additional motivation for revisiting issues related to planetesimal 148 differentiation. 149 150 In Section 2 we describe physical processes treated by our collection of models: 151 thermal conduction in a spherical body; the production and migration of partial 152 melt; and the transport of the primary heat source, Al, in that melt. In Section 3 we 153 present calculations for the thermal evolution of a planetesimal up to 154 differentiation, the influence of melt migration, and an estimate of the minimum size 155 for differentiation. In Section 4 we describe and justify the exclusion of several 156 processes from our model. Finally, in Section 5 we combine these results in a 157 synthesis of the relevant processes, time scales and sizes associated with 158 differentiated bodies, compare our results with previous studies, and propose future 159 areas of investigation. 160 161 Table 1 lists the variables and parameters employed in our calculations. 162 163 € ∂ T ∂ t = r ∂∂ r r κ∂ T ∂ r ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + Q ( t ) c , (1) 172 173 where r is the radial coordinate, Q(t) is the decay energy per unit mass per unit time 174 for Al and Fe, and c is the specific heat at constant pressure. The thermal 175 diffusivity is related to the thermal conductivity k , the density ρ and specific heat c : 176 177 € κ = k ρ c . (2) 178 179 ssuming a radially constant ρ , κ and c , and expanding Equation 1 yields the 180 conduction equation for r > 0: 181 182 € ∂ T ∂ t = kr ρ c ∂ T ∂ r + k ρ c ∂ T ∂ r + Q ( t ) c . (3) 183 184 The singularity at r =0 can be remedied by expanding the offending term in a 185 first-‐order Taylor series, yielding the conduction equation for r=0 : 186 187 € ∂ T ∂ t = k ρ c ∂ T ∂ r + Q ( t ) c . (4) 188 189 Equations 3 and 4 are solved using the explicit finite difference method (Ozisik 190 1994) with a constant temperature prescribed for the outer boundary ( r=R ). 191 Though a radiative boundary condition offers a more robust approach (Ghosh et al. 192 1998), the errors associated with a simpler fixed-‐temperature, Dirichlet boundary 193 condition are small because radiation balance for bodies in the inner Solar System 194 will always dominate interior heat flow (Hevey et al. 2006). The initial temperature 195 of the body is assumed to be constant for all radii and equal to the boundary 196 temperature. The spatial element size and time step are chosen sufficiently small 197 (e.g. 250 m and 800 yr) to ensure stability of the numerical result. For each time 198 step heating from Al and Fe decay is computed: 199 200 € Q ( t ) = f Al Al Al ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ E Al τ Al e − t / τ Al + f Fe Fe Fe ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ E Fe τ Fe e − t / τ Fe , (5) 201 202 where t is time relative to CAI formation and all other quantities are defined in 203 Table 1. 204 205 For temperatures above the silicate solidus, the effective heat capacity c is computed 206 as an average, weighted by the melt fraction, of the specific heats of the solid and 207 liquid phases (800 and 2000 J/kg/K respectively; Ghosh et al. 1998). These 208 temperature-‐independent values are upper limits for chondritic assemblages 209 (Ghosh et al. 1999; Navrotsky 1995), their adoption produces temperatures that are 210 underestimated by less than ~10% (Sec. 4.5). The melt fraction Φ is calculated from 211 an empirical fit to the fertility function of peridotite (McKenzie et al. 1988): 212 213 € Φ ( T ') = + ⋅ T ' + ⋅ T ' + ⋅ T ' (6) 214 215 and 216 217 € T ' = T − . (7) 218 219 he coefficients in these equations were calculated from the solidus and liquidus 220 temperatures of peridotite at a pressure of 1 bar (1373 and 2009 K respectively, 221 McKenzie et al. 1988). Planetesimals smaller than 500 km had internal pressures 222 less than 1 kbar (assuming a spherical body in hydrostatic equilibrium with a mean 223 density of 3300 kg/m ) and would not have dramatically different solidus and 224 liquidus points. The latent heat of fusion L for chondritic silicates is 4 × J/kg 225 (Ghosh et al. 1998) and was accounted for between the silicate solidus and liquidus 226 based on the melt fraction computed from Equation 6. 227 228 ∆ ρ are ; we adopt the lower value to minimize the rate 241 of melt migration. Experimental and theoretical work has shown that even at melt 242 fractions of a few percent, molten silicates form an interconnected network (Taylor 243 et al. 1993). Compaction of a melted region is controlled by the bulk viscosity of the 244 matrix, microscopic shear viscosity of the matrix, viscosity of the melt and 245 permeability of the matrix to fluid flow control (McKenzie 1984, 1985). Assuming 246 reasonable values for these material properties, the size of the compacted region 247 ( δ c ) can be parameterized in terms of melt fraction, grain size and viscosity of the 248 melt ( φ , a and µ respectively; McKenzie 1985): 249 250 € δ c = (10 km ) φ
3/ 2 a mm ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Pa ⋅ s µ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
1/ 2 . (8) 251 252 In general, compaction-‐driven migration will occur if the thickness of the region 253 undergoing compaction ( δ c ) is smaller than the melting region (Taylor et al. 1993). 254 The size of the melting zone for a planetesimal uniformly heated by radionuclide 255 decay is comparable to its radius R (Wilson et al. 2008). When R / δ c is much greater 256 than unity, compaction is likely to occur. 257 258 The four key parameters for compaction-‐driven melt migration ( R , µ , a and φ ) can 259 each vary by several orders of magnitude, thus making it difficult to define specific 260 conditions for compaction. However, inserting plausible values into Equation 8 261 demonstrates that compaction and melt migration were likely outcomes. 262 hondrules, which offer the best available guide for grain sizes in unaltered 263 planetesimals (Taylor et al. 1993), are typically several 100 µ m in size and are 264 rarely larger than 1 cm (Weisberg et al. 2006). The viscosity of molten silicates 265 varies from 0.001 to 1000 Pa·s, however, basaltic melts derived from chondritic 266 precursors have silica contents generally ≲ a < 1 268 cm and µ > 1 Pa·s, will expel melt up to φ =50% (Eqn. 8). Equation 8 suggests that a 269 basaltic crust can form via compaction for bodies as small as ~20 km in size, 270 assuming a 10% melt fraction (i.e. consistent with the degree of partial melting that 271 removed basalt from the lodranite meteorites; McCoy et al. 1997). This size limit is 272 similar to that for the onset of buoyant melt migration within a rigid (non-‐273 compacting) silicate matrix (Walker et al. 1978). Fully molten bodies smaller than 274 20 km may develop a basaltic crust via fractional crystallization of a magma ocean. 275 276 Employing Equations 2 and 4 of Taylor et al. (1993) and Equation 7 from McKenzie 277 (1985), the characteristic e-‐folding time for the expulsion of molten silicates from a 278 compacted region is: 279 280 € τ mig = µ π a φ (1 − φ ) Δρ G ρ , (9) 281 282 where all parameters are the same as in Equation 8 and/or defined in Table 1. This 283 equation is valid when the size of the melt zone is comparable to R , the body is of 284 uniform density, and R / δ c >> 1. Interestingly, τ mig is independent of planetesimal 285 size: the low velocity of melt migration in small bodies (due to low gravitational 286 acceleration) is counter-‐acted by the short distance over which the melt must travel. 287 For standard values of Δρ and € ρ (Table 1), µ = 1 Pa·s (typical of basaltic melts), and 288 when grains grow larger than approximately 1 mm, the migration time scale 289 becomes comparable to the Al mean-‐life (Eqn 9). 290 291 Al Via Melt Migration Al. 296 Primitive and differentiated achondrites derived from partial melt residues exhibit 297 decreasing Al abundance with increasing degrees of melting (Mittlefehdlt et al. 298 1998), a trend consistent with the early removal of Al-‐enriched partial melts. 299 300 A zero-‐dimensional box model is developed to quantify the thermal consequences of 301 the removal of Al-‐enriched partial melt. Only unmelted silicates and pore space are 302 initially present and as temperature increases from the heat of Al decay, partial 303 melt is produced and migrates out of the box on a characteristic time scale 304 (Equation 9). We assume that the planetesimal is sufficiently large such that the rate 305 f heat loss due to conduction is negligible relative to timescales of heating and melt 306 migration. 307 308 The amount of melt in the box is governed by: 309 310 € d φ dt = − φτ mig + ∂Φ ( T ) ∂ T ∂ T ∂ t , (10) 311 312 where τ mig is given by Equation 9 and all other parameters are given in Table 1. 313 Equation 10 accounts for the melt lost due to migration and the amount produced 314 due to heating. By defining 315 316 € α = µ π a G ρ Δρ , (11) 317 318 Equation 10 can be rewritten as: 319 320 € d φ dt = − φ (1 − φ ) α + ∂Φ∂ T ∂ T ∂ t . (12) 321 322 All of physical properties of this system are included in the α -‐parameter and with 323 the exception of grain size ( a ) are assumed constant. The coarsening of grains 324 associated with increasing degrees of partial melting (i.e. Ostwald ripening; Taylor 325 1992) is parameterized using a correlation between typical grain size and degree of 326 partial melting for several achondrite meteorite groups (see Mittlefehldt et al. 1998 327 and McCoy et al. 1997). The acapulcoites tend to be fine grained (150-‐230 μm) and 328 experienced low degrees of partial melting (<1%, up to a few %). The lodranites 329 have coarser grains (540-‐700 μm) and experienced higher degrees of partial 330 melting (5-‐10%). The ureilites are characterized by large grains averaging 1 mm in 331 size and represent melt fractions of 10-‐20%. From these examples we parameterize 332 the growth of silicate grains as a linear function from 100 μm (a typical size for 333 small chondrules; Weisberg et al. 2006) at 0% melt fraction, increasing with a slope 334 of 90 μm/melt %. We safely assume that grain growth is an equilibrium process: 335 grains ripen to 5 mm (the maximum for which this model is valid, i.e. φ = 50%) in 336 much less than 10 yr (Taylor et al. 1993), faster than typical migration time scales 337 (Eqn. 9). This parameterization is used to adjust the α -‐parameter for each time step. 338 The derivative of the melt fraction Φ in Equation 12 is calculated from Equation 6. 339 340 The total concentration of Al C tot is governed by the loss of Al due to melt 341 migration and the loss due to decay in both the liquid and solid phases: 342 343 € dC tot dt = − C l τ mig − C s + C l τ Al , (13) 344 345 here all quantities are defined in Table 1. C l will depend on the chemical 346 partitioning of Al into the partial melt as a function of melt fraction. It is assumed 347 that this partitioning is an equilibrium process, because the time steps in these 348 calculations (100s of years) are much longer than the multi-‐hour time scales for 349 which Al partition coefficients are measured in the laboratory (Agee et al. 1990; 350 Miyamoto et al. 1994; Pack et al. 2003). Chemical partitioning for equilibrium 351 melting is described by (Best 2003): 352 353 € C l = C tot D + φ (1 − D ) , (14) 354 355 where D is the partition coefficient for Al, defined as the ratio of the weight percent 356 of Al in the solid to the weight percent in the liquid. Irrespective of silicate 357 composition, temperature or pressure D is always much less than 1 (typical values 358 range from 0.002 -‐ 0.02; Agee et al. 1990; Miyamoto et al. 1994; Pack et al. 2003). 359 We adopt a conservative value of 0.02 to minimize partitioning into the melt and 360 thus the thermal effects of losing Al to melt migration. 361 362 Equation 13 can be rewritten using Equations 9, 11 and 14: 363 364 € dC tot dt = − φ (1 − φ ) α ( D + φ − φ D ) C tot − C tot τ Al . (15) 365 366 Temperature is controlled by Al heating and the latent heat of melting ( L ) for 367 silicates: 368 369 € dTdt = Q Al ( t ) c − Lc d Φ dt , (16) 370 371 where Q Al (t) is the energy per unit mass per unit time released by the decay of Al 372 and all other parameters are defined in Table 1. Q Al (t) can be written as: 373 374 € Q Al ( t ) = E Al τ Al C tot . (17) 375 376 Employing the chain rule and Equation 17, we can rewrite Equation 16 as: 377 378 € dTdt = E Al τ Al c + L τ Al ∂Φ∂ T ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − C tot . (18) 379 380 The three coupled equations for T , φ and C tot (Eqns. 12, 15 and 18) are solved using 381 the direct finite difference method (Ozisik 1994). Initial temperature T is set to 180 382 K (reasonable for a planetesimal in thermal equilibrium with the solar nebula), 383 initial melt fraction φ is set to 0%, and the initial Al concentration C is 384 etermined by the amount of decay before a prescribed time of instantaneous 385 accretion: 386 387 € C = f Al Al Al ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ e − t acc / τ Al . (19) 388 389 Here f Al is the number of Al atoms per kg of unaltered planetesimal assuming an 390 initial mass fraction of Al equal to 0.0113 (Ghosh et al. 1998), t acc is defined relative 391 to the formation of CAIs and the other quantities are defined in Table 1. 392 µ = 1 Pa·s, t acc = 1 Myr 413 and standard values for all other quantities (Table 1). The simulations without melt 414 migration (Fig. 2, grey curves) are identical to the temperature evolution at the 415 center of the planetesimal in Figure 1 for t < 6.0 Myr, i.e. before conduction removes 416 appreciable heat. This case results in a melt fraction of 100% and a peak 417 temperature of 2130 K at 2.5 Myr. In the scenario with melt migration (Fig. 2, black 418 curves) the concentration of Al initially decays with the half-‐life of the isotope; 419 however, once partial melting begins ( T =1373 K) Al is quickly lost from the system 420 and subsequent heating is halted within a few times 10 years. The maximum melt 421 fraction is 27%, which results (based on our parameterization) in the growth of 422 grains from 100 µ m to 2.7 mm. This is larger by a factor of two than typical grain 423 sizes for achondrite meteorites derived from ~10-‐20% melt residues (Mittlefehdt et 424 al. 1998). We discuss possible reasons for this discrepancy in Section 4.3. The peak 425 temperature in the simulation with melt migration is significantly lower (1610 K) 426 than for the non-‐migration case. 427 428 e compute peak temperatures across a grid of viscosities and times of accretion, 429 the two key parameters in this model of melt migration (Figure 3). If melt does not 430 migrate, then the peak temperature depends only on the amount of energy released 431 by the decay of Al, which would cause >50% melting for all times of accretion <1.7 432 Myr. Melt migration and removal of Al do not prevent the formation of >10% melt 433 (Figure 3). However, melt migration requires that µ > 1 Pa·s and t acc < 1.5 Myr to 434 achieve >50% melt (i.e. a magma ocean). Low viscosities (<1 Pa·s) can prevent a 435 body from reaching melt fractions >50% (Figure 3). Such viscosities have been 436 predicted (e.g. µ =0.067 Pa·s by Folco et al. 2004) and measured in the laboratory 437 (e.g. µ <<3 Pa·s by Knoche et al. 1996) for chondritic melts. 438 439 Al-‐enriched crust. 442 If partial melting and crust formation occur early, i.e. within one or two Al half-‐443 lives, this concentration could re-‐melt the crust. Analogous concentration of long-‐444 lived radionuclides following differentiation has been considered as a possible 445 contributor of pre-‐basin and/or basin lunar mare volcanism (Manga et al. 1991; 446 Arkani-‐Hamed et al. 2001) and relatively recent volcanism on Mars (Schumacher et 447 al. 2007). Assuming a planetesimal average melt fraction F , and completely efficient 448 crust formation, the crust thickness d will be RF/3 . To conservatively estimate the 449 condition for re-‐melting, we assume that the equilibrium surface temperature of the 450 planetesimal is much less than the silicate melting point, and that the eruption 451 happened slowly enough for the crust to initially and everywhere reach that 452 equilibrium temperature. In the plane-‐parallel approximation, the thickness of the 453 crust’s thermal boundary layer at any time is (Turcotte et al. 2002): 454 455 € δ ~ 2( κ t )
1/ 2 , (20) 456 457 and is independent of surface temperature or heating rate. Penetration of the cold 458 thermal boundary layer to the center of the crust (i.e. δ=d/2 ) will occur after an 459 elapsed time: 460 461 € Δ t ≈ κ RF ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (21) 462 463 The temperature elevation in the center of the crust will reach a maximum at about 464 this time and is approximately 465 466 € Δ T peak ≈ Q c p e − t '/ τ Al dt ' = t C t C + Δ t ∫ Q τ Al c p − exp − κτ Al RF ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ exp − t C τ Al ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , (22) 467 468 where t C is the formation time of the crust relative to CAIs. For standard values of 469 the parameters, re-‐melting of a crust that formed soon (<< 1 Myr) after CAIs would 470 ccur on planetesimals where RF exceeded 20 km (e.g. 200 km radius for a 10% 471 average melt fraction). No re-‐melting would occur on any planetesimal (i.e. the 472 required RF diverges) with a crust that formed later than 2 Myr after time zero. Re-‐473 melting of basalt would produce relatively Si-‐rich, Na-‐rich and Mg-‐poor rocks 474 resembling andesitic basalts or trjondhhemites (Helz 1976; Martin 1986; Petford et 475 al. 1996). 476 Al Fe was sufficient to reach this temperature following silicate melting and 483 the loss of Al via melt migration (Sec. 3.3). We use the conduction model (Sec. 2.1) 484 to compute the thermal evolution of planetesimals 100 km in diameter, a typical size 485 for the parent bodies of iron meteorites (Haack et al. 2004). We assume that once 486 temperatures reach the silicate solidus, all of the Al is instantaneously removed by 487 migration, i.e. fast (~10 years, Fig. 2) relative to the mean life of Fe. Pyroclastic 488 eruptions at the surface may have facilitated the complete removal of basaltic melt 489 from a planetesimal (Wilson et al. 1991). The initial abundance and decay 490 properties of Fe are given in Table 1. 491 492 Figure 4 shows the thermal evolution of four different planetesimals with formation 493 times ranging from 0 to 2 Myr. In all cases, the silicate solidus is reached within a 494 few times 10 years after the start of each simulation. The melt fraction does not 495 reach 50% for formation times greater than ~2 Myr, consistent with Hf-‐W ages for 496 the formation of magmatic iron meteorites (Goldstein et al. 2009). Core formation in 497 the presence of melt fractions >50% could take place in as little as a few years 498 (Stevenson 1990; Taylor 1992), thus the persistence of high temperatures over Myr 499 timescales (Fig. 4) suggests that differentiation was likely for 100 km bodies when 500 t acc < 2 Myr. 501 R / κ (Turcotte et al. 2002); however, this is a poor approximation for internally 509 heated spheres, which cool several times faster (Osuga 2000). Furthermore, the 510 correlation between effective heat capacity and melt fraction (Sec. 2.1) means that 511 R / κ is not constant for a body that reaches melting temperatures. Thus, 512 characteristic cooling times for planetesimals reflect a mix of conductive heat loss, 513 the decaying Al and Fe heat sources, variable heat capacity, and the influence of 514 the heat of fusion on the thermal budget. 515 516 E-‐folding times (defined as τ cool , the time it takes the peak temperature at the center 517 of a planetesimal to drop by a factor of 1/e) are numerically computed with the 518 conduction model (Sec. 2.1) for bodies with radii between 1 and 250 km. Internal 519 heat production is maximized by setting t acc = 0 Myr. These calculations show that 520 characteristic cooling times for planetesimals with radii >10 km are a factor of ~3 521 less than R / κ when κ is fixed at 7.9 × -‐7 m /s (i.e. c = 800 J/kg/K, ρ = 3300 kg/m k = 2.1 W/m/K; Eqn. 2). Smaller bodies ( R <5 km) require longer than R / κ to 523 cool due to the internal heat sources. However, the peak temperatures of these small 524 bodies do not become large: a planetesimal with a radius of 1 km and t acc = 0 Myr 525 will reach a peak temperature of 245 K, only 65 K above the ambient and starting 526 temperature, but then requires 1.1 Myr to cool by a factor of 1/e. Smaller melt 527 fractions from later times of accretion reduce the effective heat capacity and thus 528 decrease τ cool . 529 530 The minimum size for differentiation will occur when the heat flux is maximum, i.e. 531 t acc = 0 Myr. The approximate conductive cooling time for a planetesimal with t acc = 0 532 Myr is given by a least-‐squares fit to our numerically computed e-‐folding times: 533 534 € τ cool ( t acc = ≈ Myr R km ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (23) 535 536 Planetesimals with τ cool longer than the most important heating time scale τ Al will 537 sustain melting temperatures and differentiate. Equation 23 is equal to τ Al for 538 bodies 18 km in size. 539 540 δ (Eqn. 20) will appear and thicken. The cooler, 550 negatively-‐buoyant boundary layer may eventually become Rayleigh-‐Taylor 551 unstable, and when the viscosity contrast across the boundary layer is large (i.e. due 552 to temperature dependence), the instability manifests itself with short-‐wavelength 553 "drips" that grow from the lower surface, break off, and descend into the warmer, 554 less dense interior (Jaupart et al. 1985). This will drive a passive upwards counter-‐555 flow and maintain a subadiabatic temperature gradient below the boundary layer 556 (Moore 2008). Due to erosion at its base, the boundary layer ceases to grow and the 557 heat flow will be set by conduction through its equilibrium thickness. 558 559 n the plane-‐parallel and thin boundary layer ( δ < R ) approximations, the 560 temperature profile with depth z in a body is expressed as an integral of Green’s 561 function solutions to the thermal diffusion equation: 562 563 € T = T s + Qtc d τ⋅ erf ζ −τ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∫ , (24) 564 565 where ζ = z /( κ t ) , τ is dimensionless time, and all other parameters are given in 566 Table 1. The criterion for the onset of the Rayleigh-‐Taylor instability is usually 567 described as a local Rayleigh number Ra exceeding a critical value Ra c , where 568 569 € Ra = αρ g Δ T δ κ µ B , (25) 570 571 with Δ T the temperature drop across the boundary layer. In the case of a material 572 with a strongly temperature-‐dependent viscosity, only the lower, warmer, and less 573 viscous part of the boundary layer will participate in the instability. This complexity 574 can be accommodated by calculating an "available buoyancy" (Conrad et al. 1999) 575 but this correction is small (Korenaga et al. 2003) compared to large uncertainties 576 in the viscosity itself and we ignore this complication and instead compute a 577 maximum Ra that favors the onset of solid-‐state convection. Substituting Equation 578 20, Δ T = Qt/c , and surface gravity for a homogeneous body into Equation 25, we 579 obtain a critical time τ c for the formation of convective instabilities: 580 581 € τ c = Ra c c µ B πρ α GRQ κ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ . (26) 582 583 Ra c lies between 1000 and 2000 (Korenaga et al. 2003). The parameter here with 584 the largest uncertainty is µ B , the viscosity at the base of the lithosphere, which will 585 depend on temperature, and fall markedly as the planetesimal heats up and begins 586 to melt. The viscosity of Earth's deep mantle is ~10 Pa·s but beneath mid-‐ocean 587 ridges where partial melting is occurring it may be as low as a few times 10 to 10 µ B = 10 t acc = 0 Myr and R = 1000 km) τ c = 0.27 Myr. From the thermal conduction 590 model (Sec. 2.1), a 1000 km body with t acc = 0 Myr reaches silicate-‐melting 591 temperatures in only 0.13 Myr, well before the onset of subsolidus convection. In 592 fact, for all of the scenarios considered in Section 3 (i.e. t acc < 2.0 and R ≤ 50 km), τ c is 593 longer than the time to achieve silicate melting. Only large (>100 km), late accreting 594 (>2 Myr) bodies will be affected by subsolidus convection, thus justifying our use of 595 a conduction-‐only model. 596 597 Vigorous convection would occur at temperatures well above the solidus, i.e. in a 598 magma ocean (Taylor et al. 1993). However, our focus on constraining the 599 conditions leading up to differentiation does not require treatment of magma ocean 600 rocesses. It is sufficient to assume that differentiation will occur (either by 601 fractional or equilibrium crystallization) in the presence of high degrees of melting. 602 603 ρ and a 607 corresponding jump in thermal conductivity from initial values as low as 0.001 608 W/m/K (Hevey et al. 2006; Sahijpal et al. 2007). Computing peak central 609 temperatures with a modified conduction model demonstrates that sintering does 610 not have a significant effect on our results. The peak temperature reached at the 611 center of a 50 km body with t acc = 2.2 Myr and fixed k = 2.1 W/m/K is 1480 K. 612 However, a body originally 63 km in radius with k = 0.001 W/m/K that compacts 613 down to 50 km (i.e. the loss of 50% pore space) and k = 2.1 W/m/K at 700 K will 614 reach a nearly identical peak temperature of 1484 K. This treatment of sintering is 615 achieved by allowing κ (Eqn. 2) to vary with temperature. The primary effect of 616 sintering is that the cool outer layers remain uncompacted and thus decrease the 617 rate of interior cooling. It is the time of accretion (i.e. the initial Al and Fe 618 abundances) that largely controls peak temperature. 619 620 Although this treatment of sintering shows little effect on bodies that reach melting 621 temperatures (>>700 K), there are large uncertainties associated with the assumed 622 initial porosities (and the corresponding thermal conductivities). A value of k = 623 0.001 W/m/K is based on laboratory measurements of the lunar regolith in a 624 vacuum (Fountain et al. 1970). However, processes such as volatile release and 625 collisional packing during accretion would act to decrease vacuum space and thus 626 increase conductivity. Other differences, such as size, collisional history and 627 composition, suggest that the lunar regolith may not be an ideal analog for an 628 unsintered planetesimal. 629 630 Al), but required melt fractions >40% 651 (Stevenson 1990; Taylor 1992). Laboratory experiments produce conflicting results 652 regarding the efficiency of metal-‐silicate segregation at lower melt fractions (see 653 McSween et al. 1978; Rushmer et al. 2000; Yoshino et al. 2003; 2004; McCoy et al. 654 2006 and references within for details). In short, a complicated interplay of 655 temperature, pressure, alloy melt composition, silicate melt fraction, and rheology of 656 the matrix dictated the way in which cores formed, e.g. percolation of liquid metal 657 through interconnected pore spaces between silicate grains versus gravitational 658 settling of metal globules through a viscous silicate matrix (Stevenson 1990). 659 660 Core formation would have concentrated Fe and Al into smaller volumes (the 661 core and mantle respectively). Chondritic materials typically have ~20 weight % Fe 662 (Lodders et al. 1998), so that the volume of an Fe-‐Ni core would occupy less than 663 1/8 the total volume of a differentiated planetesimal. Thus, core formation would 664 result in an enhanced number density per kg of Al in the mantle of no more than 665 15% over chondritic concentrations. This enhancement is small relative to other 666 uncertainties in our model (e.g. grain size distribution, composition, accretion 667 scenario) and is probably insignificant. The concentration of Fe into a core can 668 increase cooling times due to the insulating effects of the surrounding silicate 669 matrix; however, the overall thermal budget of a planetesimal will be unchanged. 670 The gravitational potential released by core formation is insignificant for 671 planetesimal-‐size bodies (Moskovitz 2009). 672 673 t acc = 1.0 Myr results 681 in a peak temperature of 2850 K, an increase of ~240 K over the standard scenario 682 presented in Figure 1. This illustrates that the assumed heat capacities produce 683 temperatures that are underestimated by no more than 10%. 684 685 years to allow sufficient time for cooling (Eqn. 23). Recent dynamical 696 studies that treat the turbulent concentration of small particles in proto-‐planetary 697 disks (Johansen et al. 2007; 2009; Cuzzi et al. 2010) show that planetesimals can 698 grow to sizes of 100 km or larger on orbital timescales (typically < 100 years). Such 699 rapid accretion is nearly instantaneous relative to other relevant timescales (e.g. 700 τ mig , τ cool , τ Al ) and thus does not affect thermal evolution scenarios. 701 702 If we had considered some parameterization of prolonged accretion, our standard 703 conduction model (Fig. 1) would likely reach peak temperatures lower by a few 704 hundred K (Merk et al. 2002; Ghosh et al. 2003). Slower accretion would require 705 even earlier formation times for a body to melt and differentiate, particularly in the 706 absence of Al (Fig. 4). The consequences of melt migration (Sec. 3.2) are 707 independent of accretion scenario as long as melting temperatures are reached. It is 708 possible that melt migration would occur before a planetesimal had fully formed, 709 particularly for accretion times longer than τ mig . For a given initial planetesimal melt 710 fraction, the re-‐melting of a crust only depends on its time of formation relative to 711 CAIs (Sec. 3.3) and thus is independent of the details of accretion. The minimum size 712 for differentiation (Sec. 3.5) is unaffected by accretion scenario, because the limiting 713 case occurs with instantaneous accretion at t acc = 0. 714 715 716 Al in partial melts 723 (Sec. 2.3). This migration model was used to show that low viscosity ( µ < 1 Pa·s) 724 silicate melt can migrate on timescales less than the mean life of Al (Sec. 3.2). 725 Following such migration, the residual material becomes enriched in coarse-‐grained 726 silicates and Fe-‐Ni metal, depleted in elemental Al, and depleted in low temperature 727 phases like plagioclase. The lodranite and ureilite achondrites are examples of such 728 melt residues (Mittlefehdt et al. 1998). After removal of Al from the interior of a 729 planetesimal, subsequent heating from the decay of Fe alone was sufficient to 730 generate melt fractions >50% and complete differentiation as long as accretion 731 occurred within 2 Myr of CAI formation (Sec. 3.4). This is in contrast to previous 732 studies that either adopted lower Fe abundances (e.g. Ghosh et al. 1998), leading 733 to the conclusion that Fe did not have an appreciable effect on thermal evolution, 734 or did not consider heating by Fe following the removal of Al (e.g. Hevey et al. 735 2006). We showed that remelting of a basaltic crust enriched in Al would have 736 ccurred for bodies that accreted within 2 Myr of CAIs, reached melt fractions of 737 order 10%, and were ≳ R-t acc
R-t acc pair. The three black, solid curves in this figure represent 745 objects whose peak temperatures at 50% radius were equal to the Fe-‐S eutectic 746 (1200 K), the silicate solidus (1400 K) and the silicate liquidus (1900 K). These 747 curves segregate this figure into two primary domains. To the right of these curves, 748 no melting would occur. To the left, high degrees of melting would result in surface 749 volcanism and a vigorously convecting magma ocean (Taylor et al. 1993, Righter et 750 al. 1997). Our model predicts that silicate melting required accretion within 2.7 Myr 751 of CAIs (Fig. 5), consistent with radiometric dating of the magmatic iron meteorites, 752 whose parent bodies accreted and differentiated within 1-‐2 Myr of CAI formation 753 (Markowski et al. 2006). This constraint is slightly less than the 2.85 Myr for 25% 754 melting of the Vesta parent body calculated by Ghosh et al. (1998). Their higher 755 initial temperature (by more than 100 K) is the likely cause of this discrepancy. 756 757 The shaded regions in Figure 5 represent different fates of molten material. The 758 light grey region represents planetesimals too small (<20 km) for compaction-‐759 driven melt migration to occur, i.e. δ c is comparable to or larger than R . These bodies 760 would not differentiate by melt migration, but could form a magma ocean. This 761 region can extend to larger radii (perhaps up to R =50 km) depending on the specific 762 grain size distribution and rheologic properties of the planetesimal (Sec. 2.2). The 763 medium grey region represents bodies small enough (<100 km) that pyroclastically 764 erupted melt can escape (Wilson et al. 1991; Keil et al. 1993). This ejection of melt 765 would have precluded the formation of a basaltic crust and may be relevant to 766 understanding the scarcity of basaltic material amongst asteroids and meteorites in 767 the present day (Moskovitz et al. 2008; Burbine et al. 1996). The dark grey region 768 represents larger objects, e.g. Vesta, where melt does not escape and instead is 769 deposited in a crust. Rapid removal of Al-‐bearing melt from the interior of 770 planetesimals in the medium and dark grey regions will significantly alter their peak 771 temperatures (Sec. 3.2), though the presence of Fe may have helped these bodies 772 to fully differentiate (Sec. 3.4). The arrow at top indicates the upper limit to the time 773 of accretion (2 Myr) for which a planetesimal depleted in Al could still have 774 differentiated as a result of the decay of Fe. 775 776 Subsolidus convection did not occur for bodies smaller than ~100 km, thus 777 justifying the use of a conduction model. However, later times of accretion decrease 778 the critical radius at which subsolidus convective instabilities developed prior to 779 melting. The hatched region in Figure 5 represents this limit. The lower boundary of 780 this region is defined by the
R-t acc pairs for which τ c (Eqn. 26) is equal to the time 781 equired to reach melting, assuming a bulk viscosity of 10 Pa·s, a reasonable lower 782 limit to the viscosity of the Earth’s asthenosphere (Pollitz et al. 1998; James et al. 783 2008). Convective overturn on these bodies would act to reduce peak temperatures 784 and prevent melting. The bulk viscosity of material near partial melting 785 temperatures can vary greatly. If bulk viscosities were as low as ~10 Pa·s then 786 convective instabilities would affect over half of the melting region in Figure 5. 787 Though Hevey et al. (2006) and Sahijpal et al. (2007) parameterized fluid 788 convection in a magma ocean, we are unaware of any study that has considered the 789 thermal consequences of subsolidus convection on planetesimal-‐size bodies. 790 791 The dashed curve in Figure 5 indicates the
R-t acc pairs for which the melt migration 792 time scale ( τ mig ) and the conductive cooling time scale ( τ cool ) are equal, assuming a 793 grain size of 1 mm, melt viscosity of 1 Pa·s and melt fraction of 10%. This boundary 794 is sensitive to variations in these quantities and can shift across the full range of 795 radii. Above the dashed line, removal of Al-‐enriched melt is the dominant cooling 796 process. Below, the conductive loss of heat causes melt (if any is present) to freeze 797 in place before migration occurs. Irrespective of the dominant cooling mechanism, if 798 τ cool < τ Al then a planetesimal will not melt. 799 800 Our calculations for the rate of compaction-‐driven melt migration (Sec. 2.2) and the 801 efficiency of conductive cooling (Sec. 3.5) suggest that differentiation only occurred 802 for bodies larger than ~20 km in diameter, consistent with previous studies (e.g. 803 Hevey et al. 2006; Sahijpal et al. 2007). In the present day Solar System, the only 804 asteroids other than 4 Vesta that are >15 km in diameter (Delbo et al. 2006; Tedesco 805 et al. 2002) and have spectroscopically confirmed basaltic surfaces are 1459 806 Magnya and 1904 Massevitch. Although these bodies are near the differentiation 807 limit, it is not clear whether they could escape collisional disruption over the age of 808 the Solar System (Bottke et al. 2005). 809 810 In general, our results are most consistent with Sahijpal et al. (2007), though several 811 key differences do exist. Most notably is their treatment of melt migration velocities. 812 They prescribe migration rates of 30-‐300 m/yr, however these rates are not 813 physically motivated and instead are a reflection of stability requirements in their 814 numerical model. These rates are more than an order of magnitude faster than our 815 most optimistic scenario (i.e. 1 cm grains with 10’s of % partial melt) and thus 816 overestimate the rate of Al redistribution. Another difference with Sahijpal et al. 817 (2007) is that we do not include changes to volume and thermal conductivity 818 associated with sintering at 700 K (Hevey et al. 2006). Implementation of our 819 conduction model with a temperature-‐dependent κ shows that the high peak 820 temperatures (>700 K) in which we are interested are barely affected by sintering 821 (Sec. 4.3). The most significant effect of sintering for bodies of all sizes is an increase 822 of interior cooling rates due to the insulating effects of an unsintered, low 823 conductivity surface layer. 824 825 ur results raise several issues in need of further investigation. For instance, the 826 basaltic eucrite meteorites are thought to have formed from liquid expelled out of a 827 partial melt region (Consolmagno et al. 1977; Walker et al. 1978, Righter et al. 828 1997). This scenario probably required mm-‐size grains and melt viscosities ~1 Pa ⋅ s 829 (Sec. 3.2). Experimental measurement of the viscosity of basalts with compositions 830 similar to eucrites can test the plausibility of this model. Calculations of the chemical 831 composition of chondritic melts as a function of temperature (e.g. using the MELTS 832 software package, Ghiorso et al. 1995) could also constrain their viscosity evolution 833 and thus their resistance to migration. Lastly, our calculations suggest that bodies as 834 small as ~12 km in diameter could partially melt (Fig. 5). We do not expect melts to 835 segregate on these bodies due to their small size (Sec. 2.2). However, fractional 836 crystallization in a magma ocean could produce a differentiated structure. 837 Additional modeling of magma ocean formation and evolution on bodies with low 838 self-‐gravity would address whether small basaltic asteroids in the present day Solar 839 System, e.g. 1459 Magnya and 1904 Massevitch, are intact differentiated 840 planetesimal or relic fragments from heavily eroded parent bodies. 841 842 Acknowledgments
We are grateful to G. J. Taylor, T. McCoy and J. Sinton for their 843 insightful opinions regarding this manuscript and appreciate the opportunity to 844 have benefitted from their vast expertise in this subject matter. We thank H. 845 McSween and L. Wilson for their helpful reviews. N.M. would like to acknowledge 846 support from the Carnegie Institution of Washington, the NASA Astrobiology 847 Institute and NASA GSRP grant NNX06AI30H. 848 849
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Table 1
Symbols and definitions. 1197
Symbol Name, definition and reference Value [Units] T Temperature [K] T s Surface temperature [K] t Time [s] t acc Time of accretion [s] φ Net melt fraction by weight -‐ Φ Instantaneous amount of melt -‐ R Radius of body [m] r Radial variable [m] δ Thermal boundary layer thickness, = 2( κ t) [m] a Grain size [m] µ Liquid viscosity [Pa·s] µ B Effective bulk viscosity [Pa·s] Δρ Density contrast between solid and liquid 300 kg/m € ρ Mean density of olivine 3300 kg/m Q(t)
SLR decay energy generation rate (Eqn. 5) [J/kg/s] f Al Chondritic abundance of Al (Ghosh et al. 1998) 2.62 × kg -‐1 E Al Al decay energy per atom (Lide 2002) 6.4154 × -‐13 J [ Al/ Al] Al initial abundance (Jacobsen et al. 2008) 5 × -‐5 τ Al half-‐life (Goswami et al. 2005) 0.74 Myr τ Al Al mean life, = τ / ln(2) 1.07 Myr f Fe Chondritic abundance of Fe (Ghosh et al. 1998) 2.41 × kg -‐1 E Fe Fe decay energy per atom (Ghosh et al. 1998) 4.87 × -‐13 J [ Fe/ Fe] Fe initial abundance (Tachibana et al. 2006; Dauphas et al. 2008; Rugel et al. 2009; Mishra et al. 2010) 6 × -‐7 τ Fe Fe mean life (Rugel et al. 2009) 3.49 Myr C l Concentration of Al in the liquid phase [1/kg] C s Concentration of Al in the solid phase [1/kg] C tot Total concentration of Al, = C l + C s [1/kg] D Al partition coefficient -‐ α Thermal expansion coefficient [1/K] c Specific heat at constant pressure [J/kg/K] k Thermal conductivity [W/m/K] κ Thermal diffusivity, (Eqn. 2) [m /s] L Latent heat of fusion for silicates (Ghosh et al. 1998) 400 kJ/kg Ra Raleigh number (Eqn. 25) -‐ τ c Convective instability timescale (Eqn. 26) [s] τ mig Melt migration time scale (Eqn. 9) [s] τ cool Conductive cooling timescale (Sec. 3.5) [s] g Gravitational acceleration, = (4πGr € ρ )/3 [m/s ] G Gravitational constant 6.673 × -‐11 m /kg/s
10 20 30 40Time a f ter C A I fo rmati o n ( Myr ) R a d i u s ( k m )
180 K2700 K1440 K
Moskovitz Figure 1 200 1201 T e m p e r a t u r e ( K ) f ter C A I fo rmati o n ( Myr ) (cid:113) C A l / C f ter C A I fo rmati o n ( Myr ) V i s c o s i t y ( P a s ) % % % % % % % % % % > 50 % Melting
Moskovitz Figure 2 Moskovitz Figure 3 R a d i u s ( k m ) K ( a ) K ( b ) R a d i u s ( k m ) (c) ( d ) Time a f ter C A I fo rmati o n ( Myr ) of Acc reti o n ( Myr ) R a d i u s ( k m ) N o Melting N o pyr oc lasti c e j e c ti o nPyr oc lasti c e j e c ti o nN o sili c ate melt migrati o n ( (cid:98) c (cid:190) R ) F e - S E u t e c t i c S o l i d u s L i q u i d u s Sub - s o lidus co nve c ti o n (cid:111) mig < (cid:111) coo l (cid:111) mig > (cid:111) coo l a = 1 mm ! = 1 Pa (cid:117) s (cid:113) = 10 % Fe0 1 2 3 10100
Moskovitz Figure 4 Moskovitz Figure 5 igure Captions R = 50 km planetesimal that accreted 1.0 Myr 1208 after CAI formation. Time is reported relative to CAI formation. The body reaches its 1209 maximum temperature 10 Myr after accretion and then cools on a characteristic 1210 time scale ( τ cool ) of ~21 Myr. 1211 1212 Fig. 2. Melt migration simulation for t acc = 1 Myr. The top panel shows the 1213 temperature evolution, the bottom shows the melt fraction ( φ , solid) and the 1214 normalized Al concentration ( C Al /C , dashed). The grey curves assume no melt 1215 migration and are identical to the temperature evolution at the center of the body in 1216 Figure 1 for t < 2.5 Myr. The black curves include melt migration. At the onset of 1217 melting (T=1373 K) the two scenarios diverge. Without migration the melt fraction 1218 reaches 100% and the peak temperature is 2130 K at 2.5 Myr. With migration the 1219 melt fraction is never larger than 27% and the peak temperature is only 1610 K. The 1220 changes of inflection in these curves are due to the latent heat of melting and the 1221 parameterization of melt fraction (Eqn. 6). 1222 1223 Fig. 3. Melt fraction as a function of viscosity and time of accretion. The contours 1224 represent 10% intervals of partial melting (1407, 1452, 1522, 1707, and 1830 K; 1225 McKenzie et al. 1988). Migration of low viscosity melt acts to remove Al and 1226 reduce peak temperature. Without melt migration, all times of accretion <1.7 Myr 1227 would result in >50% melting. Combinations of viscosity and accretion time that lie 1228 above the 50% melting line would produce conditions favorable to the formation of 1229 a magma ocean. The jagged features in the melt contours are numerical artifacts. 1230 1231 Fig. 4. Temperature evolution due to Fe decay in Al-‐depleted planetesimals. The 1232 four panels correspond to different times of instantaneous accretion relative to CAI 1233 formation: 0, 1.0, 1.5, and 2.0 Myr (a-‐d respectively). In all cases, heating by the 1234 decay of Al is stopped as soon as the silicate solidus (1373 K) is reached, which 1235 typically occurs within a few times 10 years. If melt migration removed Al from 1236 the parent bodies of the magmatic iron meteorites, then they had to have accreted 1237 within 2 Myr of CAI formation in order to fully differentiate (i.e. reach 50% partial 1238 melting at 1558 K). 1239 1240 Fig. 5. Thermal evolution as a function of planetesimal radius ( R ) and time of 1241 accretion ( t acc , relative to CAI formation). The solid curves represent contours of 1242 maximum temperature at 50% radius (liquidus = 1900 K, solidus = 1400 K, Fe-‐S 1243 eutectic = 1200 K). The arrow at top indicates an upper limit to the time of accretion 1244 for Fe-‐driven differentiation to occur. The light grey region represents bodies for 1245 which compaction-‐driven melt migration will not occur because the size of the 1246 compaction region ( δ c ) is comparable to R . Bodies in the medium grey region can 1247 lose melt via pyroclastic eruption, those in the dark grey region have sufficient self-‐1248 gravity to retain melt erupted at the surface (Wilson et al. 1991; Keil et al. 1993). 1249 The dashed line represents bodies for which the melt migration time scale ( τ mig ) is 1250 qual to the conductive cooling time scale ( τ cool ), assuming melt parameters a =1 1251 mm, µ =1 Pa·s and φ =10%. The hatched region in the upper right represents bodies 1252 that experience solid-‐state convection if their subsolidus viscosity is > 1017