Wilson Et Al. (2008) Thermal Evolution and Physics of Melt Extraction on the Ureilite Parent Body

8/13/2019 Wilson Et Al. (2008) Thermal Evolution and Physics of Melt Extraction on the Ureilite Parent Body

1/23


2/23

1. INTRODUCTION

Ureilites, which form the second largest group of achon-drites (240 samples), are coarse-grained, highly-equilibratedultramafic rocks that are thought by most workers to repre-sent the residual mantle of a partially melted, carbon-richasteroid (see reviews by Goodrich, 1992; Mittlefehldt

et al., 1998; Goodrich et al., 2004). One of their distinctivecharacteristics is their primitive(i.e. chondrite-like) oxy-gen isotopic signature, which suggests that they preserve aunique stage of early planetary differentiation. This paperpresents calculations of the heating, melting, and physicsof melt extraction on the ureilite parent body (UPB) duringthis early period of its history, based on the petrologic andgeochemical model developed in Goodrich et al. (2007).Here, we briefly summarize the background and most sali-ent aspects of that model.

The majority of ureilites consist of olivine + low-Capyroxene (pigeonite and/or orthopyroxene), with intersti-tial carbon (graphite secondary, shock-produced dia-

mond) and metal (sulfide and/or phosphide) as the onlycommon accessory phases. Their olivine compositions (con-stant within each sample) show a large range of Fo (molarMg/[Mg + Fe]) values (7692) at essentially constant Mn/Mg, which indicates that they are related to one anotherprincipally by various degrees of oxidation or reductionrather than various degrees of melting (Mittlefehldt, 1986;Goodrich et al., 1987; Goodrich and Delaney, 2000). Thisredox relationship can be explained by smelting (pressure-dependent carbon redox reactions: in simplest form,FeO + C? Fe + CO) over a range of pressures in the urei-lite parent body (Berkley and Jones, 1982; Goodrich et al.,1987, 2007; Warren and Kallemeyn, 1992; Walker and

Grove, 1993; Sinha et al., 1997; Singletary and Grove,2003), an interpretation that is supported by a correlationof Fo with pyroxene/olivine ratio (Singletary and Grove,2003; Goodrich et al., 2007) and by the restriction of orth-opyroxene to the few most magnesian (FoP 86) samples(Goodrich et al., 2007). The smelting model is thus highlysuccessful at explaining many of the petrologic and chemi-

cal features of ureilites that would be difficult to explain in anormal igneous fractionation model (particularly the largerange of Fo values). It may be less successful at explainingtheir metal and siderophile element abundances. Smeltingpredicts a correlation of Fo, not only with pyroxene/olivineratio, but also with metal content or (if the metal is re-moved) siderophile element abundances. In fact, ureiliteshave uniformly low metal contents (at most a few percent)and show no correlation of siderophile elements with Fo.Although some workers (Mittlefehldt et al., 2005; Warrenand Huber, 2006; Warren et al., 2006) have argued that thisobservation is fatal for the smelting model, we have sug-gested one hypothesis that may explain it (Goodrich

et al., 2007) and believe there are other possibilities as well.The present work is thus based on a smelting model, underthe assumption that future work will be able to reconcile itwith the metal and siderophile element abundances inureilites.

In this model, the olivine + low-Ca pyroxene ureilitesrepresent the mantle of a ureilite parent body (UPB) thatwas stratified (or had a radial gradient) in mg (molar Mg/[Mg + Fe]; = Fo in referring to olivine), pyroxene/olivineratio, and pyroxene type (Fig. 1), due to the pressure depen-dence of carbon redox reactions. Ureilite smelting (i.e. finalequilibration) pressures have been estimated using a varietyof thermodynamic treatments and experimental methods

Fig. 1. Schematic cross-section of the ureilite parent body (UPB), showing relationship between depth (km), pressure (MPa), Fo (molarMg/[Mg + Fe]) of olivine, and pyroxene type (opx = orthopyroxene) in the smelting model developed in Goodrich et al. (2004, 2007)and thispaper. Here, we focus on the formation of shallow crustal basaltic intrusions (a sill ).

Thermal evolution and melt extraction on the UPB 6155


3/23

(Berkley and Jones, 1982; Goodrich et al., 1987; Warrenand Kallemeyn, 1992; Walker and Grove, 1993; Sinhaet al., 1997; Singletary and Grove, 2003), with the most reli-able results being 910 MPa for the most ferroan ureiliteand 23 MPa for the most magnesian. Here, we assumethat the total range of pressures sampled by the meteoritesis310 MPa, as determined inGoodrich et al. (2007).

What is missing from this picture, from the point of viewof planetary differentiation, is the melts. A few ureilites thatare augite-bearing appear to be ultramafic cumulates ratherthan residues (Goodrich et al., 2004, 2006), but the fact thatthey have Fo values in the same range as the olivine + low-Ca pyroxene ureilites suggests that they formed over thesame range of depths. There are no basaltic ureilites, andthus the only direct petrologic information we have aboutthe melts that were complementary to these mantle rocksis what can be gleaned from the polymict ureilites. These17 (not accounting for many likely pairings) samples areregolith breccias, and contain a few percent feldspathicmaterial (shown from O-isotopes to be indigenous to the

UPB) in the form of small lithic and mineral clasts(Ikeda et al., 2000, 2003; Ikeda and Prinz, 2001; Kita et al.,2004, 2006; Cohen et al., 2004; Goodrich et al., 2004;

Downes et al., in press). The recognition that these clastsrepresent a diversity of melt lithologies, none of which isstrictly basaltic, pointed toward the possibility that meltextraction on the UPB was a fractional process (Cohenet al., 2004; Kita et al., 2004) and was one of the main moti-vations behind our work. In addition, these clasts have per-mitted the first precise dating of ureilites using short-livedradionuclide systems, yielding ages of5 Ma after forma-tion of CAI (Goodrich et al., 2002a; Kita et al.,2003, 2007).

Goodrich et al. (2007)recognized that if melt extractionon the UPB was fractional, then it was important to exam-

ine the progress of smelting during the course of melting (asopposed to previous treatments of smelting in which thesource region for each individual ureilite was assumed tohave been pre-smelted). Thus, for a model bulk startingcomposition (given inTable 2 of that paper) we calculated(1) degree of melting, (2) the evolution ofmg, (3) production

of CO + CO2gas and (4) the evolution of mineralogy in theresidue as a function of temperature for three different pres-sures (3, 6.5 and 10 MPa) on the UPB. The starting compo-sition was determined from petrologic constraints followingthe approach ofGoodrich (1999), and is similar to oxidizedCV chondrites (all Fe assumed to be FeO) with the excep-tion of having superchondritic Ca/Al ratio (2.5CI).The latter (which derives from the requirement to producepigeonite, rather than orthopyroxene, as the dominantpyroxene) is assumed to be a post-accretionary feature, pos-sibly established by mobilization of Ca during low-T aque-ous alteration (Goodrich et al., 2002b, 2007).

Results of these calculations, which provide essential in-put for the present work, are summarized inFigs. 2 and 3.Two features deserve special note. First, these calculationsshow that although all ureilite source regions reach the sil-icate solidus at1050(10)C, and experience30% ( afew percent) total melting, their melting sequences varygreatly with depth because the temperature at which smelt-ing begins (thus low-Ca pyroxene appears and CO/CO2starts to be produced) is strongly dependent on pressure.In the shallowest source regions, smelting begins nearlysimultaneously with melting, whereas in the deepest it doesnot begin until 22% melting has already occurred (nota-bly, after plagioclase is exhausted), which has significantimplications in the case of fractional, as opposed to batch,melt extraction. Second, these calculations confirm the re-

sult ofSingletary and Grove (2003)that the UPB had an in-verse temperature gradient; i.e. peak temperatures decreasefrom the shallowest to the deepest source regions.

2. THE SIZE OF THE UPB

As a prelude to calculations of the melting history andthe extraction of melt from the UPB we first establish somebasic physical constraints on the asteroids properties. Dur-ing the hydration and dehydration phases of its develop-ment, the asteroid must have lost a mass fractioncorresponding to its initial ice mass fraction, say k. It is easyto show, by considering the partial volumes of the compo-

Table 2Excess pressure, DPl, in melt as a function of mass %, mm, andequivalent volume %,q, melted together with corresponding valuesof the critical length Lc(see text) and the vein length, Lf, requiredfor the onset of vein growth.

mm(%) q(%) DPl(MPa) Lc(lm) Lf(lm)

0.0100 0.0114 1.7 1596 175,318

0.0300 0.0341 5.1 1363 21,6630.0500 0.0569 8.4 1189 83130.0700 0.0796 11.8 1055 44130.1000 0.1138 16.8 902 22320.1200 0.1365 20.2 823 15650.1503 0.1710 25.3 726 10000.1750 0.1991 29.4 663 7330.2000 0.2275 33.6 609 5550.2300 0.2616 38.6 555 4110.2654 0.3019 44.6 502 3000.3000 0.3412 50.4 460 2270.4179 0.4752 70.1 357 1000.5000 0.5686 83.8 309 61

Table 1Variation with depth, D, below the surface of lithostatic pressure,Ps, and acceleration due to gravity, g, in the UPB assuming a radiusofR = 100 km and a mean density ofr = 3300 kg m3.

D(km) Ps (MPa) g(m s2)

0.00 0.00 0.0921.00 0.30 0.0913.00 0.90 0.0895.00 1.48 0.0887.00 2.06 0.0868.00 2.34 0.084

10.40 3.00 0.08317.50 4.86 0.07624.32 6.50 0.07032.35 8.25 0.06241.46 10.00 0.05457.77 12.50 0.03975.00 14.26 0.023

100.00 15.21 0.000

6156 L. Wilson et al. / Geochimica et Cosmochimica Acta 72 (2008) 61546176


4/23

nents, that as long as there is no significant change in thedensity of the bulk silicate component of the asteroid dur-ing the hydration and dehydration reactions, the ratio of

the final radius,rf, of the asteroid after water loss to the ini-tial radius, r i, of the asteroid before ice melting is given by

rf=ri 1kqif g= kqm 1kqif g 1=3 1

where qi is the density of ice, 917 kg m3 and qm is the

density of the silicate component, 3500 kg m3 appropri-ate to a CV-like composition. As an illustration, if the icemass fraction k takes the values 0, 0.1, 0.2 and 0.3, (rf/ri)has values 1.00, 0.89, 0.80 and 0.72. Thus, depending onthe initial ice content, the asteroid may have experienceda significant decrease in size, but without an independentestimate of its initial ice content, which we do not have,we cannot specify how large this may have been. We can,however, place quite stringent constraints on the final sizeof the asteroid after these modifications have occurred, asfollows.

We have meteorite samples that show evidence ofhaving undergone 30% ( a few percent) melting underpressure conditions ranging from 3 to 10 MPa (1MPa) (Goodrich et al., 2007, and references therein). Thusat the onset of silicate melting, the asteroid must have beenlarge enough to allow a pressure of 10 MPa to be present inits mantle and small enough that rocks at the depth corre-sponding to 3 MPa pressure were heated sufficiently to be

melted. We show inTable 1the variation of lithostatic pres-sure, Ps, and acceleration due to gravity, g, with depth, D,in a body with a radius ofR= 100 km and a mean density,r, of 3300 kg m3 (a typical density for peridotite). Therelationships for a body with uniform density are

Ps 2=3pGr2DRD2 2

and

g 4=3pGrRD 3

where G is the gravitation constant,6.671011 m3 s2 kg1. The central pressure is propor-tional to the density and to the square of the radius.

Fig. 2. Evolution of mineralogy and Fo (labelled on olivine curve)duringprogressive meltingand smelting on theUPB atthreedifferentpressures (sourceregion depths). All source regionsreach thesilicatesolidus at 1050 10 C and experience 30% total melting.However, their melting sequences vary significantly with depthbecause the temperature at which smelting begins (marked by theappearance of low-Ca pyroxene) is strongly dependent on pressure.Data fromGoodrich et al.(2007). Oliv = olivine; plag = plagioclase;aug = augite; opx = orthopyroxene; pig = pigeonite.

Fig. 3. Weight percent CO + CO2in gas plus silicate melt mixtureas a function of degree of melting on the UPB at three differentpressures (depths). The CO + CO2gas is a product of smelting andthus is not present until the onset of smelting (see Fig. 2). DatafromGoodrich et al. (2007).



5/23

The pressure range 310 MPa corresponds to depths of10to50 km in a 100 km radius body and the central pressureis15 MPa; the acceleration due to gravity at the surface is0.092 m s2. We will now show that the radius of the UPBcould not have been much smaller, and is unlikely to havebeen much larger, than this 100 km value.

This involves considering the two major differences be-

tween the onset of silicate melting in the interior of anundifferentiated asteroid and the melting process inside abody like the Earth at its present stage of development.The first is the timescale of the process: the Earth is heatedby long-lived radioisotopes of uranium, potassium and tho-rium with half-lives of4.5, 1.2 and 14.1 Ga, respectively,whereas the asteroid is dominantly heated by short-lived26Al with a half-life of 0.72 Ma. More subtle is the fact thatmelting within the Earth is triggered by pressure release inthe rising parts of convective systems, so that increasing de-grees of partial melting take place under decreasing pres-sure and adiabatically decreasing temperature conditions,whereas in the asteroid, provided that bulk convection of

the interior does not take place (an assertion that we justifylater), melting is triggered purely by temperature increase.Thus as long as the 26Al heat source is distributed uni-formly, or at least randomly, throughout the asteroid,which we consider to be the most likely circumstance, theonset of melting occurs at all except the shallowest depthsat essentially the same time. This is the consequence ofthe fact that, over a time interval t, the influence of radiativeheat loss from the surface on the evolving internal temper-ature can only penetrate to a depth of(j t)1/2, wherej isthe thermal diffusivity of rock, 106 m2 s1 for all sili-cates. Witht equal to a few Ma, the interval for which heatfrom 26Al is available, this skin depth is less than10 km. The effect can readily be illustrated in more detail

by calculating the temperature history for a spatially uni-formly-distributed heat source within a spherical bodyusing an analytical formulation like that given in section95, Eq. (6), p. 207 of Carslaw and Jaeger (1947). Fig. 4shows the variation of temperature with depth below thesurface for the times at which temperatures of 1323, 1433and 1543 K have been reached in the deep interior, thesetemperatures representing approximately 0%, 15% and30% melting at the 3 MPa depth level (Fig. 2).Fig. 4is plot-ted for an asteroid radius of 100 km, but the plot for a ra-dius of 200 km would be indistinguishable at this scale: at agiven depth temperatures differ by no more than 0.5% be-tween these two asteroid sizes. Thus, essentially indepen-dent of its size, as an asteroid heats up its temperaturewill increase nearly uniformly at all depths below 8 km,corresponding to a pressure of 2.3 MPa (see Table 1).The very low temperatures in the shallower part of theasteroid are presumably the reason that no examples ofureilites with source region pressures much less than3 MPa (other than the polymict ureilite regolith breccias)are found, because these rocks would have resided in thecold outer shell. Furthermore, in a larger asteroid the3 MPa level would be shallower. Thus in a 200 km radiusasteroid the 3 MPa level would be at a depth of 5 km,and the 8 km level where the near-uniform internal tem-perature is reached would correspond to a pressure of

4.8 MPa. Repeating this calculation for a range of aster-oid sizes shows that if the 8 km depth is to correspond toa pressure of not more than 3 MPa then the asteroid cannothave a radius significantly larger than 128 km. Note thatthe 8 km skin depth estimate used in this calculationwould decrease if the thermal diffusivity of the outer layerswere decreased, and this would lead to an increase in theestimate of the maximum asteroid radius, in proportionto the square root of the ratio by which the thermal diffu-sivity was decreased. Void-space creation during regolith

formation on the surface of the asteroid could lead to sucha decrease in thermal conductivity and hence in thermal dif-fusivity, though this would be partly offset by the corre-sponding decrease in bulk density and by the presence ofice in pore spaces as H2O molecules diffused from the dee-per interior. However, the amount of regolith formation onthe timescales of a few Ma relevant here would be small(perhaps a few cm by comparison with the 10 m of rego-lith formed in the lunar highlands during the first500 Maof the Moons history) and we do not consider this effect tobe important. Finally, we can also estimate the minimumasteroid radius: we have meteorites that experienced pres-sures of 10 MPa, which is the central pressure in an asteroidof radius 82 km. Thus unless the asteroid radius was atleast this large these meteorites could not exist. We adopt100 km as the nominal UPB radius in what follows.

3. TIMESCALE FOR HEATING AND MELTING

The thermal history of the asteroid, assumed to accreteas a mixture of chondritic silicates and water ice (consistentwith the suggestion that ureilite precursor material experi-enced pre-igneous aqueous alteration; Goodrich et al.,2002b), is modeled by following the progressive heatingdue to the decay of radioisotopes from an assumed initialtemperature below the triple point of water. The treatment

Fig. 4. The variation of temperature with depth in a 100 km radiusasteroid at the times after formation when the deep internaltemperature has reached 1323, 1443 and 1553 K. These tempera-

tures correspond to 0%, 15% and 30% partial melting,respectively.



6/23

is similar to that employed by Cohen and Coker (2000)tostudy hydration and dehydration in asteroids but the pro-cess is followed up to temperatures within the silicate melt-ing range. We find that the thermal history is controlledalmost entirely by the 26Al content with a subsidiary contri-bution from 60Fe. The inclusion or exclusion of all of theother radioisotopes considered by Cohen and Coker

(2000)leads to a less than 1% change in the time at whicha given amount of melting occurs. The assumption that26Al was alive at the time of ureilite formation is justifiedby the presence of excess 26Mg in feldspathic clasts inpolymict ureilites (Kita et al., 2003, 2007). The total Al con-tent in the UPB is specified by assuming that it had a bulkcomposition similar to that given by Wasson andKallemeyn (1988) for CV meteorites. The key parametersare then the 26Al power production at CAI time, which forthe assumed composition and the canonical 26Al/27Al ratioof 5105 is nominally 2.156107 W kg1 of ice-freeasteroid mass, and the 26Al half-life, 0.720 Ma. We appreci-ate that the half-life has an uncertainty of3% and that the

detailed chemical composition of theparent body is notactu-ally known; thespreadof compositions of CV meteorites, as-sumed to be representative of the UPB, is large enough thatthe power production rate is uncertain by at least 10%, butwe adopt these nominal values. The corresponding heat con-tribution from 60Fe is characterized by a nominal power pro-duction of 1.0108 W kg1 with a half-life of 1.04 Ma. Allother contributing isotopes (see Cohen and Coker, 2000)have initial power production values at least three orders ofmagnitude smaller than 26Al; however, almost all of themhave longer half-lives than both 26Al and 60Fe and thus con-tribute very slightly (at the 1% level) to the late thermal his-tory of melting after the 26Al and 60Fe are effectivelyexhausted.

Seven thermal stages are involved: heating of rock andice to 273.15 K; buffering of the temperature at 273.15 Kwhile ice is melting and the latent heat of 330 kJ kg1 issupplied; heating of unreacted water and silicate reactionproducts while hydration is occurring between 273.15 and300 K; heating of silicates until dehydration reactions beginat 530 K; heating of dehydration products until dehydra-tion ends at 623 K; subsequent heating of dehydrationproducts until the onset of silicate melting at 1323 K; andheating of unmelted silicates as melting progresses. Duringthe hydration (273.15300 K) and dehydration (530623 K)phases, the latent heat of reaction (we use the 249 kJ kg1

estimate fromCohen and Coker, 2000) is added to and sub-tracted from, respectively, the heat released by the radioac-tives to find the temperature rise, and it is assumed that thelatent heat transfer occurs uniformly across the relevanttemperature interval. Similarly, it is assumed that duringsilicate melting the latent heat required (taken as400 kJ kg1) is absorbed uniformly across the solidusliqui-dus temperature range, though in this case the actual melt-ing temperature range varies with pressure (and hencedepth), due to the compositional (mg) variation caused bysmelting: although the solidus temperature is the same(1323 K) at all pressures because it is reached before smelt-ing begins (Fig. 2), the liquidus temperatures at 3, 6.5 and10 MPa are 1860, 1842 and 1827 K, respectively (these are

obtained from MAGPOX for the pre-smelted mg 91,mg 86 and mg76 compositions see Goodrich et al.,2007).

The thermal history is traced using time steps of 0.5 kabefore silicate melting begins and 1 ka afterwards. Convert-ing the available heat into a temperature rise during each ofthe thermal stages involves summing the heat per unit mass

produced locally from all of the isotopes, subtracting oradding the latent heat per unit mass where relevant, anddividing the result by the weighted mean specific heat atconstant volume of whichever materials (ice, water and/orsilicates) are present at the current temperature. Duringmelting and smelting it is also necessary, as described in de-tail later, to take account of advective redistribution of heatas silicate melt and volatiles are transferred to shallowerlevels or erupted. The temperature-dependent specific heatsof ice and water were taken from standard sources, and forthe silicates a weighted average of the specific heats of themineral assemblages present was used. Above the solidusthese are dominated by olivine, augite, pigeonite and pla-

gioclase as shown inFig. 2, and data for these were takenfrom a compilation byDobran (2001); there is not a greatvariation of specific heat with composition for silicatesand so the data for the same minerals were used at lowertemperatures. Fig. 5 shows the variations of the specificheats with temperature.

Both the water vapor released during dehydration andthe gases (mainly carbon dioxide and hydrogen) releasedby the earlier, lower-temperature hydration reactions areassumed to escape efficiently through fractures to the sur-face. These fractures will have been generated by the buildup of gas pressure as the reactions proceeded. It has beenproposed that, in extreme cases, gas generation duringhydration reactions may have led to massive disruption

and even dispersal of some small asteroids (Wilson et al.,1999). However, the100 km radius of the UPB was suffi-ciently larger than the 35 km radius marking the uppersize of body that can be disrupted in this way (Wilsonet al., 1999) that neither of the gas production phases ledto disruption, but both must have contributed to the formation

Fig. 5. The variation with absolute temperature of the specificheats at constant volume of ice, liquid water and the averagesilicate assemblage in the UPB.



7/23

of a network of fractures in the asteroid. This network mayhave been at least partially removed in the deep interior ofthe asteroid by the annealing effects of the high tempera-tures as silicate melting was approached, but it will havesurvived in at least the cooler outer57 km of the asteroid(seeFig. 4) to be exploited by the silicate melts, and even inthe deep interior the pressures are low enough that the

annealing process would not have been very efficient.The transfer of melts out of the interior of the asteroid

during silicate melting plays a key role in its thermal historysince most of the Al in the rock is contained in plagioclase,a mineral that is completely consumed during the meltingprocess (Fig. 2). Hence the main heat source is progressivelyremoved from the asteroid interior. We show below in Sec-tion5.1that the large volume fraction of carbon monoxideproduced by the smelting reaction causes much of the meltto be lost to space in explosive eruptions. However, thefraction that is not lost is intruded into the shallow crustallayers where it plays a key role in controlling the tempera-ture history at the 3 MPa pressure level. This retained melt

fraction is denotedf

iand is a model parameter for which wesolve.We model the spatial variations of properties within the

asteroid by dividing it into five zones: these are the regionfrom the surface down to 7.0 km depth, representing thecold crust; the layer from 7.0 to 17.1 km, taken to be repre-sentative of the 3 MPa pressure level at 10.4 km (see Table1); the layer from 17.1 to 32.3 km, representative of the6.5 MPa pressure level at 24.3 km; the layer from 32.3 to57.8 km, representative of the 10 MPa pressure level at41.5 km and ending at the 12.5 MPa level where smeltingceases; and the spherical region between 57.8 km depthand the center. The volumes of these spherical shells are8.20105, 9.80105, 10.92105, 9.82105 and

3.15105 km3, respectively.At each time step in the model we use the temperature

increase to define the increment of melting at each of the3, 6.5 and 10 MPa pressure levels (Fig. 2; see alsoFig. 7bofGoodrich et al., 2007). The melting in the central spher-

ical region is assumed to follow the melting sequence calcu-lated for primitive (mg62) UPB precursor material(Goodrich et al., 2007), with no smelting. We track the Alremoval using polynomials fitted to the plagioclase contentsas a function of melt fraction (Fig. 2). At each time step, theheat source due to 26Al at each of the three pressure levelsin the asteroid interior is scaled down in proportion to thecurrently remaining plagioclase mass fraction. The fraction

fiof retained melt is inserted into the crust to form a grow-ing intrusion layer (i.e. a sill) centered on a depth of 7 km,essentially the base of the outer layer that never undergoesmelting as the asteroid warms up. At each time step, thenew average Al content of this intrusion is calculated fromits previous Al concentration and the Al content of the new-ly added volume. The consequence of the evolving compo-sitional difference between the melt and its source rocks isthat as melting in the source region progresses from zeroto 30%, the 26Al content of the melt in the intrusion de-creases from 85% to 42% of the initial 26Al content inUPB parent material. A corresponding tracking of the Feshows that as melting progresses the 60Fe content of theintrusion increases slightly from 1.05 to 1.09 times the ini-tial value in the parent material. The heat from the growingintrusion that is conducted upward toward the surface is

considered in Section5.4where it is used to model the cool-ing history of the sill contents. The presence of the sill heatsource greatly reduces heat transfer from deeper levels, andplays a major role in controlling the maximum degree ofmelting at the 3 MPa level.

As smelting progresses at any given level, the metalliciron that is produced is assumed to settle downward. Theprocess is not modeled here in detail but is likely to be com-plex. For example, droplets of immiscible liquid metal pro-duced at smelting sites may be locally so small that they areinitially carried upward in the surrounding silicate melt but,being so much denser (7200 kg m3), they will alwayshave significant downward terminal velocities, and so in

Fig. 7. The temperature at the 10 MPa depth in the UPB as afunction of time after CAI formation as the asteroid heats up to theonset of silicate melting. This example assumes that the asteroidaccretes 0.547 Ma after CAI with an initial temperature of 250 K

and contains 20% ice by mass.

Fig. 6. The time after CAI formation at which the UPB mustaccrete in order to achieve 30% melting in the interior. Values areshown for plausible ranges of the initial mass fraction of ice and thetemperature at accretion.



8/23

the narrowest melt veins, where the silicate melt speed isvery small, they will have the opportunity to coalesce intolarger droplets with larger terminal velocities and musteventually migrate downward. As an extreme example, ifall of the iron coalesced into a central metal core, thiswould have a radius of41 km occupying7% of the vol-ume of the UPB. The redistribution of mass would lead to a

negligible change in asteroid radius but would cause thecentral pressure to rise progressively from its initial valueof 15.2 MPa to 26.6 MPa. However, the effect on the pres-sures within the 310 MPa range of levels from which theureilite meteorites are derived would be much less dramatic.The rearrangement of materials with the UPB would meanthat silicates that completed their melting in equilibriumwith a pressure of 10 MPa would in fact have begun meltingat a slightly greater depth in the asteroid where the ambientpressure was 10.95 MPa. The pressure changes at shallowerlevels would be proportionally smaller. Since we are in anycase using the melting process at 10 MPa (and at 6.5 and3 MPa) to approximate conditions over a range of asteroid

radii, spanning about a 20% pressure range in each case, wecan safely neglect this effect.A final issue concerns the approximate method used to

model the temperature distribution within the asteroid.We have not solved the full heat diffusion equation numer-ically; this would be particularly complicated in the presentcase where heat is being advected (by moving melt), as wellas conducted, through the asteroid. Instead, we approxi-mate the evolving temperature gradient in the crust as thedifference between the intrusion temperature and the sur-face temperature (assumed to be maintained at the initialformation temperature) divided by the depth of the intru-sion to find the total amount of heat lost by conductionto the surface in each time step. We divide this by the cur-

rent total mass of the asteroid to obtain a heat loss per unitmass. This is then subtracted from the heat per unit massgenerated by radioactive decay at all levels within the aster-oid. Next, we examine the new temperatures at all depthsafter this first-order heat transfer adjustment and refinethem by allowing heat to be conducted between each ofthe three pressure levels, the crustal intrusion and the cen-tral region of the asteroid as a function of the temperaturechange that has occurred and the length of the time step.

The target for a successful run of the model is that 30%melting should be achieved at each of the 3, 6.5 and 10 MPapressure levels. As a separate phase in obtaining solutionsto the model for a given set of input parameters we exper-imented with optimizing the value used for the thermalconductivity during the second temperature adjustmentphase. We find that a value larger than the actual thermalconductivity (expected to be close to 2 W m1 K1 at thetemperatures involved), by a factor of close to 6 is neededfor two reasons: one is the crude way the temperature pro-file is being specified (we are effectively approximating asmooth profile by five data points: the temperatures at thethree internal pressure levels, the crustal intrusion and thesurface), and the other is the fact that some heat is being ad-vected through the system by the melt migration. We opti-mize this thermal conductivity factor simultaneously withoptimizing the value of fi, the fraction of melt that is re-

tained in the sill. The optimum value of fi is found to be0.147, implying that 15% of the melt is not lost to space.

The thermal calculations were implemented as a FOR-TRAN program that was used to explore the variation ofthe melting history with the key controlling parameters:the time of asteroid formation after CAI time, the initialtemperature, and the initial ice content. The outputs gener-

ated are the times at which the temperature reaches a givenvalue during the warm-up to melting; the times and temper-atures at which successive increments of partial melting oc-cur at each of the 3, 6.5 and 10 MPa pressure levels; and thetemperature history of the melt intruded into the crust. Thetime of onset of melting is the same at all depths, given ourearlier demonstration of the near-uniform temperature atdepths greater than 8 km and the fact that the solidustemperature is independent of pressure over the small pres-sure range considered. However, the time to reach a givendegree of partial melting does vary with depth due to thedepth-dependence (i.e. mg-dependence) of the liquidus.

The most important input parameter controlling the

thermal history is, as expected, the time of formation,though initial temperature and ice content also have stronginfluences. Consideration of the measured temperatures ofmain belt asteroids (Lim et al., 2005) suggests that a plau-sible accretion temperature was250 K, and measurementsof the density of the large asteroid Ceres (McCord and So-tin, 2005) imply that the present-day, and probably initial,H2O content was less than 30% by mass. Using these valuesas order of magnitude guides, we show inFig. 6the varia-tion of the time at which the asteroid must be formed toachieve 30% silicate melting at the 10 MPa depth level asa function of the initial ice mass fraction in the range 030% and the asteroid formation temperature in the range240260 K. Given that we do not in fact know how far

from the (then less luminous) Sun the UPB accreted, lowermodel accretion temperatures might be justified, andFig. 6shows that these would drive the required formation timetoward earlier times. However, the implication of the start-ing conditions adopted here is that the asteroid accreted be-tween 0.535 and 0.565 Ma after CAI formation. Wenote that this result is similar to the


9/23

was decreased from its value at CAI time using the 0.72 Mahalf-life to provide the heat production rate at the time offormation. Fig. 7 shows the temperature as a function oftime up to the onset of silicate melting and Fig. 8 showsthe entire temperature history at all three internal pressurelevels and in the melt intruded into the crust treated as aglobal sill. Finally,Fig. 9shows how the melt fractions in

the interior vary with time. Clearly a maximum amountof melting very close to 30% is achieved at all pressure lev-els, although the time at which this happens varies between4.5 and 5.8 Ma after CAI. We emphasize again thatachieving the requisite high temperatures (thus 30% melt-ing) in the shallowest source regions depends critically onthe presence of the sill, a result that was foreseen by Kitaet al. (2005).

Fig. 9can be used to find the rate of melt production, /,at each pressure level as a function of time.Fig. 10showsthe result: the main feature, controlled by the half-life of26Al, is the dramatic decrease in melt production rate fromvalues in excess of 100 m3 s1 in the very early stages ofmelting at1 Ma after CAI to values of5 m3 s1 between2 and 3 Ma after CAI, decreasing again to extremely small

values as the end of melting is approached beyond 5 Ma.The significance of this for melt extraction is explored inthe following section. Here, we point out that the bulk ofthe melt is produced quickly in the early stages of melting,before plagioclase is exhausted. The implication of this,which we will examine later, is that the melts that producedthe indigenous feldspathic clasts in polymict ureilites wereproduced in a short period of time at 0.91 Ma after CAI.

4. MELT FORMATION AND EXTRACTION FROM

THE ASTEROID INTERIOR

The onset of melting in asteroids, just as in the Earth, in-

volves the formation of melt films along graingrain con-tacts and subsequent melt migration. This can occur bypercolation of melt along grain boundaries (Maaloe andSchie, 1982; McKenzie, 1984; Richter and McKenzie,1984; Spiegelman and Elliot, 1993; Kelemen et al., 1997)and by compaction of the unmelted matrix, either in amonotonic fashion (Sleep, 1974; McKenzie, 1984; Richterand McKenzie, 1984; Ribe, 1985; Spiegelman andMcKenzie, 1987) or by the formation of propagating wavesof variation in porosity (Scott and Stevenson, 1984, 1986).Various papers address the collection of melt into veins ona larger scale than the fabric size (i.e. the mineral grain size)of the unmelted matrix (Maaloe, 1981; Spence et al., 1987;Sleep, 1988; Hart, 1993; Maaloe, 2003) and the further

interconnection of large veins to form the dikes which ulti-mately drain large volumes of partial melt to the surface(Nicolas and Jackson, 1982; Sleep, 1984; Fowler, 1985;Nicolas, 1986; Wickham, 1987; Sleep, 1988). We incorporate

Fig. 8. The temperature at various pressure levels (i.e. depths) inthe UPB as a function of time after CAI formation. Thetemperature histories at the 6.5 and 10 MPa levels are sufficientlysimilar that they are indistinguishable on this plot. Accretionconditions are the same as for Fig. 7.

Fig. 9. The fractional amount of melting that occurs at the 3, 6.5and 10 MPa pressure levels in the UPB as a function of time afterCAI formation. Accretion conditions are the same as for Fig. 7.

Fig. 10. The total silicate melt production rate in the UPB as afunction of time after CAI formation. Accretion conditions are thesame as forFig. 7.



10/23

key aspects of these treatments into a model of melt extrac-tion from the UPB as follows.

All of the treatments developed for the Earth relate toconditions within it at the present time, with the spatialand temporal scales of the melting system set by mantledynamics driven by the current heat loss rate and tempera-ture structure of the planetary interior. Strain rates are suffi-

ciently small that large deviatoric stress do not build upeasily and cannot be sustained for geologically long periods.Maaloe (2003)gives a calculation for melting beneath mid-ocean ridges on Earth showing that magma produced by24% partial melting of mantle rock would reflect the rise ofthe host material between pressure levels of 5.6 and 3 GPa,corresponding to depths between 170 and 90 km. The80 km rise would occur at a speed of0.1 m/a implyinga time interval of 0.8 Ma. The ambient temperature ofthe host rocks would decrease from1500 to1300 C overthis depth range and the effective viscosity of the host rocks(a function of both the bulk and shear viscosities) would be1018 Pa s (Sleep, 1988). In contrast, in the UPB the time re-

quired to reach24% partial melting after the onset of melt-ing would be between1 and 2 Ma (Fig. 9), up to double theterrestrial value. However, because the UPB was much smal-ler than the Earth, internal lithostatic pressures were alsomuch smaller (Table 1) so that melting started at a lowertemperature (Fig. 2), 1323 K to be compared with1620 K in the Earths mantle (Turcotte and Schubert,2002). This means that the unmelted matrix would probablyhave had a somewhat higher effective viscosity than theEarths mantle on the point of melting. Given Sleeps(1988) suggested value of 1018 Pa s for partially moltenmantle in the Earth and Spiegelmans (1993a) suggestedrange of 10181019 Pa s, we adopt 31018 Pa s for theUPB. We show below that this higher viscosity approxi-

mately compensates for the slower rate of melting in control-ling therelaxation of the excess melt pressures induced by thevolume increase on initial partial melting, and the presenceof high melt pressures would have assisted the interconnec-tion of growing melt veins. Some aspects of this behaviorwere addressed by Muenowet al. (1992) andKeil andWilson(1993)in modeling the removal of mafic mantle melts andFe,Ni,S core-forming fluids from asteroid parent bodies,but the treatment given below is more detailed in that it re-lates the onset of melt extraction to the mineralogy and thedegree of partial melting, the key factors in understandingthe chemistry of the melts.

4.1. Geometry of melting

Consider an array of mineral grains (Fig. 11), each as-sumed to be a uniform cube of size L (the exact geometryadopted is not critical). When more than one mineral spe-cies is present, melting may occur only at the interfaces be-tween some fraction of the minerals, and we idealize this byassuming that if a grain is melting then melt forms at all 12of its edges, as shown, but that only a fraction f of thegrains are in this state. The cross-sectional area A of anyone melt vein of diametera is

A p=4a2 4

The volume Vvein of one vein is

Vvein p=4a2L 5

and the volume of all 12 veins is 3pa2L. However, each veinis shared by 4 adjacent grains, so the melt volume corre-sponding to one grain is

Vmelt 3=4pa2L 6

Since a fraction fof all of the mineral grains is melting,and the volume of one mineral grain is L3, this melt volumeoccurs in a reference volume Vrefequal to (L

3/f), and so themelt volume fractionq is given by the ratio (Vmelt/Vref), i.e.

q 3fpa2

=4L2

7The number of veins per unit volume, N= (q/Vvein), is

N3f=L3 8

and the average distance, d, between veins is equal to(1/N)1/3 so that

d L=3f1=3 9

Thus the ratio r between the average separation dandthe mineral grain scale length L is

r 3f1=3 10

Fig. 12shows this relationship, and clearly many veins

will be in close proximity, i.e. r< 1, if f is greater than0.20.3 whatever the detailed geometry. At the onset ofmelting, the UPB contained three major mineral species,and at the start of melting augite and plagioclase, totaling25% of the rock, were melting together, so we adopt

f= 0.25 and infer that close proximity between meltingveins would have been very common.

4.2. Melt pressure

We next estimate the likely pressures in pockets of melttrapped between mineral grains as a function of the amountof melting that takes place and show how this rapidly

Fig. 11. Geometry of a cubic mineral grain that is melting along all12 of its edges while in contact with other grains of similar shapethat are not melting.



11/23

encourages increased connectivity between melt veins ifnone already exists. Following Muenow et al. (1992), weadopt a convenient geometry by modeling the meltinggrains as spheres, each having an initial radius x0 and uni-form density qs, which melt to form a liquid of density q l.At some general time, melting has progressed inward froma grain surface to some radius x, so that the current meltvolume is [(4/3)p(x0

3 x3)], and the pressure in the liquidphase is Pl. If a pressure increase dPl accompanies an in-ward increase dxin the radius of the melt zone, then

4px2dxqs=ql 4=3px30=fdPl=l

4=3px30x

3

dPl=b 11

Here, the left hand side is the volume increase due toconversion of a volume element of the solid to lower densityliquid. The first term on the right hand side is that part ofthe volume increase accommodated by compression of thesurrounding matrix, taking account of the fact that only afraction f of the grains are melting, and the second termis that part taken up by compression of the liquid itself, land b being the bulk moduli of the unmelted grains andthe liquid, respectively (Tait et al., 1989). Note that thistreatment assumes that only a liquid phase is producedon melting. As noted in Section1, gas production by smelt-ing eventually occurs at all depths in the UPB down to the

12.5 MPa pressure level. Gases are much more compress-ible than liquids and so the presence of gas will reducethe rate at which the pressure rises with increasing meltfraction. However, it is only in the shallowest regionsundergoing melting that any gas is produced very early inthe melting process. We shall shortly show in Section 4.3that a fully-connected vein network is likely to be generatedafter as little as 0.15% melting, and the calculations under-lying Fig. 2 imply that only within a layer about 100 mthick at the 3 MPa pressure level at a depth of1011 km will gas be present during the period of vein con-nection. Since a complex vein and dike network will alreadyhave evolved in the deeper interior by this time (melting

starts first in the deepest parts of the asteroid as will be dis-cussed in detail in Section5.2), fractures will be propagat-ing into the 100 m thick layer from below as well as beinggenerated within it. We therefore feel justified in neglectingthe details of this process.

Eq.(11)is readily integrated to give the pressure increasein the melt, DPl, in excess of the lithostatic load Ps, as

DPl qs=qlb ln 1 fl=b1 fx=x0g3

h i 12

Since only a fractionfof the mineral grains is melting, thevolume fraction of melt,q, in the entire mineral assemblageis by definition equal to [(4/3)p(x0

3 x3)f]/[(4/3)px03] =

f(1{x/x0}3), and so Eq.(12)can be written

DPl qs=qlb ln1 l=bq 13

Appropriate values of b and l are 10 and 13 GPa,respectively (Muenow et al., 1992). With a melt density ofql=2900 kg m

3 and a matrix density qs initially equalto the bulk density of the asteroid, r=3300 kg m3,(qs/ql) =1.14, the pressure increase DPis shown inTable

2. Values are given in terms of the mass fraction of melt,mm, which is related to the volume fraction of melt, q, by

mm qql=1qqsqql 14

and the corresponding values ofq are also listed. Clearly,initial pressures of a few to a few tens of MPa are possibleeven at small amounts of melting when melt is completelytrapped between bonded mineral grains. However, wenow show that these pressures rapidly lead to connectionsforming between melt pockets.

4.3. Vein connection

The mineral grain fabric scale in the UPB prior tomelting, and hence the typical length, L, of the initial meltveins, is likely to be similar to that of chondritic meteor-ites that experienced high-temperature metamorphismbut no silicate melting, or to that of the most primitiveachondrites. We therefore expect that L will lie in therange 100300lm(Brearley and Jones, 1998; Mittlefehldtet al., 1998). Although, as we showed above, the initialmelt veins are likely to be in close proximity, their degreeof interconnection will increase greatly if the stresses at thetips of veins become large enough to allow them to frac-ture the unmelted host matrix. A detailed discussion byRubin (1993) shows that the criteria for this to occurare best expressed by requiring a balance between the

external lithostatic load, Ps, the total internal melt pres-sure, (Ps+DPl), and the graingrain cohesive forces repre-sented by the tensile strength, S. In the present notation,and removing Rubins assumption that the veins are muchlarger than the mineral grain fabric, since this is not trueat the scale of the veins we are considering, the stress bal-ance defining the vein length Lf at which fracturing will

just occur is given by

DPlLf Lc1=2 81=2=pDPlPsSL

1=2c 15a

resulting in

Lf f8=p2DPlPs S=DPl

2 1gLc 15b

Fig. 12. The variation of the ratio r of the average melt veinseparationdto the mineral grain scale length L as a function of thenumber fraction of all of the mineral grains that are melting, f.



12/23

where Lc is a critical length given by

Lc p=4e=1mdc=DPlPsS 16

Here,e and m are the shear modulus and Poissons ratio,respectively, of the unmelted mineral grain matrix and dcisthe maximum distance over which graingrain cohesiveforces can act. In partially molten rock under relatively

low confining loads (the 10 MPa stress-level depth inthe UPB is the equivalent of 400 m depth on Earth), the va-lue of [e/(1m)] is probably 4 GPa, the value found byParfitt (1991) for dikes propagating within Kilauea Vol-cano, Hawaii; this is much less than the commonly citedvalues of order 30 GPa (Rubin, 1993) for the deeper litho-sphere on Earth. Experimental measurements on unfrac-tured rocks suggest that dc is 10lm (Ingraffia, 1987)and that Sis 10 MPa (Ingraffia, 1987; Jaeger and Cook,1979). Using Ps= 80 bars as an average value of the30125 bar range of lithostatic loads in the ureilite rocks,Table 2shows how Lc and hence L fvary with the amountof melting. Examination of the values ofLfshows that atthe lower end of the expected range of grain lengths,L= 100lm, as about 0.48 volume % melting would haveto occur to initiate vein tip fracturing; at the upper end ofthe expected length range, L= 300lm, the process wouldbegin at 0.30 volume % melting; and if any grains1000lm, i.e. 1 mm, long were present, only 0.17 volume% melting would be required. Eq. (7) can be used to findthe diameters of the veins assuming the tubular geometry ofFig. 11. The diameter, a, of a 100lm vein at 0.48 volume %melting is 4.5lm; the corresponding values are 10.7lmfor a 300lm long vein at q =0.30 volume % melting and26.9lm for a 1 mm long vein at 0.17 volume % melting.

As soon as one unusually large vein begins to grow, ithas a large probability of intersecting other veins, and it

seems safe to assume that rapid interconnection betweenpockets of melt trapped between grains will occur afterabout 0.15 volume % melting is reached, with typical veinlengths of 1 mm and widths of 25lm, and we adoptthese values for subsequent use. Note thatTable 2impliesthat this degree of melting would induce about a 25 MPapressure in the melt. This would mean that this initial frac-turing phase would occur with much higher melt pressuresthan the lithostatic pressures in the UPB, well above the12.5 MPa pressure above which smelting cannot occur.Thus the very earliest isolated melt pockets would be gas-free everywhere. However, as soon as vein interconnectionbecame common, the entire asteroid interior would expandvery slightly (by 0.05%) to accommodate the volume in-crease on melting and the pressures at all depths in bothmelt and matrix would quickly relax to lithostatic, withthe combination of melting and smelting appropriate toeach depth then being resumed. We note that it is possiblethat the inherent strength of the metamorphosed UPB inte-rior may have been less than that of unfractured rocks fromthe Earths interior; however, reducing the assumed inter-grain tensile strength by a factor of 2, from 10 to 5 MPa,only reduces the pressure (and melt fraction) at which frac-turing begins by 15%.

A further aspect of the metamorphic history of the UPBis the possibility that fractures formed during the hydration

and dehydration phases of heating were not completely an-nealed by the onset of silicate melting but remained asclosed but weak incipient fractures. In that case, the30 MPa melt pressure would apply only to melt completelytrapped between bonded grains. As soon as pressurizedmelt pockets became connected to these weak relict frac-tures they would utilize them in preference to creating

new fractures, and so both the timescale and the preponder-ance of excess pressures needed for production of a thor-oughly interconnected fracture network would be less.

4.4. Evolution of vein network

The way in which the size spectrum of the pathways inan interconnected network develops in a region of partialmelting is poorly understood. Recent work by Valentiniet al. (2007) suggests that beneath mid-ocean ridges onEarth fracture networks on scales between millimetersand meters are 45 times more efficiently interconnectedthan a random arrangement of fractures would be. The

presence of high-temperature planar dikes in ophioliteexposures of mantle peridotites (Nicolas and Jackson,1982) strongly suggests that vein networks can evolve intomuch larger-scale dikes in such regions, and this led Sleep(1988)to propose mechanisms for the evolution of vein net-works in which melt moves from smaller veins into progres-sively growing larger ones as a result of pressure gradientsdriven by differences in the principle stresses. Keil andWilson (1993)explored a melt migration process of this kindfor small asteroids. Onerelevant complicationis that as soonas an interconnected vein network is in place throughout theregion of melting, so that melt can begin to move, new pro-cesses influence the scales at which melt segregation occurs.Various instabilities may develop within the region of partial

melting, driven by small inhomogeneities in density andporosity. Density variations will induce RayleighTaylorinstabilities, potentially able to grow into convection cells(though we show shortly that they do not develop this far).Porosity variations will grow and become concentrated, withzones of higher than average porosity traveling upward assolitary waves (commonly called magmons) that can propa-gate fasterthanthe melt is risingthroughthe unmelted matrix(Spiegelman, 1993b). The factor controlling porosity fluctu-ations is the compaction length, d, given by

d fkf 4=3n=gg1=2 17

where f and n are the bulk viscosity and shear viscosity ofthe matrix, respectively, both taken as 3

1018 Pa s based

on the temperature at which melting begins, as discussedearlier, and g is the viscosity of the melt, taken as 1 Pa s.The quantity k is the permeability, given by

k Y2qa=b 18

where Y is the fabric scale, dictated by vein lengths so thatY=1 mm, q is porosity of the system, equal to the localmelt fraction, which by definition is at least equal to thethreshold melt volume fraction for vein interconnection,found above to be 0.002, anda and b are constants. Var-ious treatments of the geometry of melt vein systems (e.g.von Bargen and Waff, 1986) suggest that a lies between 2



13/23

and 3 andbbetween 100 and 3000. An analysis byTurcotteand Schubert (2002)for a cubical array like that ofFig. 11gives a= 2 and b= (72 p) =226. Inserting these valueswe find k=1.771014 which in turn implies d =350m.Spiegelman (1993b)showed that the size of the propagat-ing solitary waves of melt concentration is of the same orderas (and probably a little larger than)d. Thus, although melt-

ing is occurring simultaneously over a large vertical distancewithin the asteroid, there may be small variations in meltcontent within regions with this vertical extent.

There is some ambiguity about the typical horizontalscale of melt concentration variations. This could be takenas the wavelength x of density-driven RayleighTaylorconvective instabilities present in the melt zone, given by

x 1:284X 19

whereX is the characteristic vertical extent over which sig-nificant density differences exist. At one extreme this mightbe taken as the entire vertical extent of the region withinwhich melting is taking place, equal to the radius of theasteroid minus the thickness of the outer shell that is con-ductively cooled, 10 km, so that X=90 km andx=120 km. This would make the area of the potentialconvection cell [p(x/2)2] =[p602] = 11,000 km2. Formelt being extracted from a region half way between thecenter and the surface of the asteroid, where the area of aspherical shell is [4p502] = 31,400 km2, this implies thatthere will be only about 3 such unstable regions active atany one time; nearer the surface, where the area of a shellapproaches [4p902] = 101,800 km2, there could be10 suchregions at any one time and deep within the asteroid onlyone or two. However, it could be argued that density vari-ations are also present on the scale of the porosity solitarywaves, in which case with X=d=350 m,x will be450

m and the typical area of a region from which melt is beingexpelled [p(x/2)2] will be0.16 km2. In this case 200,000extraction events could be occurring within the asteroid atany one time. We showed above that the initial total rateof melt production in the asteroid would have been100 m3 s1; if 200,000 extraction regions were active,the melt volume flux through each would have been0.00045 m3 s1, whereas if 210 such regions were activeit would have been 50 to10 m3 s1. We regard the latterscenario as very much more likely, with the compactionlength controlling the organization of the vein system with-in the asteroid and the longest wavelength controlling thenumber of major melt extraction sites. We thus expect thatin the early stages of melting there will be 5 essentiallyseparate regional networks of veins in an asteroid of the sizeof the UPB, each draining a melt flux of/R=20 m

3 s1

toward the surface.The timescale for growth of RayleighTaylor instabili-

ties,s, is given by (Turcotte and Schubert, 2002)

s 26:08f=gDqX 20

where as before f is the bulk viscosity of the partially mol-ten matrix and Dq is the typical density difference drivingthe instability. For what is probably an extreme porosityvariation, from 0.005 to 0.020, between the interiors of sol-itary waves and the regions between them, the bulk density

difference would be 10 kg m3. Then using f= 3 1018 Pa s as before, we find that for the largest possible ver-tical scale, X= 90 km, s is 21.71015 s = 58 Ma,whereas for the solitary wave scale X=350 m, s is4.51017 s = 15 Ga (3 times the age of the solar sys-tem!). Both of these timescales are so much longer thanthe5 Ma duration of melting (Fig. 9) that clearly neither

large-scale nor small-scale convection of the melting regionever developed to a significant extent. Furthermore, if wesubstitute for [gDq] in Eq.(20)a typical pressure gradientwithin the asteroid due to the high pressures postulated tobe present at the onset of melting, say a few tens of MPa act-ing over a few tensof km, i.e. 103 Pa m1, we findrelaxationtimes of 0.03 and 7.4 Ma. These are to be compared with thetimes neededto achieve pressuresof a fewtens of MPa, whichTable 2shows to require 0.2% melting.Fig. 9shows that0.2% melting is reached at all depths at about 0.015 Ma aftermelting starts. Again thetimescales for relaxation are greaterthan the timescale for generation of the relevant stresses (al-beitonlyby a factorof 2 inthe case ofthe larger length scale),

thus justifying our earlier assertion that pressures of a fewtens ofMPa are indeedgenerated for a short time atthe onsetof silicate melting and are available to drive the initial crackand vein interconnection processes.

4.5. Melt extraction

The final stage in the evolution of each regional vein net-work must be the formation of at least one giant vein,which will have the geometry of a planar dike. This is likelyto be several km long if it is transferring melt into one of thecrustal sills and must be at least 10 km long if it avoidsintersecting a sill and instead transfers melt directly to thesurface through the permanently cold outer shell of the

asteroid. The dike could be up to 40 km long, since thisis the depth of the region down to the 10 MPa pressurelevel from which we have meteorites, but to obtain a con-servative estimate of the melt transit time we use the mini-mum value, 10 km, as our upper dike length limit, Ymax,showing later that the exact value adopted is not critical.We postulate that the vein network evolves into a fractal-like structure, so that everywhere within the region wheremelting is taking place a hierarchy of vein lengths exists be-tween the largest and the smallest, which we showed earliershould have a length Ymin= 1 mm. Melt migrates along agiven-sized vein or veins until it encounters the next largestsize of vein into which it is driven by either buoyancy or thepressure differential due to the difference in size. Pressuredifferentials due to vein size differences will relax quicklyonce a continuously flowing melt network is established,and so we assume, again conservatively, that only meltbuoyancy is involved in driving the motion. The density dif-ference, Dq, providing the buoyancy will be the differencebetween the melt density and the bulk asteroid density,r= 3300 kg m3. If smelting is not occurring, the melt den-sity will be ql=2900 kg m

3 and so Dq will be400 kg m3; we saw earlier that if smelting is taking placethe melt will contain 83% to more than 90% by volumegas bubbles and up to 30% by mass liquid iron metaldroplets, making the bulk density in the range 100



14/23

300 kg m3 so that Dq is 30003200 kg m3. The meltflow speed though a vein, um, is obtained by balancingthe buoyancy force against wall friction drag. The standardfluid dynamic relationship for a Newtonian liquid flowingthrough a circular tube of diameter Zis

um gDqZ2=32g 21

if the melt motion in the vein is laminar, which we find to bethe case for all permutations of the parameters relevant here.

To determine the equilibrium conditions for melt extrac-tion from the vein network, we assume that there are Mclasses of vein in the length range between Ymaxand Ymin,the veins in each class being longer than the veins in theclass beneath it by the factor (Ymax/Ymin)

(M1). Meltingtakes place throughout the asteroid from the center to aradius Rm which, given the 10 km thickness of the coldouter shell, is 90 km. Consider the melt originating atany radial distance Rx from the center of the asteroid andbeing collected into one of the regional vein networks. Sincethe total flux through the regional network is /R, the flux

passing through this radius is/

x where/x Rx=Rm

3/R 22

However, the flux can also be defined as the product of thenumberNxof veins cutting through the radiusRx, the areaAx of each vein, (pZ

2)/4, and the flow speed of melt in thevein,um. The number of veins can be found by noting thatveins occupy the edges of cubes of length Ymin. Each cubecross-section contains 4 veins, but each vein is shared with3 adjacent cubes, so that on average there is one vein percube. The number of cubes of cross-sectional area Y2minintersected by the spherical surface of radiusRxisNxwhere

Nx 4pR2x

=Y2min 23

and so we have

Rx=Rm3/R 4pR

2x=Y

2minpZ

2=4um 24

Substituting for um from Eq. (21) and rearranging, thetypical vein diameter is given by

Z 32Rx/RY2ming

= p2gDqR3m 1=4

25

and substituting this expression into Eq. (21) we find the

melt speed to be

um gDqRx/RY2min=32p

2R3mg1=2 26

Meltflows through veins with diameter Zat this speed un-til it intersects one of the next largest size of veins in the hier-archy. This larger vein has a length (Ymax/Ymin)

(M1) timeslarger thanYmin, and since we are assuming that these veinsare separated by distances equal to their lengths, the time ta-ken by the melt to reach the larger vein is swhere

s 3=2Ymax=YminM132p2R3mg=gDqRx/R=

1=2

27

The factor (3/2) here allows for the fact that melt in gen-

eral hasto travellaterally as well as vertically toreachthe nextlargest vein. Note that this timescale depends on the ratio(Ymax/Ymin)

(M1) but not on the explicit value of the veinlengthinvolved. Thus when we consider themeltin this largervein draining into the next largest vein, the timescale for thisstep will be essentially the same. As melt moves into ever lar-ger veins there will be an adjustment due to the fact that themelt will be moving closer to the surface of the asteroid andso both the radial distance from the center, Rx, and the accel-eration due to gravity,g, will progressively increase, slightlydecreasing the transit time. However, since the total melttransit time is the sum of the times involved in each of theMsteps this is a relatively small effect.

Table 3shows examples of the pattern of melt migrationfor two values of the length multiplication factor

Table 3Examples of distribution of melt flow properties in interconnected tubular vein network in asteroid. Values follow shortest path throughsuccessively larger veins for two values of the length multiplication factor, the factor by which the vein length increases from each class to thenext.

Tubular veinlength

Distance to asteroidcenter (km)

Acceleration dueto gravity (m s2)

Tubular veindiameter

Melt flowspeed

Reynolds numberof melt

Melt transittime

Melt volumefraction

(a) Number of tube size classes = 3, length multiplication factor = 3162

1.0 mm 52.0 0.0508 73lm 26 nm/s 5.7109 5.7 years 6.9103

3.16 m 52.0 0.0509 4.1 mm 83lm/s 1.0103 5.7 years 2.2106

10.0 km 62.0 0.0606 0.23 m 0.31 m/s 216 0.62 days 6.91010

Total melt transit time = 12 years(b) number of tube size classes = 10, length multiplication factor = 5.995

1.00 mm 52.0 0.0508 73lm 26 nm/s 5.7109 3.9 days 1.2102

5.99 mm 52.0 0.0508 179lm 158 nm/s 8.4108 3.9 days 2.1103

3.59 cm 52.0 0.0508 438lm 943 nm/s 1.2106 3.9 days 3.5104

0.22 m 52.0 0.0508 1.07 mm 5.6lm/s 1.8105 3.9 days 5.8105

1.29 m 52.0 0.0509 2.62 mm 34lm/s 2.6104 3.9 days 9.7106

7.74 m 52.0 0.0509 6.43 mm 203lm/s 3.9103 3.9 days 1.6106

46.4 m 52.1 0.0509 15.7 mm 1.22 mm/s 5.7102 3.9 days 2.7107

278 m 52.3 0.0512 38.5 mm 7.35 mm/s 0.84 3.9 days 4.5108

1.67 km 54.0 0.0528 94.3 mm 4.55 cm/s 12.8 3.8 days 7.5109

10.0 km 64.0 0.0626 0.23 m 0.31 m/s 223 0.41 days 1.3109

Total melt transit time = 35.4 days



15/23

(Ymax/Ymin)(M1) for the cases where M= 3 andM= 10. In

the M= 3 case there are only 2 intermediate sizes of veinbetween the smallest and largest, whereas in the M= 10case there are 9 progressively increasing sizes. Table 4shows how the transit time varies with Mover a wide range:the minimum melt extraction time occurs between M= 16and 17 with a length multiplication factor of2.8. Clearly,

melt extraction times will be very long if there are only afew classes in the vein hierarchy, so that melt travels mostof the way to the base of the crust through the narrowestveins, but for any value ofMgreater than7, i.e. for lengthmultiplication factors less than 1015, the typical meltextraction time through the vein network will be just lessthan 1 month. The volume fraction of the asteroid interioroccupied by melt at any one time is given by the sum of thecomponent volume fractions in the last column ofTable 3and is close to 2%. We stress that the above timescales re-late to melt extraction early in the melting process, whenthe total melt production rate is100 m3 s1.Fig. 10showsthat much of the melting takes place at production rates of

5 m

3

s

1

, so that with 5 extraction zones the flow ratethough each is 1 m3 s1. A repeat of all of the above cal-culations then gives melt transit times of4 months. Thevolume fraction of the asteroid interior occupied by meltat any one time during this late stage is then 0.4%. Clearlyvery nearly all of the melt ever produced in the asteroidinterior is extracted to shallow levels.

These melt transit times refer to the time needed for meltto be drained upward out of the zone of melting. We showin the next section that melt is not likely to be erupted di-rectly to the surface but instead is expected to accumulatein a sill near the base of the crust and be erupted episodi-cally to the surface. Thus, technically speaking, transittimes to the surface may be much longer. We note, how-

ever, that residence in the sill does not need to be includedin estimates of the time that the melt was in contact with themantle residues (hence available for isotopic and geochem-ical exchange with the residues), and thus does not changethe conclusions ofGoodrich et al. (2007).

5. SMELTING AND ITS SIGNIFICANCE FOR THE

FATE OF THE MELT

5.1. Explosive volcanism

Next we explore the consequences of smelting for theway in which melt is removed from the asteroid interior

and whether it is retained as part of a crust or lost to space.It has previously been suggested (Warren and Kallemeyn,1992; Scott et al., 1993) that the absence of basaltic ureilitescan be explained if melts generated on the UPB had highcontents of CO + CO2 gas derived from smelting, andtherefore erupted explosively at velocities sufficient to es-cape their parent body (Wilson and Keil, 1991). We nowreevaluate this conclusion in light of the result (Goodrichet al., 2007) that melting and smelting do not begin simul-taneously with one another in all source regions (Fig. 2).

Fig. 3 shows that the onset of smelting, at whateverpoint it occurs in the course of melting, causes a very largeamount of gas to be generated at each pressure level and

hence depth. The data inFig. 3have been used to evaluatethe gas mass fraction,n, and the corresponding gas volumefraction,v, in each 1% increment of smelting at each of the3, 6.5 and 10 MPa pressure levels. The calculations assumea perfect-gas relationship between pressure and volume, anadequate approximation for the present purpose. It is alsoassumed that gas and melt are vented to the surfacethrough fractures that form early in the (s)melting process,as discussed below, so that the pressure at all depths re-mains close to lithostatic. In all cases the gas volume frac-tion v is greater than 83% at the depth of formation, andis generally greater than 90%; furthermore, the gas volumefraction in a given batch of melt will increase as it decom-presses on migrating upward toward lower pressure re-

gions. These values of v are significantly greater than thegenerally accepted 7080% threshold in explosive volcaniceruptions for disruption of melt into a spray of liquid drop-lets transported in the gas phase (Sparks, 1978; Vergniolleand Jaupart, 1988), so it is clear that, as long as no signif-icant net separation of gas and liquid occurs during trans-port, an issue to which we return later, the release at thesurface of melt formed during smelting will be vigorouslyexplosive. The gas mass fraction n can be used to estimatethe speed u at which the spray of gas and magma dropletswill emerge at the surface in such an eruption using themost conservative of the equations given by Wilson andKeil (1991, 1996):

u2 2nQT=mc=c1 1 Pf=Pdc1=cn o

28

wherem= 28 kg kmol1 is the molecular mass of the dom-inant volatile component, CO (Goodrich et al., 2007); Tisthe melt temperature in the range 13231553 K ( Fig. 2); Qis the universal gas constant, 8.314 kJ K1 kmol1; c is theratio of the specific heats of the gas, very close to 1.30; andPf/Pdis the ratio of the final pressure in the expanding gasto the pressure at which disruption of the melt into a sprayof droplets takes place. In practice,Pf/Pd is several ordersof magnitude less than unity and can be neglected, resultingin the eruption speeds given in Fig. 13. Even though ini-tially high values decline, due to the decline in gas produc-

Table 4Values of the time needed for melt to travel through the veinnetwork within the asteroid as a function of the number of veinlength classes, M, and the corresponding factor by which the veinlength increases between classes.

Number of classes, M Vein length scalefactor

Melt transit timein veins

2 1107 18,000 years3 3162 11.5 years4 215.4 1.14 years5 56.23 149 days7 14.68 58.1 days8 10.00 46.2 days

10 5.995 35.5 days13 3.831 30.2 days15 3.162 29.0 days16 2.929 28.8 days17 2.738 28.7 days20 2.336 29.0 days25 1.957 30.6 days



16/23

tion at all levels (Fig. 3), all obtained values are greater than300 m s1. These can be compared with the escape speed,E, from the asteroid given by

E 8=3pGr1=2R 29

Using R= 100 km and r= 3300 kg m3 as before,E= 136 m s1, so clearly all of the melt reaching the surfacedirectly during the smelting phase at each depth in theasteroid should be lost to space, rather than being retainedon (or beneath) the surface. Thus, our results support thesuggestion (Warren and Kallemeyn, 1992; Scott et al.,

1993) that explosive volcanism may largely explain theabsence of basaltic ureilites in the meteorite collection.However, we now must now consider the consequences ofmelts produced during the non-smelting phase.

5.2. Mixing of melts during ascent

Goodrich et al. (2007)recognized that a significant frac-tion of melts generated on the UPB, particularly those fromdeeper source regions, were produced during periods inwhich their source regions were not being smelted (Fig. 2and 3), and suggested that these may have erupted to forma thin crust (estimated 3.3 km thick). However, in that anal-ysis, we overlooked the likelihood that melts derived fromall depths would be mixed as they rose. Thus, if gas produc-tion is taking place somewhere in the asteroid at all timesonce melting begins (as one might initially assume fromFigs. 2 and 3), then no melt reaching the surface shouldbe gas-free. In fact, this is not quite the case:Fig. 4showsthat, although a given temperature is reached at nearlythe same time at all depths greater than 8 km, there is,nevertheless, a small temperature gradient, such that melt-ing starts first at the deeper levels and is progressivelydelayed at the shallower levels. The temperature differencebetween the 6.5 and 10 MPa levels is very much less thanthat between the 3 and 6.5 MPa levels and so we use the

latter in the following calculation. When the temperatureat the 3 MPa level reaches 1323 K, the temperatures atthe 6.5 and 10 MPa levels are very close to 1336 K, higherby 13 K.Fig. 2then implies that about 2.5% melting willhave occurred at the 6.5 MPa level and 1.1% melting willhave occurred at the 10 MPa level, in each case with nosmelting having taken place. These melts will have been

able to rise through their unmelted host rocks as a resultof their buoyancy alone, without the added buoyancy pro-vided by accompanying gas bubbles. However, that absenceof gas means that, as they pass through the 8 km depthlevel where the temperature decreases rapidly and theyencounter first thermally less-altered crust and eventuallyprimitive crust, still containing ice, and probably exten-sively fractured as a result of the net expansion of the inte-rior of the asteroid caused by the temperature rise duringsilicate melting. The melts are thus very likely to reach adensity trap where they are neutrally buoyant, the condi-tions being similar to those that appear to have led to vol-canic intrusions into the bases of the fractured breccia

zones beneath some impact crater floors on the Moon(Wichman and Schultz, 1995). The excess pressure at theupper tip of a column of buoyant melt is given by the inte-gral of the density difference between the melt and its hostrocks multiplied by the local acceleration due to gravity, gi-ven as a function of depth by Eq.(3). Evaluating this for thelikely difference between the densities of the host rocks(3300 kg m3) and the mafic melt (taken as 2900 kg m3)yields an excess pressure of at most 2.5 MPa. This willprobably be less than the tensile strength of the crust, whichwould be 10 MPa (Jaeger and Cook, 1979; Ingraffia,1987) for coherent rock, and somewhat smaller if the porespaces in the crust were largely filled with cold ice. Thusthese early gas-free melts are far more likely to be intruded,

either as dikes or sills, somewhere within the outer severalkm of the crust than to be erupted at the surface.

To estimate the volume of these early intrusions we uti-lize the volumes of the spherical regions representing thevarious pressure levels defined in Section 3. Adding 2.5%of the 10.92105 km3 volume of the 6.5 MPa sphericalshell to 1.1% of the 9.82 105 km3 volume of the 10 MPaspherical shell we find a total intruded volume of38,100 km3, which is 3.8% of the total melt volume(almost exactly 1106 km3) produced over the entire030% melting range. If this volume were intruded uni-formly at a depth of, say, 7 km, it would form a global sill350 m thick. However, as we showed in Section 4.4, theinternal structure of the asteroid is likely to evolve to pro-duce a relatively small number (5) of focused volcaniccenters. We therefore expect that this same number of earlyintrusions will form, each containing 70008000 km3 ofmelt. We address their subsequent evolution below.

Once this short initial stage is over, however, it does be-come the case that gas production is taking place some-where in the asteroid at all times (i.e. at the 3 MPa level itbegins simultaneously with melting at 1323 K, eventhough at deeper levels it is delayed). Thus, for the bulkof the asteroids melting history, when the remaining96.2% of the total melt is being produced, melts from deepparts of the asteroid where gas production has not yet

Fig. 13. The speeds at which melts would be erupted at the surfacefrom each of the 3, 6.5 and 10 MPa pressure levels in the UPB as afunction of the amount of partial melting that has taken place,during the period in which smelting is taking place. In all cases the

speed greatly exceeds the escape velocity.



17/23

started would be expected to mix, as they rise, with gas-con-taining melts being produced in shallower parts of thebody. The calculations underlyingFigs. 2 and 3 allow usto find the fraction of the melt from each pressure level thatis produced before smelting starts during this period. Mul-tiplying each of these fractions by the volume of the zonecentered on the appropriate pressure level from which it

comes yields the amount of melt produced without smelt-ing. The values found are 10% of the 2.83 105 km3 of meltfrom the 9.80105 km3 volume of the 3 MPa zone, 53% ofthe 3.15105 km3 of melt from the 10.92105 km3 vol-ume of the 6.5 MPa zone, 79% of the 2.83 105 km3 ofmelt from the 9.82105 km3 volume of the 10 MPa zone,and 100% of the 0.91105 km3 of melt from the3.15105 km3 volume of the central zone. Thus the totalvolume of melt produced without smelting during thisphase is 5.1105 km3. The thermal calculations of Sec-tion 3 showed that 15% of this melt, 0.77105 km3,was retained in the sill intrusions, the rest being lost intospace. The mixing of this retained melt with the

0.3810

5

km

3

volume of melt intruded into the sill beforeany smelting started in the asteroid yields a total of1.15105 km3. Since the total volume of sill magma was1.52105 km3, this implies that 76% of the melt inthe sill was derived from source regions in which smeltingdid not take place. This result is consistent with oxygen iso-topic data for feldspathic clasts in polymict ureilites, whichshow a strong bias toward material derived from deeper,more ferroan source regions (Kita et al., 2004, 2006).

The average gas content of the mixture of melt and gasbeing produced within the UPB can be found, as a functionof the temperature history during melting, by combiningthe gas and melt production functions at all depths withinthe asteroid. The temperature-dependent patterns of gas

production at the 3, 6.5 and 10 MPa levels ( Fig. 3) wereused to interpolate the corresponding functions at interme-diate pressure levels at 1 MPa intervals and the masses ofmelt and gas produced in each 5 K temperature intervalwere summed separately. The ratio of these summed massesgives the mean gas content of melt reaching the surface, andFig. 14shows the result. The production functions of boththe melt and the CO gas depend non-linearly on tempera-ture, and this is reflected in the very non-uniform variationof mean gas content with temperature, and hence with time,values varying between extremes of 15 and 35 wt% overmuch of the melting temperature range.

This generally large mass fraction of gas in the risingmelt has fundamental consequences for the way the twocomponents, gas and liquid, interact as they rise towardthe surface through the network of interconnected veinsthat develops in the interior of the asteroid to feed dikespenetrating through the unmelted crust to the surface.The relatively low pressures in the asteroid interior meanthat the gas occupies a large fraction of the volume of thevein and dike network. In volcanic systems on planets aslarge as Mars, Venus and Earth, the pressures are such that,except very near the surface, exsolved gases form bubblesdistributed relatively uniformly throughout the liquid; thegas volume fraction is much less than the 7090% rangeover which foams of bubbles distributed in liquids become

unstable (Jaupart and Vergniolle, 1989). In contrast, the gasvolume fraction represented by 15 and 35 wt% of gas in amelt at pressures in the range 310 MPa is readily shown,by evaluating the gas volume using the perfect gas law, torange between 0.95 and more than 0.99. Thus foam stabilityis violated at essentially all depths, and the fluid cannot flowas a uniform dispersion of bubbles distributed throughoutthe liquid; the mode of transport of the fluid in veins anddikes in the UPB must consist of some kind of two-phaseflow (Wallis, 1969). The basic options are either slug flow,

in which a series of long, closely-spaced gas bubbles, eachnearly as wide as the available pathway, move upward in-side a thin film of liquid that slides up the wall, or annularflow, in which a continuous central gas stream replaces theslugs. Intermediates between the various phases exist(Wallis, 1969), and in the case of the UPB, where new gasbubbles are constantly being created by smelting, pure slugflow is impossible instead bubbly slug flow, in which theliquid surrounding the slugs still contains some gas bubbles,is required. Similarly pure annular flow is unlikely, becausethe formation, shearing and rupturing of newly formedbubbles will make the interface between the gas core andthe liquid film on the walls unstable, and the drop-annular,or annular mist, flow mode, in which liquid droplets are en-trained in the gas core, is more likely. In the deep interior,where Section 4.3 shows that the mean vein size is verysmall, surface tension effects should ensure that liquid filmswill intermittently break the continuity of the central gasstream if any trend toward annular flow occurs, and sothe transport mode will be bubbly slug flow.

However, in the largest veins and pathways into whichmelt is focused, the flow mode is much more likely to beannular flow. This is particularly true of the final path tothe surface in a dike breaching the cold unaltered crust.We have already shown that the small amount of gas-freemelt generated in the deep parts of the asteroid before

Fig. 14. The variation of the proportion of gas in the mixture ofmelts and gas from all levels within the UPB as a function ofincreasing temperature as melting progresses up to 30%. Thescatter of points reflects all of the accumulated errors in the

methods used to calculate the amounts of liquid and gas producedin the asteroid.



18/23


19/23

intrusion inflates, the inflow speed will decrease and thepressure will rise toward the 1215 MPa level consistentwith the magma buoyancy. At some point the crust overly-ing the intrusion fails and an eruption starts. Magma ini-tially flows to the surface at a rate consistent with theexcess pressure in the reservoir; as this pressure decreasesthe buoyancy of the melt-gas mixture takes over control

of the eruption rate. The gas bubbles in the melt expandas the pressure in the reservoir decreases and continue todo so, driving out both gas and melt, until either the pres-sure in the intrusion decreases to the ambient lithostaticload,1 MPa at the roof of the intrusion, or the gas bubblevolume fraction in the melt reaches the critical value atwhich the foam collapses and gas escapes without furthersignificant liquid loss. If the critical bubble volume fractionfor foam collapse isfcand gas mass fraction isn, it is readilyshown that the fraction of the melt that is retained at theend of an eruptive event, mr, is equal to [(1fc)/(1n)].Fig. 15 shows the variation of mr with n for three valuesoffccovering the 0.70.9 range generally judged to be rele-

vant to magmatic foams (Jaupart and Vergniolle, 1989).With an uncertainty of about a factor of 2, we expect that25% of all of the melt from the interior of the asteroid willbe retained in the crustal intrusions, a total volume of253,000 km3. This result may be compared with the find-ing from the thermal modeling of the asteroid interior de-scribed in Section3where it was estimated that 15% ofthe melt was retained in the intrusion. The estimates fromthese two approaches would agree exactly if the criticalbubble volume fraction for foam collapse were 0.9, at theupper end of the likely range.

5.3. Loss of asteroid mass due to explosive volcanism

We now address the concern raised by Warren andHuber (2006)that the loss of asteroid mass due to explosive

volcanism on the UPB would have led to a diminution inpressure that would have caused a runaway smelting pro-cess (although there is no evidence for this in ureilites). If75% of the melt is lost through explosive eruptions in theway described in the previous section, this corresponds to17% of the asteroid volume, which produces a reductionin its radius from 100 to94 km. This6% change in body

radius due to the loss of the

Date post:	04-Jun-2018
Category:	Documents
Upload:	clarklipman
View:	216 times
Download:	0 times

Wilson Et Al. (2008) Thermal Evolution and Physics of Melt Extraction on the Ureilite Parent Body

Documents