Mixing and Transport of Short-Lived and Stable Isotopes and Refractory Grains in Protoplanetary Disks
aa r X i v : . [ a s t r o - ph . E P ] J un Mixing and Transport of Short-Lived and Stable Isotopes andRefractory Grains in Protoplanetary Disks
Alan P. Boss
Department of Terrestrial Magnetism, Carnegie Institution for Science, 5241 Broad BranchRoad, NW, Washington, DC 20015-1305 [email protected]
ABSTRACT
Analyses of primitive meteorites and cometary samples have shown that thesolar nebula must have experienced a phase of large-scale outward transport ofsmall refractory grains as well as homogenization of initially spatially heteroge-neous short-lived isotopes. The stable oxygen isotopes, however, were able toremain spatially heterogenous at the ∼
6% level. One promising mechanism forachieving these disparate goals is the mixing and transport associated with amarginally gravitationally unstable (MGU) disk, a likely cause of FU Orionisevents in young low-mass stars. Several new sets of MGU models are presentedthat explore mixing and transport in disks with varied masses (0.016 to 0.13 M ⊙ )around stars with varied masses (0.1 to 1 M ⊙ ) and varied initial Q stability min-ima (1.8 to 3.1). The results show that MGU disks are able to rapidly (within ∼ yr) achieve large-scale transport and homogenization of initially spatiallyheterogeneous distributions of disk grains or gas. In addition, the models showthat while single-shot injection heterogeneity is reduced to a relatively low level( ∼ ∼ Subject headings: accretion, accretion disks — hydrodynamics — instabilities —planets and satellites: formation 2 –
1. Introduction
The short-lived radioisotope (SLRI) Al was alive during the formation of the firstrefractory solids in the solar nebula, the Ca-, Al-rich inclusions (CAIs) found in primitivechondritic meteorites. This means that at least some of the solar system’s SLRIs may havebeen injected into either the presolar cloud (e.g., Boss & Keiser 2012; Boss 2012) or thesolar nebula (Ouellette et al. 2007, 2010; Dauphas & Chaussidon 2011) by a supernovaor AGB star shock wave. In either case, injection occurred as a single event that wasspatially heterogeneous, which would potentially reduce the usefulness of Al as a spatiallyhomogeneous chronometer (Dauphas & Chaussidon 2011) for precise studies of the earliestphases of planet formation (MacPherson et al. 2012; cf. Krot et al. 2012). Previous models(e.g., Boss 2011, 2012) have shown how such initial spatial isotopic heterogeneity can besubstantially reduced in a marginally gravitationally unstable (MGU) disk, as a result ofthe large-scale inward and outward transport and mixing of gas and particles small enoughto move with the gas (e.g., Boss et al. 2012). Other elements and their isotopes suggest asimilarly well-mixed solar nebula (e.g., Os: Walker 2012; Fe: Wang et al. 2013). The stableoxygen isotopes, on the other hand, appear to have been spatially heterogeneous in thesolar nebula during the early phases of planet formation; e.g., small refractory particles fromComet 81P/Wild 2 have normalized , O/ O ratios that span the entire solar system rangeof ∼
6% variations (Nakashima et al. 2012). The leading explanation for generating theseoxygen anomalies is UV photodissociation of CO molecules at the surface of the outer solarnebula (e.g., Podio et al. 2013), where self-shielding could lead to isotopic fractionationbetween gas-phase and solid-phase oxygen atoms (e.g., Lyons & Young 2005; Krot et al.2012). CO self-shielding on the irregular, corrugated outer surface of the disk would alsolead to initial spatial heterogeneity, though the process would be continuous in time, ratherthan a single-shot event like a supernova shock wave. Furthermore, the very existenceof refractory particles in Comet 81P/Wild 2 (Brownlee et al. 2006; Simon et al. 2008;Nakamura et al. 2008), which are thought to have formed close to the protosun, implies thatthese small particles experienced large-scale outward transport from the inner solar nebula tothe comet-forming regions of the outer solar nebula. MGU disks offer a means to accomplishthis early large-scale transport (e.g., Boss 2008, 2011; Boss et al. 2012).Marginally gravitationally unstable disks are likely to be involved in the FU Orionisoutbursts experienced by young solar-type stars (e.g., Zhu et al. 2010b; Vorobyov & Basu2010; Martin et al. 2012). MGU disk models (e.g., Boss 2011) can easily lead to the high massaccretion rates ( ∼ − M ⊙ yr − ) needed to explain FU Orionis events. FU Orionis outburstsare believed to last for about a hundred years and to occur periodically for all low massprotostars (Hartmann & Kenyon 1996; Miller et al. 2011). MGU models are also capableof offering an alternative mechanism (disk instability) for gas giant planet formation (e.g., 3 –Boss 2010; Meru & Bate 2012; Basu & Vorobyov 2012). However, the magnetorotationalinstability (MRI) is likely to be involved in FU Orionis outbursts as well (Zhu et al. 2009a),with MRI operating in the ionized innermost disk layers as well as at the disk’s surfaces. Zhuet al. (2009c, 2010a,b) have constructed one- and two-dimensional (axisymmetric) modelsof a coupled MGU-MRI mechanism, with MGU slowly leading to a build-up of mass in theinnermost disk, which then triggers a rapid MRI instability and an outburst. Alternatively,MRI may operate in the outermost disk, partially ionized by cosmic rays, leading to a build-up of mass in the dead zone at the intermediate disk midplane, thus triggering a phaseof MGU transport. Such a coupled mechanism may be crucial for achieving outbursts inT Tauri disks, where the disk masses are expected to be smaller than at earlier phases ofevolution.We present here several new sets of MGU disk models that examine the time evolutionof isotopic heterogeneity introduced in either the inner or outer solar nebula, by either asingle-shot event or a continuous injection process, for a variety of disk and central protostarmasses, including protostars with M dwarf masses. Low mass exoplanets are beginning to bediscovered around an increasingly larger fraction of M dwarfs (Bonfils et al. 2013; Dressing& Charbonneau 2013; Kopparapu 2013), with a number of these being potentially habitableexoplanets, elevating the importance of understanding mixing and transport processes in Mdwarf disks.
2. Numerical Methods
The numerical models were computed with the same three dimensional, gravitationalhydrodynamics code that has been employed in previous MGU disk models (e.g., Boss 2011).Complete details about the code and its testing may be found in Boss & Myhill (1992).Briefly, the code performs second-order-accurate (in both space and time) hydrodynamicson a spherical coordinate grid, including radiative transfer in the diffusion approximation. Aspherical harmonic ( Y lm ) expansion of the disk’s density distribution is used to compute theself-gravity of the disk, with terms up to and including l = m = 32. The radial grid contains50 grid points for the 10 AU disk models and 100 grid points for the 40 AU disk models.All models have 256 azimuthal grid points, and effectively 45 theta grid points, given thehemispherical symmetry of the grid. The theta grid is compressed around the disk’s midplaneto provide enhanced spatial resolution, while the azimuthal grid is uniformly spaced. TheJeans length constraint is used to ensure adequate resolution. The inner boundary absorbsinfalling disk gas, which is added to the central protostar, while the outer disk boundaryabsorbs the momentum of outward-moving disk gas, while retaining the gas on the active 4 –grid. The central protostar wobbles in such a manner as to preserve the center of mass ofthe entire system.The time evolution of a color field is calculated (e.g., Boss 2011) in order to follow themixing and transport of isotopes carried by the disk gas or by small particles, which shouldmove along with the disk gas. The equation for the evolution of the color field density ρ c (e.g., Boss 2011) is identical to the continuity equation for the disk gas density ρ∂ρ c ∂t + ∇ · ( ρ c v ) = 0 , where v is the disk gas velocity and t is the time. The total amount of color is conserved inthe same way that the disk mass is conserved, as the hydrodynamic equations are solved inconservation law form (e.g., Boss & Myhill 1992).
3. Initial Conditions
The initial disk density distributions are based on the approximation derived by Boss(1993) for a self-gravitating disk orbiting a star with mass M s ρ ( R, Z ) γ − = ρ o ( R ) γ − − (cid:16) γ − γ (cid:17)h(cid:16) πGσ o ( R ) K (cid:17) Z + GM s K (cid:16) R − R + Z ) / (cid:17)i , where R and Z are cylindrical coordinates, G is the gravitational constant, ρ o ( R ) is themidplane density, σ o ( R ) is the surface density, K = 1 . × (cgs units) and γ = 5 /
3. Theinitial midplane density is ρ o ( R ) = ρ oi (cid:16) R oi R (cid:17) / , while the initial surface density is σ o ( R ) = σ oi (cid:16) R oi R (cid:17) / . The parameters ρ oi and σ oi and the reference radius R oi are defined in Table 1 for the variousdisk models explored in this paper. The total amount of mass in the models does not change 5 –during the evolutions; the initial infalling disk envelope accretes onto the disk, and no furthermass is added to the system across the outer disk boundary at R o . The outer disk surfacesare thus revealed to any potential source of UV irradiation.For the 10 AU outer radius disks listed in Tables 2 and 3, the initial disk temperatureprofiles (Figures 1 and 2) are based on the Boss (1996) temperature profiles, with variations inthe assumed outer disk temperature T o , chosen in order to study the effect of varied minimumvalues of the Q stability parameter. Values of Q > . Q stable, with Q >>
1. For the 40 AU outerradius disks listed in Table 4, the initial disk temperatures are uniform at the specified outerdisk temperature T o , leading to similar initial Q values throughout the disks. For all of themodels, the temperature of the infalling envelope is 50 K.The initial color field is added to the surface of the initial disk in an azimuthal sectorspanning either 45 degrees (10 AU outer radius disks) or 90 degrees (40 AU outer radius disks)in a narrow ring of width 1 AU, centered at the injection radii listed in the Tables. Thesemodels are intended to represent one-time, single-shot injections of isotopic heterogeneity,such as supernova-induced Rayleigh-Taylor fingers carrying live Al (e.g., Boss & Keiser2012). Table 4 lists both single-shot and continuous injection models, where in the lattercase the color is added continuously to the same location on the disk surface throughout theevolution, crudely simulating ongoing photodissociation of CO (e.g., Lyons & Young 2005)possibly leading to stable oxygen isotope fractionation between the gas and solid phases. Thecolor field in the latter case is intended to represent isotopically distinct gas or small particlesresulting from the UV photochemistry. Note that in both the single-shot and continuousinjection models, the total amount of color added is arbitrary (e.g., the color field in theinjection volume is simply set equal to 1), and is intended to be scaled to whatever valueis appropriate for the isotope(s) under consideration. The color field is a massless, passivetracer that has no effect on the disk’s dynamics, so the total amount of color added isirrelevant for the disk’s subsequent evolution. The models seek to follow the deviations fromuniformity of the color field, not the absolute amounts of color added; the evolution of thedispersion of the color field about its mean radial value, divided by the mean radial value ateach instant of time, is the goal of these models.Observations of the DG Tau disk by Podio et al. (2013) have shown that DG Tau itselfirradiates its disk’s outer layers from 10 AU to 90 AU with a strong UV flux, sufficient forsignificant UV photolysis and the formation of observable water vapor. Much higher levelsof UV irradiation can occur for protoplanetary disks that form in stellar clusters containingmassive stars (e.g., Walsh et al. 2013), an environment that has been suggested for our ownsolar system (e.g., Dauphas & Chaussidon 2011) in order to explain the evidence for live 6 –SLRIs found in primitive meteorites.The fact that molecular hydrogen constitutes the great majority of a disk’s mass, yetcannot be directly detected, except at the star-disk boundary region, means that estimatesof disk masses are uncertain at best (e.g., Andrews & Williams 2007), as they are typicallybased on an assumed ratio between the amount of mm-sized dust grains and the total diskmass. Isella et al. (2009) estimated that low- and intermediate-mass pre-main-sequence starsform with disk masses ranging from 0.05 to 0.4 M ⊙ . DG Tau’s disk has a mass estimatedto be as high as 0.1 M ⊙ (Podio et al. 2013). Recently, the mass of the TW Hydra diskwas revised upward to at least 0.05 M ⊙ (Bergin et al. 2013). These and other observationssuggest that the MGU disk masses assumed in these models may be achieved in some fractionof protoplanetary disks, and perhaps in the solar nebula as well. In fact, Miller et al. (2011)detected a FU Orionis outburst in the classical T Tauri star LkH α < . M ⊙ for such stars. The fact that a FU Orionis event occurredin LkH α
4. Results
We present results for a variety of protostellar and protoplanetary disk masses, variedinitial minimum Q stability parameters, and varied injection radii, for disks of two differentsizes. Table 2 shows the initial conditions for the models with a 0 . M ⊙ disk in orbit arounda 1 . M ⊙ protostar. The disks extend from 1 AU to 10 AU, as in the models by Boss (2008,2011). The main difference from these previous models is that the disk mass (0 . M ⊙ )is considerably lower than that of the previous models (0 . M ⊙ ). As a result, the initialminimum Q values are considerably higher than in the previous models, ranging from 2.2to 3.1, compared to the previous range of 1.4 to 2.5. The present models are thus lessgravitationally unstable initially than the disks previously considered, with the goal beingto learn whether or not the previous results will change for higher values of min Q i . Figure1 displays the initial midplane temperature profiles for these models. Only the outermostregions of the disks are cool enough to be gravitationally unstable, but the models show that 7 –this is sufficient to result in qualitatively similar behavior for all of the Table 2 models.Figures 3 and 4 show the equatorial plane distribution of the color/gas ratio ( ρ c /ρ ) formodel 1.0-2.6-9. This ratio is plotted, as it is equivalent to the Al/ Al and , O/ Oratios measured by cosmochemists, i.e., the abundance of an injected or photolysis productspecies, divided by that of a species that was prevalent in the pre-injection disk. Figure 3shows that the initial disk surface injection at 9 AU has resulted in the rapid transport ofthe color field downward to the disk’s midplane, as well as inward to close to the inner diskboundary at 2 AU. The vigorous three dimensional motions of a MGU disk are responsiblefor this large-scale transport in just 34 yr. At this time, the color/gas ratio is still highlyheterogeneous, but Figure 4 shows that only 146 yr later, the color/gas ratio has beenstrongly homogenized throughout the entire disk midplane.Figure 5 shows the evolution of the dispersion of the ratio of the color density to thegas density for models 1.0-2.6-9 and 1.0-2.6-2 at two times. These models differ only inthe injection radius, either 9 AU or 2 AU. The dispersion plotted in Figure 5 is defined tobe the square root of the sum of the squares of the color field divided by the gas density,subtracted by the azimuthal average of this ratio at a given orbital radius, divided by thesquare of the azimuthal average at that radius, normalized by the number of azimuthal gridpoints, and plotted as a function of radius in the disk midplane. Figure 5 shows that theisotopic dispersion is a strong function of orbital radius and time, with the dispersion initiallybeing relatively large (i.e., at 180 yr, in spite of the apparent homogeneity seen in Figure 4at the same time) as a result of the isotopes traveling downward and radially inward andoutward. However, the dispersion decreases dramatically in the outer disks for both modelsby 777 yr to a value of ∼
1% to 2%. In fact, the dispersion in both models 1.0-2.6-9 and1.0-2.6-2 evolves toward essentially the same radial distribution by this time, showing thatthe exact location of the injection location has little effect on the long term evolution of thedistribution: that is controlled solely by the evolution of the underlying MGU disk, which isidentical for these two models (i.e., the color fields are passive tracers, and have no effect onthe disk’s evolution). Note that any small refractory grains present in the initial disk willbe carried along with the disk gas, so that some of the grains that start out at 2 AU will betransported to the outermost disk, in the same manner that some of the gas is transportedoutward. Most of the gas and dust, however, is accreted by the growing protostar.Figure 6 shows the results for three models with varied min Q i , i.e., models 1.0-2.6-9,1.0-2.9-9, and 1.0-3.1-9, all after 1370 yr. It can be seen that in spite of the variation inthe initial degree of instability, the dispersions in the outermost disks all converge to similarvalues of ∼
1% to 2%. This suggests that MGU disk evolutions are not particularly sensitiveto the exact choice of the initial Q profile, a result that was also found by Boss (2011) for 8 –somewhat more massive disks. As also found by Boss (2011), the dispersions in the innermostdisks are significantly higher ( ∼
10% to 20%) than in the outermost disks, a direct resultof the stronger mixing associated with the cooler outer disks, in spite of the longer orbitalperiods in the outer disks.
Table 3 shows the initial conditions for the models with either 0 . M ⊙ disks around0 . M ⊙ protostars, or 0 . M ⊙ disks around 0 . M ⊙ protostars. In either case, the disksextend from 1 AU to 10 AU. These models are of interest for exploring how conditionsmight vary between disks around G dwarfs and M dwarfs, with possible ramifications for thehabitability of any rocky planets that form (e.g., Raymond et al. 2007) around M dwarfs.Figure 2 shows the initial midplane temperature profiles for these models.Figure 7 shows the time evolution of the dispersion for model 0.1-1.8-2, appropriate fora late M dwarf protostar. As in all the models, it can be seen that the initially highly het-erogeneous disk becomes rapidly homogenized, in this case by about 5000 yr. Note that thistime scale is considerably longer than that for G dwarf disk mixing and transport processes,as a result of the longer Keplerian orbital periods for lower mass, M dwarf protostars. Asin the G dwarf protostar models (e.g., Figure 5), the inner disk dispersion is higher thanin the outer disk, though in these models (with a lower initial min Q = 1 .
8) the inner diskdispersion drops to ∼
5% to 10%, compared to ∼
1% to 2% in the outer disk. Figure 8shows the same behavior for model 0.1-1.8-9, which differs from the previous model shownin Figure 7 only in having the injection occur at 9 AU instead of 2 AU. As in the G dwarfdisks, the dispersions for both of these models evolve toward essentially identical radial dis-tributions: the underlying MGU disk evolution determines the outcome for the dispersions.Similar results hold for the models with 0 . M ⊙ protostars, i.e., early M dwarf disks. We now turn to a consideration of the consequence of single-shot versus continuousinjection at the surface of much larger outer radius (40 AU) disks than have been consideredto date for G dwarf stars; Boss (2007) considered disks extending from 4 AU to 20 AU inradius. Table 4 shows the initial conditions for the models with a 0 . M ⊙ disk around a1 . M ⊙ protostar, with the disks extending from 10 AU to 40 AU. Because of the much largerinner and outer disk radii for this set of models, these models can be calculated for times 9 –as long as ∼ × yr (Table 4). Such times are still considerably less than the typicalages ( ∼ yr) of T Tauri stars, implying that in order for MGU disks to occur at such latephases, a prior phase of coupled MRI-MGU evolution might be required to make the presentresults relevant.Figures 9 and 10 display the evolution of the dispersions for models 1.0-1.1-40-20 and 1.0-1.1-40-20c, differing only in that the former model has single-shot injection while the lattermodel has continuous injection, intended to simulate a disk with ongoing UV photolysisand fractionation at the outer disk surface. For model 1.0-1.1-40-20, it can be seen thatthe evolution is similar to that of the previous single-shot models: a rapid drop in thedispersion, followed by homogenization to ∼
1% to 2% away from the inner disk boundary.The higher dispersions seen near the outer disk boundaries ( ∼
40 AU) are largely causedby the unphysical pile-up of considerable disk mass at 40 AU and should be discounted.However, for the continuous injection model shown in Figure 10, it can be seen that thedispersions throughout the disk even after ∼ yr can be as high as ∼ ∼
90 compared to the single-shot total, and by another factorof ∼
60 after 27000 yr.Figures 11 and 12 compare the results for continuous injection at either 20 AU or 30AU, respectively, i.e., for models 1.0-1.1-40-20c and 1.0-1.1-40-30c. In spite of the differentinjection radii, Figures 11 and 12 show that even at a relatively early phase (405 yr) ofevolution, the midplane color/gas ratios look somewhat similar; as before, the MGU diskevolution is the same for both models, and that is the primary determinant of the long termevolution.Finally, similar results as those shown in Figures 9-12 were obtained for the other modelslisted in Table 4. These models show that the main factor in determining the radial dispersionprofile is whether the injection occurs in a single-shot or continuously; in the latter case, theMGU disk does its best to homogenize the color field, but the fact that spatial heterogeneityis being continuously injected limits the degree to which this heterogeneity can be reduced. 10 –
5. Discussion
While dust grains in the interstellar medium are overwhelmingly amorphous, crystallinesilicate grains have been found in a late M dwarf (SST-Lup3-1) disk at distances rangingfrom inside 3 AU to beyond 5 AU, in both the midplane and surface layers (Mer´ın et al.2007). Such crystalline silicate grains are likely to have been produced by thermal annealingin the hottest regions of the disk, well inside of 1 AU (Sargent et al. 2009). Again, outwardtransport seems to be required to explain the observations, and the results for the modelswith a 0.1 M ⊙ protostar suggest that MGU phases in low mass M dwarf disks may beresponsible for these observations. In fact, crystalline mass fractions in protoplanetary disksdo not appear to correlate with stellar mass, luminosity, accretion rate, disk mass, or thedisk to star ratio (Watson et al. 2009). These results also appear to be consistent withthe results of the present models, which show that MGU disk phases are equally capable ofrelatively rapid large-scale mixing and transport, regardless of the stellar or disk mass, orthe exact value of the Q stability parameter.
6. Conclusions
These models have shown a rather robust result, namely that a phase of marginalgravitational instability in disks and stars with a variety of masses and disk temperaturescan lead to relatively rapid inward and outward transport of disk gas and small grains, asrequired to drive the protostellar mass accretion associated with FU Orionis events, as wellas to explain the discovery of refractory grains in Comet 81P/Wild 2. A MGU disk phasedriving a FU Orionis outburst is astronomically quite likely to have occurred for our protosun,and cosmochemically convenient for explaining the relative homogeneity of Al/ Al ratiosderived from a supernova injection event, and the range of , O/ O ratios derived fromsustained UV self-shielding at the surface of the outer solar nebula. Low-mass stars, from Gdwarfs to M dwarfs, may well experience a similar phase of MGU disk mixing and transport.In this context, it is worthwhile to note that FU Orionis itself, the prototype of the FUOrionis outburst phenomenon, has a mass of only ∼ . M ⊙ (Zhu et al. 2007, 2009b; Beck& Aspin 2012), i.e., the mass of a M dwarf, suggesting that M dwarf protoplanetary disksmay experience evolutions similar to that of the solar nebula, with possible implications forthe habitability of any resulting planetary system (e.g., Raymond et al. 2007; Bonfils et al.2013; Dressing & Charbonneau 2013; Kopparapu 2013).I thank Jeff Cuzzi for his comments, the referee for a number of suggested improve-ments, and Sandy Keiser, Michael Acierno, and Ben Pandit for their support of the cluster 11 –computing environment at DTM. This work was partially supported by the NASA Originsof Solar Systems Program (NNX09AF62G) and is contributed in part to the NASA Astro-biology Institute (NNA09DA81A). Some of the calculations were performed on the CarnegieAlpha Cluster, the purchase of which was partially supported by a NSF Major ResearchInstrumentation grant (MRI-9976645). REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
13 –Table 1. Density ( ρ oi ) and surface density ( σ oi ) parameters at a reference radius ( R oi ) forvaried disk masses ( M d ) in orbit around varied mass protostars ( M s ). R i and R o are theinner and outer disk boundaries, respectively. M s ( M ⊙ ) M d ( M ⊙ ) ρ oi σ oi R oi (AU) R i (AU) R o (AU)1.0 0.019 4 . × − . × . × − . × . × − . × . × − . × t f ) for models with a 10 AU outer radius,0 . M ⊙ disk in orbit around a 1 . M ⊙ protostar. T o is the outer disk temperature, min Q i is the minimum value of the initial Q disk parameter, and r inject is the radius where thecolor is injected at the disk’s surface.model T o (K) min Q i r inject (AU) t f (yr)1.0-2.2-2 15 2.2 2 25201.0-2.2-9 15 2.2 9 25201.0-2.6-2 20 2.6 2 20431.0-2.6-9 20 2.6 9 20431.0-2.9-2 25 2.9 2 12001.0-2.9-9 25 2.9 9 14001.0-3.1-2 30 3.1 2 11001.0-3.1-9 30 3.1 9 1300 15 –Table 3. Initial conditions and final times for models with 10 AU outer radius disks ( M d )in orbit around lower mass protostars ( M s ), as in Table 2.model M s ( M ⊙ ) M d ( M ⊙ ) T o (K) min Q i r inject (AU) t f (yr)0.1-1.8-2 0.1 0.016 40 1.8 2 80400.1-1.8-9 0.1 0.016 40 1.8 9 80400.5-2.4-2 0.5 0.018 25 2.4 2 46600.5-2.4-9 0.5 0.018 25 2.4 9 4660 16 –Table 4. Initial conditions and final times for models with a 40 AU outer radius, 0 . M ⊙ disk in orbit around a 1 . M ⊙ protostar, as in Table 2.model T o (K) min Q i r inject (AU) injection mode t f (yr)1.0-1.1-40-20 30 1.1 20 single-shot 250001.0-1.1-40-30 30 1.1 30 single-shot 245001.0-1.4-40-20 50 1.4 20 single-shot 240001.0-1.4-40-30 50 1.4 30 single-shot 150001.0-1.1-40-20c 30 1.1 20 continuous 195001.0-1.1-40-30c 30 1.1 30 continuous 198001.0-1.4-40-20c 50 1.4 20 continuous 270001.0-1.4-40-30c 50 1.4 30 continuous 27000 17 –Fig. 1.— Initial midplane temperature distributions (Boss 1996) for models 1.0-2.2-2 and1.0-2.2-9 (blue), 1.0-2.6-2 and 1.0-2.6-9 (red), 1.0-2.9-2 and 1.0-2.9-9 (green), and 1.0-3.1-2and 1.0-3.1-9 (black). 18 –Fig. 2.— Initial midplane temperature distributions (Boss 1996) for models 0.1-1.8-2 and0.1-1.8-9 (red) and 0.5-2.4-2 and 0.5-2.4-9 (black). 19 –Fig. 3.— Logarithm of the ratio of the color field to the gas density (log ρ c /ρ ) in the disk’smidplane (arbitrary units) after 34 yr for model 1.0-2.6-9, with single-shot color injection atthe initial disk’s surface at 9 AU. Region shown is 20 AU in diameter. A 1.0 M ⊙ protostarlies at the center of the MGU disk. The color field has been transported inward to close tothe inner boundary at 1 AU. The color to gas ratio is still highly heterogeneous. 20 –Fig. 4.— Same as Figure 1, but after 180 yr. The color field has been transported throughoutthe disk and the color to gas ratio is now relatively homogeneous. 21 –Fig. 5.— Time evolution of the azimuthal dispersion of the color to gas ratio as a function ofdisk radius for models with single-shot surface injection at either 9 AU or 2 AU, respectively:model 1.0-2.6-9 (red at 180 yr, blue at 777 yr) and model 1.0-2.6-2 (black at 180 yr, greenat 777 yr). While initially quite different, the dispersions converge by 777 yr. 22 – Fig. 6.— Comparisons of the azimuthal dispersions after 1370 yr of the color to gas ratio asa function of disk radius, for models with single-shot surface injection at 9 AU, but variedinitial minimum Q values of 2.6, 2.9, and 3.1: models 1.0-2.6-9 (red), 1.0-2.9-9 (green), and1.0-3.1-9 (blue), respectively. The outer disks have become relatively homogenized. 23 –Fig. 7.— Time evolution of the azimuthal dispersion of the color to gas ratio as a function ofdisk radius for model 0.1-1.8-2, with single-shot injection at 2 AU and a 0.1 M ⊙ protostar,at times: black - 71 yr, red - 320 yr, green - 430 yr, blue - 5300 yr, and cyan - 8040 yr. 24 –Fig. 8.— Time evolution of the azimuthal dispersion of the color to gas ratio as a function ofdisk radius for model 0.1-1.8-9, with single-shot injection at 9 AU and a 0.1 M ⊙ protostar,at times: black - 140 yr, red - 320 yr, green - 430 yr, blue - 5300 yr, and cyan - 8040 yr. 25 – Fig. 9.— Time evolution of the azimuthal dispersion of the color to gas ratio as a functionof disk radius for model 1.0-1.1-40-20, with single-shot injection at 20 AU on the surface of a40 AU radius disk around a 1.0 M ⊙ protostar, at times: black - 200 yr, red - 5200 yr, green- 12400 yr, and blue - 19700 yr. 26 –Fig. 10.— Time evolution of the azimuthal dispersion of the color to gas ratio as a functionof disk radius for model 1.0-1.1-40-20c, with continuous injection at 20 AU on the surfaceof a 40 AU radius disk around a 1.0 M ⊙ protostar, at times: black - 200 yr, red - 5200 yr,green - 12600 yr, and blue - 19600 yr. 27 –Fig. 11.— Log ρ c /ρ in the midplane after 405 yr for model 1.0-1.1-40-20c, with continuouscolor injection at the disk’s surface at 20 AU. Region shown is 80 AU in diameter. A 1.0 M ⊙⊙