document: the short - lived radioisotope ( slri ) @xmath4al was alive during the formation of the first refractory solids in the solar nebula , the ca- , al - rich inclusions ( cais ) found in primitive chondritic meteorites . 
   this means that at least some of the solar system s slris may have been injected into either the presolar cloud ( e.g. , boss & keiser 2012 ; boss 2012 ) or the solar nebula ( ouellette et al . 2007 , 2010 ; dauphas & chaussidon 2011 ) by a supernova or agb star shock wave . in either case , 
   injection occurred as a single event that was spatially heterogeneous , which would potentially reduce the usefulness of @xmath4al as a spatially homogeneous chronometer ( dauphas & chaussidon 2011 ) for precise studies of the earliest phases of planet formation ( macpherson et al . 
   2012 ; cf . 
   krot et al . 
   previous models ( e.g. , boss 2011 , 2012 ) have shown how such initial spatial isotopic heterogeneity can be substantially reduced in a marginally gravitationally unstable ( mgu ) disk , as a result of the large - scale inward and outward transport and mixing of gas and particles small enough to move with the gas ( e.g. , boss et al . 
   2012 ) . other elements and their isotopes suggest a similarly well - mixed solar nebula ( e.g. , os : walker 2012 ; fe : 
   wang et al . 2013 ) . 
   the stable oxygen isotopes , on the other hand , appear to have been spatially heterogeneous in the solar nebula during the early phases of planet formation ; e.g. , small refractory particles from comet 81p / wild 2 have normalized @xmath5o/@xmath6o ratios that span the entire solar system range of @xmath0 6% variations ( nakashima et al . 2012 ) . 
   the leading explanation for generating these oxygen anomalies is uv photodissociation of co molecules at the surface of the outer solar nebula ( e.g. , podio et al . 
   2013 ) , where self - shielding could lead to isotopic fractionation between gas - phase and solid - phase oxygen atoms ( e.g. , lyons & young 2005 ; krot et al . 
   co self - shielding on the irregular , corrugated outer surface of the disk would also lead to initial spatial heterogeneity , though the process would be continuous in time , rather than a single - shot event like a supernova shock wave . 
   furthermore , the very existence of 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 that these small particles experienced large - scale outward transport from the inner solar nebula to the comet - forming regions of the outer solar nebula . 
   mgu disks offer a means to accomplish this 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 orionis outbursts experienced by young solar - type stars ( e.g. , zhu et al . 2010b ; vorobyov & basu 2010 ; martin et al . 
   mgu disk models ( e.g. , boss 2011 ) can easily lead to the high mass accretion rates ( @xmath7 yr@xmath8 ) needed to explain fu orionis events . 
   fu orionis outbursts are believed to last for about a hundred years and to occur periodically for all low mass protostars ( hartmann & kenyon 1996 ; miller et al . 
   mgu models are also capable of offering an alternative mechanism ( disk instability ) for gas giant planet formation ( e.g. , boss 2010 ; meru & bate 2012 ; basu & vorobyov 2012 ) . 
   however , the magnetorotational instability ( 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 . 
   zhu et al . ( 2009c , 2010a , b ) have constructed one- and two - dimensional ( axisymmetric ) models of a coupled mgu - mri mechanism , with mgu slowly leading to a build - up of mass in the innermost 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 phase of mgu transport . 
   such a coupled mechanism may be crucial for achieving outbursts in t tauri disks , where the disk masses are expected to be smaller than at earlier phases of evolution . 
   we present here several new sets of mgu disk models that examine the time evolution of isotopic heterogeneity introduced in either the inner or outer solar nebula , by either a single - shot event or a continuous injection process , for a variety of disk and central protostar masses , including protostars with m dwarf masses . 
   low mass exoplanets are beginning to be discovered around an increasingly larger fraction of m dwarfs ( bonfils et al . 
   2013 ; dressing & charbonneau 2013 ; kopparapu 2013 ) , with a number of these being potentially habitable exoplanets , elevating the importance of understanding mixing and transport processes in m dwarf disks . 
   the numerical models were computed with the same three dimensional , gravitational hydrodynamics 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 ) hydrodynamics on a spherical coordinate grid , including radiative transfer in the diffusion approximation . 
   a spherical harmonic ( @xmath9 ) expansion of the disk s density distribution is used to compute the self - gravity of the disk , with terms up to and including @xmath10 . 
   the radial grid contains 50 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 the hemispherical symmetry of the grid . 
   the theta grid is compressed around the disk s midplane to provide enhanced spatial resolution , while the azimuthal grid is uniformly spaced . 
   the jeans length constraint is used to ensure adequate resolution . 
   the inner boundary absorbs infalling disk gas , which is added to the central protostar , while the outer disk boundary absorbs the momentum of outward - moving disk gas , while retaining the gas on the active grid . 
   the central protostar wobbles in such a manner as to preserve the center of mass of the entire system . 
   the time evolution of a color field is calculated ( e.g. , boss 2011 ) in order to follow the mixing and transport of isotopes carried by the disk gas or by small particles , which should move along with the disk gas . 
   the equation for the evolution of the color field density @xmath11 ( e.g. , boss 2011 ) is identical to the continuity equation for the disk gas density @xmath12    @xmath13    where @xmath14 is the disk gas velocity and @xmath15 is the time . 
   the total amount of color is conserved in the same way that the disk mass is conserved , as the hydrodynamic equations are solved in conservation law form ( e.g. , boss & myhill 1992 ) . 
   the initial disk density distributions are based on the approximation derived by boss ( 1993 ) for a self - gravitating disk orbiting a star with mass @xmath16    @xmath17 @xmath18,\ ] ]    where @xmath19 and @xmath20 are cylindrical coordinates , @xmath21 is the gravitational constant , @xmath22 is the midplane density , @xmath23 is the surface density , @xmath24 ( cgs units ) and @xmath25 . 
   the initial midplane density is    @xmath26    while the initial surface density is    @xmath27    the parameters @xmath28 and @xmath29 and the reference radius @xmath30 are defined in table 1 for the various disk models explored in this paper . 
   the total amount of mass in the models does not change during the evolutions ; the initial infalling disk envelope accretes onto the disk , and no further mass is added to the system across the outer disk boundary at @xmath31 . 
   the outer disk surfaces are 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 temperature profiles 
   ( figures 1 and 2 ) are based on the boss ( 1996 ) temperature profiles , with variations in the assumed outer disk temperature @xmath32 , chosen in order to study the effect of varied minimum values of the @xmath2 stability parameter . 
   values of @xmath33 indicate marginally gravitationally unstable disks . 
   the inner disks are all highly @xmath2 stable , with @xmath34 . 
   for the 40 au outer radius disks listed in table 4 , the initial disk temperatures are uniform at the specified outer disk temperature @xmath32 , leading to similar initial @xmath2 values throughout the disks . 
   for all of the models , 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 sector spanning 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 . 
   these models are intended to represent one - time , single - shot injections of isotopic heterogeneity , such as supernova - induced rayleigh - taylor fingers carrying live @xmath4al ( e.g. , boss & keiser 2012 ) . table 4 lists both single - shot and continuous injection models , where in the latter case the color is added continuously to the same location on the disk surface throughout the evolution , crudely simulating ongoing photodissociation of co ( e.g. , lyons & young 2005 ) possibly leading to stable oxygen isotope fractionation between the gas and solid phases . 
   the color field in the latter case is intended to represent isotopically distinct gas or small particles resulting from the uv photochemistry . 
   note that in both the single - shot and continuous injection models , the total amount of color added is arbitrary ( e.g. , the color field in the injection volume is simply set equal to 1 ) , and is intended to be scaled to whatever value is appropriate for the isotope(s ) under consideration . 
   the color field is a massless , passive tracer that has no effect on the disk s dynamics , so the total amount of color added is irrelevant for the disk s subsequent evolution . 
   the models seek to follow the deviations from uniformity of the color field , not the absolute amounts of color added ; the evolution of the dispersion of the color field about its mean radial value , divided by the mean radial value at each 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 itself irradiates its disk s outer layers from 10 au to 90 au with a strong uv flux , sufficient for significant uv photolysis and the formation of observable water vapor . 
   much higher levels of uv irradiation can occur for protoplanetary disks that form in stellar clusters containing massive stars ( e.g. , walsh et al . 
   2013 ) , an environment that has been suggested for our own solar system ( e.g. , dauphas & chaussidon 2011 ) in order to explain the evidence for live slris found in primitive meteorites . 
   the fact that molecular hydrogen constitutes the great majority of a disk s mass , yet can not be directly detected , except at the star - disk boundary region , means that estimates of disk masses are uncertain at best ( e.g. , andrews & williams 2007 ) , as they are typically based on an assumed ratio between the amount of mm - sized dust grains and the total disk mass . 
   isella et al . 
   ( 2009 ) estimated that low- and intermediate - mass pre - main - sequence stars form with disk masses ranging from 0.05 to 0.4 @xmath1 . 
   dg tau s disk has a mass estimated to be as high as 0.1 @xmath1 ( podio et al . 
   recently , the mass of the tw hydra disk was revised upward to at least 0.05 @xmath1 ( bergin et al . 
   2013 ) . these and other observations suggest that the mgu disk masses assumed in these models may be achieved in some fraction of 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@xmath35 188-g4 . 
   disk masses are typically thought to be @xmath36 for such stars . 
   the fact that a fu orionis event occurred in lkh@xmath35 188-g4 shows that even the disks around class ii - type objects can experience instabilities leading to rapid mass accretion , e.g. , mgu disk phases . 
   we present results for a variety of protostellar and protoplanetary disk masses , varied initial minimum @xmath2 stability parameters , and varied injection radii , for disks of two different sizes . 
   table 2 shows the initial conditions for the models with a @xmath37 disk in orbit around a @xmath38 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 ( @xmath37 ) is considerably lower than that of the previous models ( @xmath39 ) . as a result , 
   the initial minimum @xmath2 values are considerably higher than in the previous models , ranging from 2.2 to 3.1 , compared to the previous range of 1.4 to 2.5 . 
   the present models are thus less gravitationally unstable initially than the disks previously considered , with the goal being to learn whether or not the previous results will change for higher values of @xmath40 . 
   figure 1 displays the initial midplane temperature profiles for these models . 
   only the outermost regions of the disks are cool enough to be gravitationally unstable , but the models show that 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 ( @xmath41 ) for model 1.0 - 2.6 - 9 . 
   this ratio is plotted , as it is equivalent to the @xmath4al/@xmath42al and @xmath5o/@xmath6o ratios measured by cosmochemists , i.e. , the abundance of an injected or photolysis product species , divided by that of a species that was prevalent in the pre - injection disk . 
   figure 3 shows that the initial disk surface injection at 9 au has resulted in the rapid transport of the color field downward to the disk s midplane , as well as inward to close to the inner disk boundary at 2 au . 
   the vigorous three dimensional motions of a mgu disk are responsible for this large - scale transport in just 34 yr . at this time 
   , the color / gas ratio is still highly heterogeneous , but figure 4 shows that only 146 yr later , the color / gas ratio has been strongly homogenized throughout the entire disk midplane . 
   figure 5 shows the evolution of the dispersion of the ratio of the color density to the gas density for models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 at two times . 
   these models differ only in the injection radius , either 9 au or 2 au . 
   the dispersion plotted in figure 5 is defined to be 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 the square of the azimuthal average at that radius , normalized by the number of azimuthal grid points , and plotted as a function of radius in the disk midplane . 
   figure 5 shows that the isotopic dispersion is a strong function of orbital radius and time , with the dispersion initially being relatively large ( i.e. , at 180 yr , in spite of the apparent homogeneity seen in figure 4 at the same time ) as a result of the isotopes traveling downward and radially inward and outward . 
   however , the dispersion decreases dramatically in the outer disks for both models by 777 yr to a value of @xmath0 1% to 2% . 
   in fact , the dispersion in both models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 evolves toward essentially the same radial distribution by this time , showing that the exact location of the injection location has little effect on the long term evolution of the distribution : that is controlled solely by the evolution of the underlying mgu disk , which is identical for these two models ( i.e. , the color fields are passive tracers , and have no effect on the disk s evolution ) . 
   note that any small refractory grains present in the initial disk will be carried along with the disk gas , so that some of the grains that start out at 2 au will be transported to the outermost disk , in the same manner that some of the gas is transported outward . 
   most of the gas and dust , however , is accreted by the growing protostar . 
   figure 6 shows the results for three models with varied @xmath40 , 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 in the initial degree of instability , the dispersions in the outermost disks all converge to similar values of @xmath0 1% to 2% . 
   this suggests that mgu disk evolutions are not particularly sensitive to the exact choice of the initial @xmath2 profile , a result that was also found by boss ( 2011 ) for somewhat more massive disks . 
   as also found by boss ( 2011 ) , the dispersions in the innermost disks are significantly higher ( @xmath0 10% to 20% ) than in the outermost disks , a direct result of the stronger mixing associated with the cooler outer disks , in spite of the longer orbital periods in the outer disks . 
   table 3 shows the initial conditions for the models with either @xmath43 disks around @xmath44 protostars , or @xmath45 disks around @xmath46 protostars . in either case 
   , the disks extend from 1 au to 10 au . 
   these models are of interest for exploring how conditions might vary between disks around g dwarfs and m dwarfs , with possible ramifications for the habitability 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 for a late m dwarf protostar . as in all the models 
   , it can be seen that the initially highly heterogeneous disk becomes rapidly homogenized , in this case by about 5000 yr . 
   note that this time 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 . 
   as in the g dwarf protostar models ( e.g. , figure 5 ) , the inner disk dispersion is higher than in the outer disk , though in these models ( with a lower initial @xmath47 ) the inner disk dispersion drops to @xmath0 5% to 10% , compared to @xmath0 1% to 2% in the outer disk . 
   figure 8 shows the same behavior for model 0.1 - 1.8 - 9 , which differs from the previous model shown in figure 7 only in having the injection occur at 9 au instead of 2 au . as in the g dwarf disks , 
   the dispersions for both of these models evolve toward essentially identical radial distributions : the underlying mgu disk evolution determines the outcome for the dispersions . 
   similar results hold for the models with @xmath46 protostars , i.e. , early m dwarf disks . 
   we now turn to a consideration of the consequence of single - shot versus continuous injection at the surface of much larger outer radius ( 40 au ) disks than have been considered to date for g dwarf stars ; boss ( 2007 ) considered disks extending from 4 au to 20 au in radius . 
   table 4 shows the initial conditions for the models with a @xmath48 disk around a @xmath38 protostar , with the disks extending from 10 au to 40 au . because of the much larger inner and outer disk radii for this set of models , 
   these models can be calculated for times as long as @xmath49 yr ( table 4 ) . 
   such times are still considerably less than the typical ages ( @xmath50 yr ) of t tauri stars , implying that in order for mgu disks to occur at such late phases , a prior phase of coupled mri - mgu evolution might be required to make the present results 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 latter model has continuous injection , intended to simulate a disk with ongoing uv photolysis and fractionation at the outer disk surface . for model 1.0 - 1.1 - 40 - 20 
   , it can be seen that the evolution is similar to that of the previous single - shot models : a rapid drop in the dispersion , followed by homogenization to @xmath0 1% to 2% away from the inner disk boundary . 
   the higher dispersions seen near the outer disk boundaries ( @xmath0 40 au ) are largely caused by 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 the dispersions throughout the disk even after @xmath3 yr can be as high as @xmath0 20% , consistent with the much larger variation in stable oxygen isotope ratios , compared to slri ratios . in a calculation with finer spatial grid resolution , as well as perhaps sub - grid mixing processes , one might expect even stronger homogenization to occur , so the dispersion levels obtained from the present models should be considered to be upper bounds . 
   the total amount of color added during the continuous injection models is large , compared to single - shot injection models : for model 1.0 - 1.4 - 40 - 20c , for example , after 200 yr , the total amount of color injected has increased by a factor of @xmath0 90 compared to the single - shot total , and by another factor of @xmath0 60 after 27000 yr . 
   figures 11 and 12 compare the results for continuous injection at either 20 au or 30 au , respectively , i.e. , for models 1.0 - 1.1 - 40 - 20c and 1.0 - 1.1 - 40 - 30c . in spite of the different injection radii , figures 11 and 12 
   show that even at a relatively early phase ( 405 yr ) of evolution , the midplane color / gas ratios look somewhat similar ; as before , the mgu disk evolution is the same for both models , and that is the primary determinant of the long term evolution .    finally , similar results as those shown in figures 9 - 12 were obtained for the other models listed in table 4 . 
   these models show that the main factor in determining the radial dispersion profile is whether the injection occurs in a single - shot or continuously ; in the latter case , the mgu disk does its best to homogenize the color field , but the fact that spatial heterogeneity is being continuously injected limits the degree to which this heterogeneity can be reduced . 
   while dust grains in the interstellar medium are overwhelmingly amorphous , crystalline silicate grains have been found in a late m dwarf ( sst - lup3 - 1 ) disk at distances ranging from inside 3 au to beyond 5 au , in both the midplane and surface layers ( mern et al . 
   such crystalline silicate grains are likely to have been produced by thermal annealing in the hottest regions of the disk , well inside of 1 au ( sargent et al . 
   2009 ) . again 
   , outward transport seems to be required to explain the observations , and the results for the models with a 0.1 @xmath1 protostar suggest that mgu phases in low mass m dwarf disks may be responsible for these observations . in fact , crystalline mass fractions in protoplanetary disks do not appear to correlate with stellar mass , luminosity , accretion rate , disk mass , or the disk to star ratio ( watson et al . 
   these results also appear to be consistent with the results of the present models , which show that mgu disk phases are equally capable of relatively rapid large - scale mixing and transport , regardless of the stellar or disk mass , or the exact value of the @xmath2 stability parameter . 
   these models have shown a rather robust result , namely that a phase of marginal gravitational instability in disks and stars with a variety of masses and disk temperatures can lead to relatively rapid inward and outward transport of disk gas and small grains , as required to drive the protostellar mass accretion associated with fu orionis events , as well as to explain the discovery of refractory grains in comet 81p / wild 2 . 
   a mgu disk phase driving a fu orionis outburst is astronomically quite likely to have occurred for our protosun , and cosmochemically convenient for explaining the relative homogeneity of @xmath4al/@xmath42al ratios derived from a supernova injection event , and the range of @xmath5o/@xmath6o ratios derived from sustained uv self - shielding at the surface of the outer solar nebula . 
   low - mass stars , from g dwarfs 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 fu orionis outburst phenomenon , has a mass of only @xmath51 ( zhu et al . 
   2007 , 2009b ; beck & aspin 2012 ) , i.e. , the mass of a m dwarf , suggesting that m dwarf protoplanetary disks may experience evolutions similar to that of the solar nebula , with possible implications for the 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 improvements , and sandy keiser , michael acierno , and ben pandit for their support of the cluster computing environment at dtm . 
   this work was partially supported by the nasa origins of solar systems program ( nnx09af62 g ) and is contributed in part to the nasa astrobiology institute ( nna09da81a ) . 
   some of the calculations were performed on the carnegie alpha cluster , the purchase of which was partially supported by a nsf major research instrumentation grant ( mri-9976645 ) . 
   1.0 & 0.019 & @xmath52 & @xmath53 & 1.0 & 1.0 & 10.0 + 0.5 & 0.018 & @xmath54 & @xmath55 & 1.0 & 1.0 & 10.0 + 0.1 & 0.016 & @xmath56 & @xmath57 & 1.0 & 1.0 & 10.0 + 1.0 & 0.13 & @xmath58 & @xmath59 & 4.0 & 10.0 & 40.0 +      1.0 - 2.2 - 2 & 15 & 2.2 & 2 & 2520 + 1.0 - 2.2 - 9 & 15 & 2.2 & 9 & 2520 + 1.0 - 2.6 - 2 & 20 & 2.6 & 2 & 2043 + 1.0 - 2.6 - 9 & 20 & 2.6 & 9 & 2043 + 1.0 - 2.9 - 2 & 25 & 2.9 & 2 & 1200 + 1.0 - 2.9 - 9 & 25 & 2.9 & 9 & 1400 + 1.0 - 3.1 - 2 & 30 & 3.1 & 2 & 1100 + 1.0 - 3.1 - 9 & 30 & 3.1 & 9 & 1300 +          1.0 - 1.1 - 40 - 20 & 30 & 1.1 & 20 & single - shot & 25000 + 1.0 - 1.1 - 40 - 30 & 30 & 1.1 & 30 & single - shot & 24500 + 1.0 - 1.4 - 40 - 20 & 50 & 1.4 & 20 & single - shot & 24000 + 1.0 - 1.4 - 40 - 30 & 50 & 1.4 & 30 & single - shot & 15000 + 1.0 - 1.1 - 40 - 20c & 30 & 1.1 & 20 & continuous & 19500 + 1.0 - 1.1 - 40 - 30c & 30 & 1.1 & 30 & continuous & 19800 + 1.0 - 1.4 - 40 - 20c & 50 & 1.4 & 20 & continuous & 27000 + 1.0 - 1.4 - 40 - 30c & 50 & 1.4 & 30 & continuous & 27000 +
summary: analyses of primitive meteorites and cometary samples have shown that the solar nebula must have experienced a phase of large - scale outward transport of small refractory grains as well as homogenization of initially spatially heterogeneous short - lived isotopes . 
   the stable oxygen isotopes , however , were able to remain spatially heterogenous at the @xmath0 6% level . 
   one promising mechanism for achieving these disparate goals is the mixing and transport associated with a marginally gravitationally unstable ( mgu ) disk , a likely cause of fu orionis events in young low - mass stars . 
   several new sets of mgu models are presented that explore mixing and transport in disks with varied masses ( 0.016 to 0.13 @xmath1 ) around stars with varied masses ( 0.1 to 1 @xmath1 ) and varied initial @xmath2 stability minima ( 1.8 to 3.1 ) . 
   the results show that mgu disks are able to rapidly ( within @xmath3 yr ) achieve large - scale transport and homogenization of initially spatially heterogeneous distributions of disk grains or gas . in addition , the models show that while single - shot injection heterogeneity is reduced to a relatively low level ( @xmath0 1% ) , as required for early solar system chronometry , continuous injection of the sort associated with the generation of stable oxygen isotope fractionations by uv photolysis leads to a sustained , relatively high level ( @xmath0 10% ) of heterogeneity , in agreement with the oxygen isotope data . 
   these models support the suggestion that the protosun may have experienced at least one fu orionis - like outburst , which produced several of the signatures left behind in primitive chondrites and comets .