Forming chondrules in impact splashes. I. Radiative cooling model

0.3 MB·English

by  Cornelis Petrus Dullemond

The Astrophysical Journal,794:91(12pp),2014October 10
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
FORMING CHONDRULES IN IMPACT SPLASHES
I. RADIATIVE COOLING MODEL
Cornelis Petrus Dullemond, Sebastian Markus Stammler
InstituteforTheoretical Astrophysics,HeidelbergUniversity,Albert-Ueberle-Strasse2,69120Heidelberg,Germany
Anders Johansen
5 LundObservatory,DepartmentofAstronomyandTheoretical Physics,LundUniversity,Box43,22100Lund,Sweden
1 The Astrophysical Journal, 794:91 (12pp), 2014 October10
0
2 ABSTRACT
The formation of chondrules is one of the oldest unsolved mysteries in meteoritics and planet for-
n
mation. Recently an old idea has been revived: the idea that chondrules form as a result of collisions
a
J betweenplanetesimalsinwhichthe ejectedmoltenmaterialforms smalldroplets whichsolidify to be-
comechondrules. Pre-meltingofthe planetesimalsbyradioactivedecayof26Alwouldhelp producing
3
spraysofmeltevenatrelativelylowimpactvelocity. Inthis paperwestudy the radiativecoolingofa
2
ballistically expandingsphericalcloudof chondruledroplets ejected fromthe impactsite. We present

results from a numericalradiative transfer models as well as analytic approximate solutions. We find
]
P that the temperature after the start of the expansion of the cloud remains constant for a time t
cool
E andthen dropswith time t approximatelyas T ≃T [(3/5)t/t +2/5]−5/3 for t>t . The time at
0 cool cool
. which this temperature drop starts t depends via an analytical formula on the mass of the cloud,
h cool
the expansion velocity and the size of the chondrule. During the early isothermal expansion phase
p
the density is still so high that we expect the vapor of volatile elements to saturate so that no large
-
o volatile losses are expected.
r Subject headings: chondrules, radiative transfer
t
s
a
[ 1. INTRODUCTION ofmeltaremuchlower,andthusmoreconsistentwithex-

pectationsoftheaveragerelativevelocitiesbetweenplan-
1 The formation of chondrules is one of the funda-
etesimals. Sanders & Taylor (2005), Hevey & Sanders
v mental questions of meteoritics and planet formation.
(2006)andSanders&Scott(2012)followuponthisidea
1 Chondrules are the 0.1···1 mm-size once-molten sili-
9 cate spherules found abundantly in primitive meteorites ofpre-moltenimpactorsandbackit upwith models and
7 known as chondrites (Jones, Grossman & Rubin 2005; meteoritic evidence. Sanders & Scott (2012) argue that
5 Sears 2004; Davis et al. 2014). Most of these chondrites this scenario is hard to avoid, given that in the first 2.5
0 in fact consist predominantly of these chondrules,so the million years most planetesimals were internally nearly
. fully molten by 26Al decay heat, and that collisions be-
1 melt-producingeventsthatcreatedthemmusthavebeen
tween planetesimals were extremely common. Each col-
0 extremely common during the first few million years of
lisionwouldalmostcertainlyreleasesubstantialamounts
5 the solar system. Yet there is confusing and conflicting
of molten rock into the nebula in the form of sprays of
1 evidence as to what these events might have been. Boss
lavadroplets,anditisnaturalto assumethatthese may
: (1996)published anoverviewofthe status of the discus-
v be chondrules. They argue that the near-solar composi-
sion at that time, though significant new developments
i tionofchondrules canbe explained by the vigorouscon-
X have occurred since then.
vection inside the molten planetesimals that may slow
Manytheorieshavebeenputforwardoverthelasthalf
r
down iron/nickel-core formation, thus keeping the melt
a a century. One theory involves impact melt sprays. Put
solar. Some degree of differentiation would then in fact
forwardbyUrey(1953)andrefinedbyKieffer(1975)this
explain the low iron-content L and LL chondrites.
model states that high-velocity impacts (& 3···5 km/s)
Recently, Asphaug et al. (2011) performed Smooth
couldleadtospraysofimpactmeltthatproducedroplets
Particle Hydrodynamics simulations for such impacts to
whichsolidifyintochondrules. Whiletheserequiredcolli-
demonstrate the dynamics of this scenario. In addition
sionvelocitiesarehigh,asmallfractionofimpactsmight
theyproposedasimplewaytocalculatethemeltdroplet
acquire such velocities (Bottke et al. 1994). Perhaps the
size by equating the released enthalpy after the collision
so-called “Grand Tack”-scenario of Walsh et al. (2011),
to the surface energy of the droplets.
inwhichJupitertemporarilyenteredtheasteroidbeltre-
There are many alternative scenarios proposed in the
gionbeforemigratingbackoutward,mightproducesuch
literature. Perhaps the most popular model is the flash
highvelocities. However,Tayloretal.(1983)putforward
heating by nebular shocks (Hood & Hora´nyi,1991). De-
anumberofargumentsagainsttheimpact-meltscenario,
sch & Connolly (2002) and Ciesla & Hood (2002) de-
some of which were based the differences between chon-
veloped detailed 1-D radiative shock models with dust
drites and lunar impact regolith.
particlesandchondrulesinteractingbothradiativelyand
Zook(1980)suggestedthatiftheinteriorsofthecollid-
frictionally with the gas. They showed how the radia-
ing bodies are already in a molten state due to 26Al de-
tive shocks exhibit temperature spikes of mere tens of
cay, then the required impact velocities to create sprays
2 Dullemond, Stammler, Johansen
seconds (with cooling rates > 104 K/hr) that would be so high that the cloud is completely optically thick and
goodcandidates for chondrule-formingevents. After the no radiation can escape from its interior. There is also
mainshocktemperaturespiketheirmodelexhibitsafur- no adiabatic cooling because of the lack of pressure. So
ther cooling at intermediate cooling rate (∼50 K/hr), thetemperatureofthemagmadropletswillinitiallystay
which cools the chondrules to sufficiently low tempera- roughly constant in time. As the cloud expands, how-
ture in a sufficiently short amount of time. This model ever, the optical depth drops and eventually the cloud
seems to produce the flash-heating events required for willstartto coolradiatively. During the earlystagesthe
turning “dust bunnies” into chondrules. However,so far density is very high and the evaporation of volatile ele-
theshockscenariostillhasseveralunresolvedissues(e.g., ments will be saturated (as was argued by Alexander et
Desch et al. 2012; Boley et al. 2013;Stammler & Dulle- al.2008). Oncethetemperaturedropsbelowthesolidus,
mond 2014). volatiles can no longer escape.
Also some issues are raised about whether large scale In this first paper we intend to compute how the tem-
shocksintheopticallythicksolarnebulacanexplainthe perature behaves as a function of time after the impact.
time scales involved in chondrule formation (Stammler To do this we set up a simple model: that of a ho-
& Dullemond 2014). It is noteworthy that there exist mologously ballistically expanding homogeneous spher-
severalothernebularflashheatingmodels,mostnotably icalcloudoflavadroplets. We computethe temperature
flash heating by nebular lightning (Gibbard et al. 1997) ofthechondrulesasafunctionoftimetandcomovingra-
andflash heating by energydissipationin currentsheets diallocationinsidethecloudr/r bysolvingthetime-
cloud
forming in MHD turbulence (Hubbard et al. 2012). dependent radiative transfer equation. We also present
One important constraint on chondrule formation an analytic approximate solution. While the spherical
modelsis thattheseheating eventswereveryshort-lived cloud model is not an accurate model of the complex
compared to any other nebular time scales. By compar- shapeofanimpactsplash,itcanberegardedasamodel
ing textures of chondrule to those obtained in furnace of part of the impact splash. As such we believe that
experimentsitcanbeinferredthatchondrulesmusthave it will give reasonable estimates of the radiative cool-
cooledfromthe liquidus temperature downto below the ing behavior of the impact splashas a function of model
solidus temperature in a matter of hours (Hewins 1983; parameterssuchasthetotalmassofthecloudanditsex-
Hewins &Connolly1996;see alsoreferencesinMorris& pansionvelocity. The totalmass of the cloudtells some-
Desch 2010 and the excellent review paper of Desch et thing about the masses of the two colliding bodies while
al. 2012). In other words: the chondrule forming events the expansion velocity of the cloud tells us something
must have been flash-events. On the other hand, un- about the impact velocity.
der optically thin conditions a molten chondrule would
2. EXPANDINGCLOUDMODEL
radiatively cool in about a second, which would be too
fast. Whenlavadropletsareproducedinacollisionbetween
Another constraint that a chondrule formation model two planetesimals, they will disperse away from the im-
must fulfill is the retention of volatile elements such as pact site in a ballistic way. Let us call this the “impact
Fe, Mg, Si, Na and K. Chondrules are not observed splash”. If the impact velocity is larger than the escape
to have low abundances of these elements. In particu- velocity of the two colliding planetesimals this ballistic
lar for the highly volatile elements Na and K this is a (pressureless) expansion will be linear (i.e. the velocity
puzzle, because any heating event that heats a (proto- of expansion will not change with time). We regard the
)chondrule above ∼ 1700 K and keeps these chondrules impact splash as an expanding cloud of lava droplets
above that temperature for more than a few minutes (chondrules-to-be). This cloud is not necessarily cen-
will cause most of the Na and K to evaporate out of teredonthe impact site; it canalsomove awayfrom the
the chondrule. Alexander et al. (2008) argue that this impact site. The radius of the cloud increases linearly
means thatchondrulesmust haveformedinregionsthat with time t and thus the density of the cloud will drop
are extremely dense in solids, much more so that typi- as1/t3. Dependingonthecomplexityofthegeometryof
caldustconcentrationmechanismsintheprotoplanetary the impact splash (see e.g. Asphaug et al. 2011) we can
disk can achieve. According to their calculations such also consider the splash to be multiple smaller clouds.
regions are so dense in solids that they must be self- In either case, each of these clouds will expand linearly.
gravitating. Morris et al. (2012) instead propose that The scenario is pictographically shown in Fig. 1.
chondrulesformedbehindthebowshockofafastmoving If the impact is at low velocity, then the gravitational
planetary embryo, and that the embryo’s atmosphere is pull of the surviving (or merged) body can cause some
rich in such volatile elements caused by outgassing from (or most) of the material to re-accrete. In this case the
theembryo’sinterior,thusprovidingthenecessaryvapor 1/t3 expansion will cease and turn into recompression.
pressure to keep the chondrules volatile-rich. Initially, however, the linear expansion and 1/t3 density
In this paper we will focus on the impact splash hy- drop will still be a good description.
pothesis,eitherthe low-velocitypre-moltenplanetesimal To compute the temperature of the droplets in these
versionor the high-velocity impact-melt version. We as- clouds as a function of time after impact we must solve
sume that after the collision between two planetesimals the time-dependent equation of radiative transfer. This
acloudofmoltendropletsofmagmawasreleased. Since is a very non-local problem, because a cooling droplet
soon after the ejection of this cloud of magma droplets at location ~x radiates away some heat into the form of
1
the internal pressure of the cloud would have dropped infrared radiation, which can then be absorbed by an-
to near-zero (there is, to good approximation, only vac- other droplet at location ~x 6= ~x which is then heated.
2 1
uum between the droplets), the cloud would simply ex- The droplets are thus radiatively coupled over large dis-
pand ballistically. Initially the density of the cloud is tances. Solving this problem in 3-D for complex cloud
Forming chondrules in impact splashes 3
Pre−collision After collision: Some parts survive. The debris clouds (here modeled as
Debris clouds (e.g. clouds of molten spherical clouds) expand linearly as
lava droplets) get ejected. they move away from the impact site.
Fig.1.—Pictographicrepresentationofthemodel. Left: Twoplanetesimalsapproacheachotherandareabouttocollide. Middle: After
thecollisionsomepartsoftheplanetesimalsmaysurvive,butsomepartsaredestroyedanddispersedinacloudofdebris. Wemodelthis
dispersingcloud as a set of spherical clouds. If the debris consists of melt, the cloud will consist of molten droplets (chondrules). Right:
As the clouds move away from the point of impact they expand linearly. The average distance between the chondrules increases linearly
withtime.
geometries is somewhat challenging. the cloud is constant and outside of the cloud is zero.
Webelieve,however,thatmuchcanalreadybelearned The density of the cloud as a function of time is then
fromasimplesphericalcloudmodel,forwhichtheradia-
tivetransferproblemcanbesolvedtohighprecisionand ρcloud(r,t)= 4π3RM3cclolouudd(t) forr ≤Rcloud(t) (5)
reliability with the 1-D tangent-ray variable eddington 0 forr >R (t)
(cid:26) cloud
factor method. We will describe the method in Section
4. From here on we write only ρcloud(t) instead of
The three main parameters of the model are: ρcloud(r,t). The time-dependence of ρcloud(t) is
3M 1
M , T , v (1) cloud
cloud 0 exp ρ (t)= (6)
cloud 4πv3 t3
exp
where M is the totalmass worthof chondrules(lava
cloud
droplets)inoursphericalcloud,T0istheinitialtempera- Letusconsiderchondrules(lavadroplets)ofradiusachon
ture ofthe chondrulesand vexp is the expansionvelocity and material density ξchon. We take ξchon = 3.3 g/cm3
of the outer radius of the cloud. To get a feeling for the for our model. The mass of a chondrule is then
numbersitismoreconvenienttoexpressthe massofthe
4π
oclfotuhdeisnamteermmsasosfMthe rad:ius Rmelt,0 of a ball of magma mchon = 3 ξchona3chon (7)
cloud
The number density of chondrules is then
4π
M = ξ R3 (2)
cloud 3 chon melt,0 ρ (t)
cloud
n (t)= (8)
chon
where ξchon is the material density of the chondrule mchon
droplets and thus, by definition, the material density of Thegeometriccrosssectionofachondruleisπa2 . Let
thehypotheticalballofmagma. Foralinearlyexpanding chon
us assume that the chondrule has zero albedo. Then
cloud the radius of the cloud is
the absorption cross section equals the geometric cross
section. The opacity (i.e. the cross section per gram) is
R (t)=v t (3)
cloud exp
then
where the time t is the time since the impact and the πa2 3 1
κ= chon = (9)
ejectionofthecloudofmeltdroplets,assumingperfectly 4πξ a3 /3 4ξ a
chon chon chon chon
ballistic (pressureless) expansion and of course assum-
ing large enough t that R (t) ≫ R so that the We will assume that the opacity of any possible vapor
cloud melt,0
droplets are clearly separated from each other. The ve- between the chondrules will be low compared to that of
locity profile inside the cloud is the chondrules so that it can be ignored. The optical
depth from the center of the cloud to the edge is
r
v(r,t)=v (4)
expR (t) 3 Mκ 1
cloud τ(t)=ρ (t)κR (t)= (10)
cloud cloud 4πv2 t2
exp
which is of course only valid for r≤R (t).
cloud
Let us define a coordinate r centered on the center of The rate of cooling will strongly depend on this optical
the spherical cloud. We assume that the density within depth.
4 Dullemond, Stammler, Johansen
Model Rmelt,0 vexp T0 tcool τcool becomes smaller. It is therefore to be expected that at
F1 1km 100m/s 2000K 27min 92 early times the temperature of the chondrules remains
F2 0.1km 1000m/s 2000K 16sec 9.2 constant. Note that since the cloud consists of liquid
F3 10km 100m/s 2000K 7h 366 drops that stay at a constant volume and move away
F4 0.01km 1000m/s 2000K 1s 2.3 from each other, there is no adiabatic cooling involved
here. Any potential vapor may adiabatically cool, but
wewillignorethiseffectinthispaper,assumingthatthe
TABLE 1
totalmassinvaporisalwayssmallcomparedtothemass
The model parameters(first three columns) forthe main
(“fiducial”)models presented in this paper. The cloud in droplets.
mass Mcloud can be derived fromMcloud= 43πξchonR3melt,0 At some point in time, however, trad becomes smaller
(cf.Eq. 2). The lasttwo columnsare computed from than t and the chondrules start to cool. Based on the
these parameters: the cooling time tcool (the time after results of the true radiative transfer calculations of Sec-
the impactwhen the cloud startsto cool, see Eq.16) and
tion 4 it turns out that near the center of the cloud the
the opticaldepth atthEaqt.t2i0m).eτcool ≡τ(t=tcool) (see transitionfromtheinitialconstanttemperaturephaseto
the temperature-decline phase occurs roughly at a time
t defined by
In this paper we will present our results based on a cool
set of fiducial models as well as parameter scans. The t (t )=5t (15)
rad cool cool
parameters of the fiducial models are listed in Table 1.
Inserting Eqs.(14,12,13,10) into Eq. (15) yields
3. ANALYTICESTIMATEOFTHECOOLINGBEHAVIOR
OFTHECHONDRULECLOUD 1 3 M2 c κ 1/5
t = cloud m (16)
As the cloud expands it can start to radiate away en- cool 5(4π)2 v4 σT3
(cid:18) exp 0 (cid:19)
ergy. A fully fledged time-dependent radiative cooling
computationwillyieldtemperatureasafunctionoftime Note that in Eq. (15) the factor of 5 is purely empirical.
and space: T(r,t). We will compute this in Section 4. The estimate we make here is essentially a dimensional
A first estimate of the cooling behavior of the cloud analysis in which proportionality factors have to be cal-
canalreadybemadewithpenandpaper. Thecloudwill ibrated against more exact calculations (in our case the
initiallybeopticallythick: τ ≫1. Theluminosityofthe full radiative transfer calculation).
cloud is then Roughly for t < tcool the temperature stays constant
L=4πRc2loudσTe4ff (11) while for t > tcool the temperature drops with time.
From now on we shall define the t as the one cal-
cool
where σ is the Stefan-Boltzmann constant and Teff is culated for the center of the cloud. Near the surface the
the effective surface temperature at r = Rcloud. In the cooling sets in earlier.
above equation, and from here onward, we omit the (t)
The temperature decline with time can also be es-
foraestheticreasons,butthe time-dependenceisstillas- timated, at least up until the point where the cloud
sumed. During the cooling phase the effective surface becomes optically thin, after which our approximation
temperature will be lower than the central temperature.
breaks down. The way to make this estimate is to solve
A commonly used estimation of this effect is: the cooling equation
1
L≃4πR2 σT4 (12) dT(t) T(t)
cloud τ =− (17)
dt t (t)
cool
where T is now the central temperature and 1/τ is the
where t (t) is the cooling time given by Eq. (16) but
correction factor to account for the lower temperature cool
with T replaced by T(t). The reasoning behind this is
of the surface T , and is based on radiative diffusion 0
eff
that if we would instead use t (t) (which might look
theory which states that T ≃ τ1/4T (for τ ≫ 1). To rad
eff more reasonable at first sight) we will quickly cool to
computethechangeofthetemperatureasaresultofthe
temperatures for which condition t . t is again no
radiative loss given by Eq. (12) we must use the heat longerfulfilled. Thisisbecauset ∝radT−3 andthusvery
capacityformula. Thetotalthermalenergystoredinthe rad
rapidly rises with declining temperature. If we insert
chondrule cloud (approximating it, for the moment, as
Eq. (16) with T replaced by T into Eq. (17) we obtain
an isothermal cloud) is: 0
a differential equation that can be solved by separation
E =M c T (13) of variables. The solution is
cloud m
where cm is the mass-weighted specific heat of the lava 3 t 2 −5/3
droplet. Avalueofc =107ergg−1 K−1 isareasonable T(t>tcool)=T0 + (18)
m 5t 5
value which we will adopt here. The radiative loss time (cid:20) cool (cid:21)
scale can now be defined as where here t is again the original one with T . We
cool 0
assume that for t<t we have:
E M c τ cool
cloud m
t (t)= = (14)
rad L 4πR2 σT3 T(t<t )=T (19)
cloud cool 0
This time scale varies with time: it is very long at early The solution Eqs.(18,19) holds true for the centraltem-
times because the cloud is then still extremely optically perature. But most of the mass of a homogeneous
thick. At late times the cloud is optically thinner and sphere resides in the outer parts. For the temperature
the radiation can more freely escape (Eq. 12) and t at r = 0.8R and r = 0.9R (roughly the radii
rad cloud cloud
Forming chondrules in impact splashes 5
which is the time between the start of the cooling at
T =T andthe time whenthe temperature hasdropped
0
below the solidus temperature 1400 K. To first order we
can write
T −1400K 1400K
0
t ≃ = 1− t (22)
1400 |T˙(t=tcool)| (cid:18) T0 (cid:19) cool
The quantities T˙ and t can be compared to the con-
1400
straints coming from the analysis of textures of chon-
drules, which put time limits on the cooling process of
order of hours, or more precisely: cooling rates in the
range 10···∼3000 K/hr (see Morris & Desch 2010 and
Deschetal. 2012andreferencestherein, thoughseealso
Miura & Yamamoto 2014 for a different view based on
theoreticalmodeling of crystallization),indicated by the
grey area in Fig. 3. According to Eq. (16) t goes as
cool
M2/5 and as v−4/5. There is not very much freedom
cloud exp
of choice of T : it must lie somewhere between 1770and
Fig.2.— The temperature evolution of the expanding cloud 0
2120K(seeMorris&Desch2010andreferencestherein).
model accordingtothe analytical estimate ofEqs.(18,19),shown
inblack. Ingreyare,forcomparison,thefullradiativetransferso- The κ does not have too much wiggle room either: the
lutionsfromSection4forthefiducialmodelF1(seeTable1). The radii of chondrules are known. This means that con-
temperaturesareshownatthreepositionsinthecloud: thecenter,
straints on t from textural analysis directly set lim-
at80%oftheradiusandat90%oftheradius(nearthesurfaceof cool
its on the ratio M /v2 or equivalently on the ratio
thecloud). Thetimescaleisscaledtotcool (Eq.16). Theanalytic cloud exp
solutionsareonlyvalidaslongasτcool≫1(cf.Eq.20). Rm3elt,0/ve2xp. If, for example, we choose achon = 0.03
of half mass and of 75% mass respectively) the best fit- cm and T = 2000 K we obtain κ = 7.6 cm2/g. If
0
tingsolutiontotherealsolutionsofSection4aresimilar we require, for instance, the cooling time scale to be
to Eq. (18) but with the term 2/5 replaced by 3/5 and t = 10 minutes, then we find that M /v2 ≃ 107
3.8/5 respectively. These analytic estimates of the tem- gcoso2lcm−2. This wouldbe fulfilled e.g.focrlovud =exp1 km/s
perature as a function of time are plotted in Fig. 2. As exp
and M = 1017 g (a mass corresponding to a ball of
onecansee,althoughthetime afterthe impactatwhich cloud
magma of roughly R =2 km radius). It would also
the cooling stars is different depending on whether you melt,0
be fulfilled by a smaller mass and smaller velocity: e.g.
lookatthe centeroratthe edgeofthecloud,the typical
for v =10 m/s and M =1013 g (R =0.1 km
coolingrateafterthecoolingbeginsissimilarthroughout exp cloud melt,0
radius). Such small mass could either mean that upon
the cloud.
impact only a small fraction of the debris is in the form
It is important to remind ourselves that the cooling
of chondrule droplets, or it could simply mean that the
normally starts well before the cloud becomes optically
largest closed unit of the droplet splash (subcloud) is so
thin. Optically thin cooling would be extremely fast: a
small, but many such expanding cloudlets of chondrules
single chondrule would cool within about 1 second. The
are ejected. The elongated streams of debris found in
protracted cooling over a time scale of hours occurs be-
Asphaug et al. (2011) could be regarded as a string of
cause the high optical depth (even after t = t ) acts
cool
such smaller mass cloudlets.
as a kind of blanket that keeps the chondrules warm, al-
This is shown more directly in Fig. 4 which shows the
beit a blanket that is thinning over time as the optical
parameter space of the model, with the grey area again
depth drops. To get a feeling for this we can calculate
showing the models which give cooling rates between 10
the optical depth at the time t=t from Eq. (10):
cool
and 3000 K/hr.
33/552/5M1/5 κ3/5σ2/5T6/5
τcool ≡τ(tcool)= (4π)c1l/o5uvd2/5c2/5 0 (20) 4. TIME-DEPENDENTRADIATIVETRANSFER
exp m So far we have only made an estimate of the cooling
So Eqs. (18,19), which are based on the assumption of behavior of the cloud of liquid chondrules. Let us now
the “blanket effect” of high optical depth, are expected calculatethetemperatureprofilewithafulltreatmentof
to be valid for the case τ ≫1. In table 1 one can see time-dependentradiativetransfer. Thetime-dependence
cool
that τ is well above 1 for all models, except the most comesinonlyduetothetimeittakesthelavadropletsto
cool
extreme model F4 which has anyway a much too small converttheirheatintoradiation,notintheformoflight-
cooling time to be consistent with chondrule textures. travel time (which typically only plays a role for media
TheanalyticmodelofEqs.(18,19)tellsusthatfort> attemperatures above105K).This canbe easilyverified
t thetemperaturedropsoffsuddenlyonatypicaltime by comparing the radiative energy density aT4 to the
cool
scale of t . The temperature decay rate at t=t is: material energy density ρc T, which in our case always
cool cool m
satisfies aT4 ≪ ρc T. Therefore the radiative transfer
dT T m
T˙(t=t )≡ =− 0 (21) equation itself is stationary, and the time-dependence is
cool
dt (cid:12)t=tcool tcool in the equation for heating/cooling of the chondrules.
(cid:12)
This is plotted, together with(cid:12) t , in Fig. 3. It is also
(cid:12) cool 4.1. Equations of time-dependent radiative transfer
useful to define another kind of cooling time scale t
1400
6 Dullemond, Stammler, Johansen
tothe geometriccrosssection,independent offrequency,
because the chondrules are much larger than the wave-
length of infrared radiation. In this case the frequency-
dependence of the radiative transfer does not need to
be considered. Eq. (23) is called the “formal transfer
equation”,anddescribesthetransportofradiationalong
rays of direction~n. It is easy to integrate if the value of
B(~x)=B(T(~x))isknowneverywhere,butsincewewant
to calculate the temperature T(~x), this function is part
of the thing we want to solve.
In our 1-D spherically symmetric setting Eq. (23) can
be written as
dI (r) 1−µ2dI (r)
µ µ
µ + =α(B(T(r))−I (r)) (25)
µ
dr r dµ
where we shortened
ρκ=:α (26)
Fig.3.—Thecoolingtimetcool (Eq.16)asafunctionofparam- Here µ = cos(θ) where θ is the angle between the ray
eters of the model for T0 = 2000 K and achon = 0.03 cm. Note
that the time to cool down from 2000 K to 1400 K (t1400) is, ac- alongwhichtheradiationisfollowedandtheradiallyout-
cording to Eq. (22), about four times smaller than tcool. On the wardpointingunitvector. TheEddingtonfactormethod
rightaxisthecorrespondingcoolingrate|dT/dt|inK/hrisplotted.
ofsolvingthisequationreliesonthedefinitionofthefirst
Thegreyareaistheobservedrangeofchondrulecoolingrates(see
maintext). Notethatthevaluesoftcool are(bydefinition)forthe three angular moments of the intensity at each location
center of the cloud. The regions of the cloud more closely to the r:
surfacewillstarttocoolearlier,butthecoolingrateisalmostthe
same. 1 +1
J(r)= I (r)dµ (27)
µ
2Z−1
1 +1
H(r)= I (r)µdµ (28)
µ
2Z−1
1 +1
K(r)= I (r)µ2dµ (29)
µ
2Z−1
Here J is called the mean intensity and H is the flux
divided by 4π. By integrating Eq. (25) over 1dµ after
2
multiplyingitby1andbyµrespectivelyoneobtainsthe
first two moment equations:
1 d(r2H)
=α(B(T)−J) (30)
r2 dr
dK 3f −1
+ J=−αH (31)
dr r
These are two equations with three unknowns. To close
this set of equations we write
Fig.4.— The parameter space of the model. The grey area
showswherethesolutionshavecoolingratesintherangeobserved K(r)=f(r)J(r) (32)
for chondrules. In the bottom-right the region is shown where
τcool<10,wheretheanalyticsolutionbecomes invalid. where f is called the “Eddington factor”. If f(r) is
known, then the two moment equations are in closed
The formal radiative transfer equation is (see e.g. Mi-
form and can be solved for J(r) and H(r). From the
halas & Mihalas 1999):
moment equations alone we cannot know what f(r) is
~n·∇~I(~x,~n)=ρκ(B(~x)−I(~x,~n)) (23) for all r, but for certain limiting cases we know their
values: For optically thick media f = 1/3. Outside of
whereI isthefrequency-integratedmeanintensityinerg the cloud for r → ∞ we will get f → 1, which is the
s−1 cm−2 ster−1, ~x is the position vector in space, ~n is free-streaminglimit. However,for generalr andfor gen-
the unit vector of direction of the radiation, and finally eralvalues of the optical depth we must employ another
B the frequency-integratedPlanck function given by method of calculating f(r). We do this by integrating,
for the current value of T(r) (and thus B(T(r))), the
σ
B(T)= T4 (24) formaltransfer equationEq.(25) using the “tangentray
π
method” (see e.g. Mihalas & Mihalas 1999). By inte-
We will use frequency-integrated quantities because the grating the resulting intensities over angle we calculate
opacity of a chondrule is expected to be roughly equal thecurrentestimatesofJ(r)andK(r). Letuscallthese
Forming chondrules in impact splashes 7
J (r)andK (r)(where“fte”standsfor“formaltrans- obtain Jn+1. The problem with this scheme is that the
fte fte i
fer equation”). Our current estimate of f(r) is then required time steps can become very small due to nu-
merical stiffness. A more robust method is to integrate
K (r)
fte the complete system Eqs.(35,36) using an implicit inte-
f(r)= (33)
J (r) gration scheme. This is discussed in appendix A.
fte
This is then the f(r) function we stick into Eq. (32), so 4.2. Results of the time-dependent radiative transfer
that the moment equations can be solved for J(r) and models
H(r). Wealsoneedtoimposeboundaryconditionsatthe
The results of the time-dependent radiative transfer
inner and outer edge. At the inner edge we take H = 0
calculations are shown in Fig. 5. These results confirm
(zeroflux). AttheouteredgewecouldsetH =J,which
the estimates made in Section3. We find thatthe shape
is valid if the outer edge of our computational domainis
of the cooling curves is mostly the same for all models,
at r →∞. For finite outer radius we set instead
exceptforthetimescaling,whichweknowfromtheana-
H (r ) lytically expressionof t (Eq. 16). Only for the rather
fte out cool
H(r )= J(r ) (34)
out out extreme (and presumably unrealistic) case such as fidu-
J (r )
fte out
cial model F4 with R = 0.01 km and v = 1000
melt,0 exp
Note that we can choose to set the outer edge of our m/s we find from the full radiative transfer model that
computational domain r at r =R , but we can the cooling is slower than the analytic solution. This is
out out cloud
alsosetitatr >R ,aslongasweproperlysetthe because in that case the condition that τ (Eq. 20) is
out cloud cool
boundary condition at r=r according to Eq. (34). ≫1isbrokenandthecoolingtimescalewillthennotbe
out
We can now combine the two moment equations into determinedbytheopticaldepthdeclinebutbytheeffec-
the following form: tiveness by which a chondrule can convert its heat into
radiation. These cases, however, imply typically time
1 1 d r2 d(fJ) 3f −1 σ
+ J =J − T4 (35) scales of seconds rather than hours, and these are any-
αr2dr α dr r π way too short to be consistent with chondrule textures.
(cid:26) (cid:20) (cid:21)(cid:27)
It seems that the analytic solutions are fairly accurate
This can be solved, together with the boundary condi-
when τ &10, which is always true for the interesting
tions, using a matrix equation. The details of this are cool
regime of parameter space.
discussed in Appendix A.
From Figs. 2 and 5 we see the model systematically
Next we must compute how the temperature T(r) re-
predicts a period of constant temperature with a dura-
acts to the radiation field, or in other words, how T(r)
tion similar to the later cooling phase. This is true at
radiatively cools with time. The energy equation is
least near the center of the cloud. Near the surface this
dT σ temperature plateau is shorter, but the cooling time is
ρcm =4πα J − T4 (36) similar.
dt π
In Fig. 6 we show, for model F1, how the temperature
(cid:16) (cid:17)
where ρ is given by Eq. (6). The d/dt operator is the looks as a function of radial position in the cloud, for
comovingderivative,i.e.the time derivativeiscomputed different times after the collision. Since the cloud is ex-
for a given chondrule. Eq. (36) is a local equation at panding, which would make it difficult to compare the
each location. The non-locality of the heating/cooling temperature profiles atdifferent times, we use the radial
is only through the non-locality of the equation for J(r) coordinate relative to the outer cloud radius. One sees
(Eq. 35). that the outer regions start cooling earlier because the
The way we solve these equations numerically is by optical depth to the surface is smaller. Also one sees
setting up a radial grid of N gridpoints r (with 1 ≤ that a negative radial temperature gradient is produced
i
i≤ N) upon which we define the mean intensity J and which causes the radiative diffusive energy transport to
i
the temperature T . Since the cloud is expanding we let the surface.
i
thegridpointsmovealongwiththematerial(Lagrangian The cooling curve from our model is significantly dif-
approach)sothatthe locationofgridpointiattime tis ferent from the ones predicted from shock heating (e.g.
Morris & Desch 2010, Morris et al. 2012). The shock
r (t)=η v t (37)
i i exp heating models predict a radiative preheating phase, a
where η is the dimensionless radial coordinate temperature spike with rapid cooling, and depending on
geometric conditions followed by a slower cooling phase.
r r
η = = (38) In our model the temperature is high from the start (or
Rcloud(t) vexpt for high-speed collisions: upon impact), stays relatively
constant for some time and then drops over a similar
Thevaluesofη donotchangewithtime. Eachchondrule
i time scale.
stays at constant η, and so the comoving time deriva-
tive d/dt used in e.g. Eq. (36) becomes simply the time 5. DISCUSSION
derivative of that quantity at a given η grid point.
i The scenario of splashes of melt droplets originating
Inprincipleonecouldimagineasimpletime-dependent
from colliding pre-molten planetesimals has been dis-
numerical integration method for the combination of
cussedatlengthinseveralpapersbySandersandcowork-
Eqs.(35,36). We know Jn and Tn at time t = t , and
i i n ers(e.g.Sanders&Scott2012)andthepaperbyAsphaug
we can solve for Tn+1 by taking an explicit Euler inte- et al. (2011). These papers discuss how this scenario
i
grationstep of Eq. (36). Then we can solve the moment holds up agains the many meteoritic constraints. We
equations (because now we have Bn+1 = B(Tn+1)) and will not repeat these arguments here, but refer for those
i i
8 Dullemond, Stammler, Johansen
Fig.5.— The temperature evolution of the fiducial expanding cloud models F1 (top-left), F2 (top-right), F3 (bottom-left) and F4
(bottom-right). See Table 1 for the parameters. The temperatures are shown at three positions in the cloud: the center, at 80% of the
radiusandat90%oftheradius(near thesurfaceofthecloud). Theverticallineshowsthetimetcool ofEq.(16).
5.1. Splash geometry
Our model describes a simple spherical expanding
cloud as a model for (part of) the splash resulting from
collidingbodies. Itisobviousthatthisisadrasticsimpli-
fication. The SPHmodel ofAsphauget al.(2011)shows
a much more realistic geometry involving, among other
complexities, tube-like geometries. A proper 3-D time-
dependent radiative transfer simulation done in concert
with the gravito-hydrodynamicsimulationof the impact
splashing is required to know precisely what the cooling
behavior is. Nevertheless we predict that our result can
be used to estimate the results of those detailed compu-
tations: the tube-like geometry can be approximated as
a homologously expanding cylinder, expanding not only
inthe 2 perpendicular directions but also inthe longitu-
dinaldirection. Its propertiesarethennottoomuchdif-
ferent from a chain of spherically expanding clouds with
diameters similar to the diameter of the tube, except
Fig. 6.—Theradialtemperaturetemperature structureoffidu- that these spherical cloud cannot cool in all directions
cialmodelF1fordifferenttimes. Thehorizontalaxisistheradius (4π) but only in partofthe sky (in the other partit will
relativetotheoutercloudradius. Notethatthetemperaturepro- heat its neighbors and be heated by its neighbors). We
file extends beyond the outer radius of the cloud because even if
would then expect that the cooling time t will be a
thatregionisempty,achondruleresidingtherewouldhaveawell- cool
definedtemperature. Forourpurposes,however,onlythetemper- bit longer but not by a large factor.
aturewithinthecloudisrelevant. Also, one can expect some parts of the splash to be
ejected at slow speed compared to the escape speed,
discussionstothosepapers. Instead,wewillfocusonthe
so that it may fall back to the remaining (unsplashed)
discussionofaspectsrelatedtooursplashcoolingmodel.
Forming chondrules in impact splashes 9
bodies and accrete on them. This evidently breaks the However, a middle way is if planetesimals were pre-
assumption of homologous expansion, and as a result heatedtoelevatedtemperaturesbelowthemeltingpoint.
will also deform any initially spherical sub-cloud into This is to be expected in particular for later generation
a more pancake-like shape. The idea of regarding the planetesimals (i.e. planetesimals formed at a time when
splashcloudasacollectionofsphericalsub-cloudsisthen a substantial fraction of the 26Al has already decayed).
clearlynolongervalid,noteveninanapproximativeway. Lower velocity collisions might then be sufficient to add
In such cases a full 3-D radiation-gravito-hydrodynamic theremainingenergyneededtoproducemelt. Theprob-
simulation of the splash is unavoidable. However,multi- lemwiththisscenarioisthatitrequiresfine-tuning: The
dimensional radiation-hydrodynamics is an enormously planetesimal must have formed at a time when the re-
challengingproblem,inparticularforproblemsinvolving maining abundance of 26Al is low enough not to melt
such extremely complex geometries. It is an interesting the planetesimal but high enough to heat it to tempera-
challenge for the future. For getting order-of-magnitude tures only slightly below the melting temperature. This
estimates we may be forced to stick, for the time be- might be too much coincidence.
ing,withsimplemodelssuchastheonepresentedinthis On the other hand, we know that some planetesimals
paper. were strongly differentiated: the parent bodies of iron
and basaltic meteorites. If chondrules formed from col-
5.2. Chondrule sizes lisions between planetesimals, one would expect some
chondrulestohavebeenformedfromcollisionswithsuch
According to the model by Asphaug et al. (2011), by
differentiated planetesimals. Some chondrules should
which the lava drop size is calculated from the initial
have compositions that are very iron poor, presumably
pressure under which the magma was in the pre-molten
moreiron-poorthanLL-chondrites. Thisraisestheques-
planetesimals before the collision, the planetesimal sizes
tion where are these? One possible answer is that chon-
must have been 10 km or larger. Since our cloud of lava
drules produced in these very early phases (< 1 million
droplets can be one of a number of such clouds ejected
years after CAI formation) were either accreted into the
from the impact, the R of our model cannot be di-
melt,0 sunorreincorporatedintootherbodies,wheretheywere
rectly compared to that > 10 km size lower limit from
again molten. But if this process of elimination is so ef-
Asphaug. Butitdoesgiveanindicationthat,ifweadopt
ficient, why did some CAIs survive andgetincorporated
theirmodelofthechondrulesizes,itseemsreasonableto
intothe (later)chodrites? Thisshowsthatwhile the im-
be looking to values of roughly the order of R ≃10
melt,0 pactsplashingmodel is appealing, there is more workto
km or larger. To elucidate our reasoning: If we collide
be done to answer such questions.
two fully molten planetesimals of 20 km radius, and we
model the splash with 10 spherically expanding droplet 5.4. Compound chondrules
clouds, then we would get R ≃ 12 km. In Fig. 3
melt,0 In our homologously expanding cloud model the melt
this means we are looking at the region of the diagram
droplets move away from their nearest neighbor at ex-
above the dotted line.
tremely low velocity (millimeters per minute). Small in-
Thereisacaveat,however: Whattellsusthatitisjust
homogeneities(whichareofcourseunavoidableinsucha
10clouds,nota1000? Thisbringsustothe complicated
messyprocessasanimpact)wouldleadtosomedroplets
issue of the actual geometry of the splash. If we “de-
actually colliding. When they do so during the initial
compose”the splashgeometryintomanysmallspherical
constant temperature phase, they presumably coalesce
clouds we must assure that these clouds can radiatively
andformalargermeltdroplet. However,whenthey col-
cool to the outside. In other words: we cannot consider
lideduringthecoolingphase,theymayhavebecomecool
a big spherical cloud as consisting of many smaller sub-
enough to be plastic but not liquid anymore. They will
clouds, because they will irradiate each other and not
then produce a compound chondrule.
coolmuch. But if the cloud geometryis a long extended
It would be useful if we could quantify the percentage
streak, it can be considered as consisting of a chain of
of chondrules that become compound in this way. How-
spherical subclouds. We are here confronted with the
ever, at present we see no way how we can estimate the
limitation of our spherical expanding cloud model.
random relative velocities between adjacent chondrules
in the expanding clouds.
5.3. Partial pre-melting
Manycompoundchondrules,however,clearlyshowone
The reasonwhy pre-melting seems to be necessary for chondrule having remained rigid, while the other having
theimpactsplashscenariotoworkisthatveryhighspeed been indented. So clearly they were of different temper-
collisions (& 5 km/s) needed to produce melt from cold ature. This is difficult to achieve in our simple spherical
planetesimalsarepresumablyrare. However,pre-melting expanding cloud model. For this it will be necessary to
may lead to differentiation of the planetesimals, causing have other debris (e.g. another cloud-fragmentemerging
the iron and nickel to sink into the core and produc- from the impact site) to interdisperse with our cloud of
ing a basaltic mantle. Chondrules formed from such ob- droplet, so as to have chondrules from different thermal
jects would be very non-solar in composition. Although environments to mix. And this must happen on a time
Sanders & Scott (2012) argue that convection reduces scaleshorterthanthetimescaleofthermalequilibration
theefficiencyofthedifferentiationandthatsomeamount between the chondrules from these different origins. To
of differentiation is in fact consistent with e.g. L and LL make this more quantitative would be, however, a chal-
chondrites,itseemsthatthisissueisnotyetconclusively lenge. It wouldlikely requireus to contemplate complex
solved and more detailed and quantitative modeling of splash geometries.
the differentiation process in these molten planetesimals Metzler(2012)foundthatsomechondritesinfactcon-
may need to be done. tain entire large scale clusters of compound chondrules.
10 Dullemond, Stammler, Johansen
A possible explanation could be that some of the im- An important question in this scenario is how aerody-
pact debris may have fallen back onto the remainder of namic drag and the resulting radial drift affects this:
the biggest of the two colliding planetesimals. If this willthechondrulesstaylongenoughintheasteroid-belt-
happens quickly enough, so that this re-accretionoccurs regionbeforetheydiftawayornot? Jacquetetal. (2012)
beforethechondruleshavecooledbelowthetemperature study this question in detail.
atwhichthe chondrulesarestillsticky,thenit isreason- Perhaps more likely is a combination of the two. In
abletoassumethatsuchclustersmayformaschondrules particular the cluster chondrites of Metzler (2012, see
simply fall onto each other. Section 5.4) can be easily understood in terms of the
reaccretion scenario while the multiple ages seems more
5.5. What about relict grains and rims? topointtowardadispersionandlateraccretionscenario.
Another problem with our simple spherically symmet-
5.7. Volatile elements
ricexpandingplumemodelisthatitdoesnotaccountfor
any possible “pollution” of the melt droplets with other One of the main characteristics of most chondrules is
(non-molten) debris. Some chondrules contain relict their “normal” abundance of volatile elements such as
grains(Jones1996),whichmusthavebeeninsertedwhile Na, K etc. If a chondrule would stay at a high tempera-
the chondrule was still liquid, but the chondrule must ture (e.g. 2000K) for more than a few minutes, these el-
have cooled very soon thereafer. As in the case with the ements would have evaporated out of the chondrule and
compound chondrules (Section 5.4) this would require leave a volatile-depleted chondrule behind. We believe
debrisfromadifferentpartoftheimpactdebriscloudto that the impact splash scenario may naturally resolve
interdispersewithourcloudofdroplets. Thisforeignde- this issue, because during the hot phases of the expand-
bris must have originated from non-molten parts of the ingimpactsplashthedensityofthe cloudissohighthat
original planetesimals, e.g. the regolith or pristine dust the vapor quickly reaches the saturation pressure and
covering the crust of the original planetesimals. Only evaporation stops. However, this idea must be verified
those grains from that non-molten debris that entered by actual modeling: Will the evaporation indeed satu-
late enoughcanenterthe chondrulesshortly before they rate? Will the temperature drop below the evaporation
cool below the solidus, and thus survive. Like with the temperature before the density of the expanding cloud
compound chondrules, however, getting quantitative es- becomes too low to saturate the pressure? Answering
timates of this process is challenging. these questions with explicit calculations is the topic of
paper II in this series.
5.6. Producing chondritic parent bodies
5.8. What kind of chondrules qualify for this scenario?
Theimpactmodelmayprovideaplausiblescenariofor
theformationofchondrules,buthowdothesechondrules Sofarwehavenotmadeanydistinctionbetweendiffer-
(and matrix) accrete to form another planetesimal (the ent kinds of chondrules or chondrites. Given that chon-
parent body of the later chondritic meteorite)? A nice drule properties vary considerably between chondritic
aspect of the low-velocity collision of pre-molten plan- classes, the natural question arises: do all chondrules
etesimals model is that much of the debris may, in fact, formthroughthe same mechanismor aredifferent chon-
reaccreteontooneofthesurvivingoriginalplanetesimals drule formation scenarios responsible for different kinds
(or what remains of it). The idea is that due to the of chondrules? And for which kind, if at all, could the
high gas density in the protoplanetary disk the stochas- impact splash scenario of this paper be applicable?
ticplanetesimalmotionsareoflowvelocitysothatwhen A special class of chondrites and chondrules is the CB
theycollide,they collideessentiallyatvelocitiesthatare andCH class. It has been recognizedfor some time that
only just above their mutual escape speed. Since much thesechondrulesmayindeedhaveoriginatedfromcollid-
of the energy is dissipated in the inelastic impact, most ing planetesimals or planets (see Desch et al. 2012 for a
of the debris will not have enough kinetic energy to es- discussion). The evidence for this includes (1) the cryp-
cape the system and will (after perhaps a flyby or two) tocrystallinestructureofCBchondrules,(2)theirstrong
eventually accrete onto a single body or a binary body. depletion of volatile elements, (3) zoned metal spherules
However,there is evidence that inside a single chondrite thatappeartohavebeencondensedoutofthe gasphase
there exist chondrules of different ages (Villeneuve et in a highly energetic single-staged process (Krot et al.
al. 2009). And some chondrules experienced multiple 2005) and (4) their young age, 5 Myr after CAIs. One
heating events. Somehow chondrules produced in multi- can interpret these properties as meaning (1) that these
plechondrule-formingimpactsmusthavemixedtogether chondrules must have cooled more rapidly (seconds to
and form a single chunk of rock that later is observedas minutes) than “normal” chondrules (hours) to explain
a chondritic meteorite. The immediate re-accretion sce- their textures, (2) that the vapor-melt plume of the im-
nariomaymakethissomewhatdifficultbecauseoncethe pact must have rapidly become of low enough density
collisionreleasesmostofthe magma,andmuchofthisis to allow volatile elements to get out of the chondrules
reaccreted as chondrules (for the molten part) and ma- and not recondense, (3) the impact must have been of
trix (for the non-molten part) the remaining body will veryhighvelocity(severalkm/s)toproducemetalvapor,
no longer be a sphere of magma. (4) which appears consistent with the young age which
If, however,the collisions would be high-speed enough means low gas densities in the protoplanetary disk and
thatmuchof the debris escapes,the newly formedchon- thus allows for high-speed impacts. Arguments (1) and
drules will disperse into the protoplanetary disk. There (2) seem to require a rapid reduction of the density and
they mix with the pool of other chondrules of various optical depth, which can be achieved by a high impact
ages,andthenlateraccreteintoachondriteparentbody. velocity (also required for (3) and consistent with (4))