Title
Natural separation of two primordial planetary reservoirs in an expanding
solar protoplanetary disk

Authors
Beibei Liu1∗,3 Anders Johansen2,3, Michiel Lambrechts2,3, Martin Bizzarro2,
and Troels Haugbølle4

Affiliations
1 Zhejiang Institute of Modern Physics, Department of Physics & Zhejiang
University-Purple Mountain Observatory Joint Research Center for Astronomy,
Zhejiang University, 38 Zheda Road, Hangzhou 310027, China
2Center for Star and Planet Formation, GLOBE Institute, University of
Copenhagen, Øster Voldgade 5-7, 1350 Copenhagen, Denmark
3Lund Observatory, Department of Astronomy and Theoretical Physics, Lund
University, Box 43, 22100 Lund, Sweden
4Niels Bohr Institute, University of Copenhagen, Øster Voldgade 5-7, 1350
Copenhagen, Denmark

Abstract
Meteorites display an isotopic composition dichotomy between non-carbonaceous
(NC) and carbonaceous (CC) groups, indicating that planetesimal formation in
the solar protoplanetary disk occurred in two distinct reservoirs. The
prevailing view is that a rapidly formed Jupiter acted as a barrier between
these reservoirs. We show a fundamental inconsistency in this model: if
Jupiter is an efficient blocker of drifting pebbles, then the interior NC
reservoir is depleted by radial drift within a few hundred thousand years. If
Jupiter lets material pass it, then the two reservoirs will be mixed. Instead,
we demonstrate that the arrival of the CC pebbles in the inner disk is delayed
for several million years by the viscous expansion of the protoplanetary disk.
Our results support that Jupiter formed in the outer disk (${>}10$ AU) and
allowed a considerable amount of CC material to pass it and become accreted by
the terrestrial planets.

Introduction

The chronological ages of meteorites are inferred either by dating the
formation of their main components (e.g., chondrules), or by dating specific
physicochemical processes (e.g., core-mantle segregation or aqueous
alteration) to derive the parent-body accretion time through thermal modeling.
Such studies indicate that the accretion of meteorite parent-bodies in the
Solar System started already within a few hundred thousand years from the
formation of the CAIs (?) (generally taken to be the starting time of Solar
System formation) and lasted for several Myr (?, ?, ?). The difference in
composition between the NC group, originating in the inner Solar System, and
the CC group, originating in the outer Solar System, needs to be present in
the very early disk phase, within a few $10^{5}$ yr to explain the formation
of primitive achondrites and iron meteorites of NC composition. These two
reservoirs must remain radially separated at least for $2{-}3$ Myr in order
for the primitive NC and CC parent-bodies to form around that time with
distinct isotopic compositions (?).

Disk thermal processing has been proposed to explain the origin of these two
disk reservoirs (?, ?). In this scenario, Solar System dust is made up of
refractory presolar grains and more volatile mantles grown in the interstellar
medium (ISM) (?). The presolar grains are s-process dominated and likely
originate from asymptotic giant branch stars, whereas the ISM grains have a
near-solar isotopic composition (CI-like) condensed from the gas that is
already well-mixed from different isotopic sources (e.g., supernovae). The NC
composition is then a result of the selective thermal destruction of either
ISM-grown mantles or fragile presolar carriers (?, ?). During the early infall
phase, the episodic accretion associated with an FU Orionis-type outburst
temporally raises the disk temperature and moves the water ice line to the
$20{-}30$ AU disk region (?). This would have naturally resulted in the
separation of the NC component from the bulk CI component out to those
distances, provided that the supernova-derived material resides chiefly in the
fragile mantle of ice and organics. Alternatively, the initial two distinct
disk reservoirs could also be inherited from isotopically heterogeneous
accretion of molecular cloud gas onto the disk (?, ?).

Previous studies have suggested that an early and rapid growth of Jupiter is
necessary to explain the subsequent $2{-}3$ Myr separation of these two
distinct reservoirs (?, ?, ?). Here, we find a fundamental inconsistency with
the Jupiter barrier scenario, which requires an alternative model. We
demonstrate that, after the formation of the protoplanetary disk, the
combination of viscous spreading and radial drift of dust particles can
consistently explain the nucleosynthetic isotopic compositions in both the
asteroid and planetary record – without invoking Jupiter to block the inwards
flow of solid material.

Results

We provide a simple estimation of the solid particles’ radial drift timescale.
Their radial drift velocity is given by (See Materials and Methods)

$v_{\rm drift}=-2\tau_{\rm s}\eta v_{\rm K}+u_{\rm g},$ (1)

where $v_{\rm K}{\equiv}\Omega_{\rm K}r$ is the Keplerian velocity, $r$ is the
orbital distance, $\Omega_{\rm K}$ is the angular velocity,
$\eta{=}-\frac{1}{2}\frac{\partial\ln P}{\partial\ln r}\left(\frac{H_{\rm
g}}{r}\right)^{2}$ is the headwind prefactor (a measure of how large the gas
rotation velocity deviates from the Keplerian velocity), $P$ is the gas
pressure, $H_{\rm g}$ is the disk gas scale height and $u_{\rm g}$ is the gas
velocity at the disk midplane, which is proportional to the midplane turbulent
viscous parameter $\alpha_{\rm t}$. The Stokes number of the particle is
defined as $\tau_{\rm s}{\equiv}\Omega_{\rm K}t_{\rm stop}$ where $t_{\rm
stop}$ is the stopping time (a measure of how fast the particle adjusts its
momentum to the surrounding gas).

The particle’s Stokes number in the fragmentation-limit growth regime is

$\tau_{\rm s,F}=\frac{v_{\rm F}^{2}}{\alpha_{\rm t}c_{\rm s}^{2}}\simeq
2\times 10^{-3}\left(\frac{\alpha_{\rm t}}{2\times
10^{-3}}\right)^{-1}\left(\frac{v_{\rm F}}{1.5\rm\
m/s}\right)^{2}\left(\frac{c_{\rm s}}{0.8\rm\ km/s}\right)^{-2},$ (2)

where $v_{\rm F}$ is the fragmentation threshold velocity. In the inner disk
regions with a moderate midplane turbulence (e.g., at $5$ AU), $\alpha_{\rm
t}{\lesssim}\tau_{\rm s,F}{\ll}1$, and the drift timescale can approximately
expressed as

$t_{\rm drift}\simeq\frac{1}{2\eta\tau_{\rm s}\Omega_{\rm K}}\simeq
0.06\left(\frac{\eta}{0.007}\right)^{-1}\left(\frac{\alpha_{\rm t}}{2\times
10^{-3}}\right)\left(\frac{v_{\rm F}}{1.5\rm\
m/s}\right)^{-2}\left(\frac{c_{\rm s}}{0.8\rm\
km/s}\right)^{2}\left(\frac{r}{5\ \rm AU}\right)\ \rm Myr.$ (3)

The above timescale strongly depends on the fragmentation threshold $v_{\rm
F}$. For a high fragmentation threshold ($v_{\rm F}{\gg}1\,{\rm m/s})$, the
larger particle sizes lead to more rapid orbital decay of drifting particles,
which results in a faster mixing between NC and CC reservoirs. On the other
hand, for disks with a high turbulent level $\alpha_{\rm t}{\gg}\tau_{\rm
s,F}$, solid particles are well coupled to the gas and their radial drift is
determined by the gas accretion flow.

We construct a $1$-dimensional model for the gas and solid components in a
protoplanetary disk. The disk is evolved using the conventional
$\alpha$-viscosity approach. Most of the gas in the expanding disk is accreted
onto the central star while the rest spreads outwards to conserve the angular
momentum (?). The solid particles are allowed to grow to their fragmentation-
limited size (?), yielding particle sizes that are smaller further from the
star and later in time. The effects of the gas spreading and dust radial drift
are included in the calculations.

We show in Figure 1a that the particles initially at $r{\gg}r_{\rm exp}$ move
outward together with the expanding gas, where $r_{\rm exp}$ is the separation
radius between the gas inflow and outflow. Their motion then becomes inward
when the particles’ radial drift overwhelms the outward spreading of the gas
at later times. We find that dust particles starting at $1{-}10$ AU drift into
the inner disk within $1$ Myr, whereas those initially located in the
$20{-}30$ AU region sweep the asteroid belt $2{-}4$ Myr later, ending up with
a millimeter characteristic size. This is also the case when we consider a
disk with less vigorous turbulence in the midplane than the upper surface
layer. In this so-called dead zone model the local midplane turbulence further
depends on the column density of the gas in the protoplanetary disk (see
Materials and Methods). We additionally account for the early disk building
phase in Figure 1c, where the angular momentum is dominated by the infall
process with gas only being accreted inward. After a typical infall time of
$0.3$ Myr, the disk viscosity dominates angular momentum transport and leads
the outward spreading of the gas. Our result implies that the inner disk can
preserve its NC nature until the CC dust particles drift through, as long as
the initial boundary of NC/CC reservoirs ($r_{\rm CC}$) is larger than the
original viscous expansion radius of gas ($r_{\rm exp}$).

In the prevailing view, the early and rapid formation of Jupiter has been
invoked to explain the NC-CC separation (?, ?, ?). When the core of Jupiter
grows to approximately $10{-}20\ \rm M_{\oplus}$ (?), it starts carving a gap
in the gas disk, preventing the outer CC solids from drifting inwards and
mixing with the inner NC reservoir. Importantly, this model requires proto-
Jupiter to form early within $0.5{-}1$ Myr in order to explain the formation
of differentiated planetesimals in both reservoirs. We identify an
inconsistency in this model in Figure 2a: if Jupiter appears early in the $5$
AU region and totally blocks the flow of solids, then the solids of NC
composition interior of Jupiter would drift into the Sun within a few hundred
thousand years (Figure 2a). This would then preclude the formation of the
primitive NC parent-bodies after $2{-}3$ Myr.

Thus, a second scenario is considered in Figure 2b where Jupiter allows
material to permeate past the gap and replenish the inner reservoir. However,
in this circumstance the CC reservoir quickly overtakes the inner regions of
the protoplanetary disk and there would be no compositional dichotomy
preserved in the primitive NC/CC parent bodies formed after 2–3 Myr. Reducing
or stopping the radial drift of solids interior of Jupiter could solve this
inconsistency (?, ?), but this suggestion comes at the expense of ignoring or
reducing the radial drift of solids – a fundamental physical process operating
in protoplanetary disks – in order to separate the reservoirs.

In this context, it has then been suggested that the NC/CC reservoirs were
isolated due to trapping of pebbles in speculative long-lived pressure bumps
in the inner disk (?) or through continuous and efficient planetesimal
formation around the water ice line (?, ?). The latter model requires that the
inner disk experiences either intensive irradiation or strong viscous heating
to keep the water ice line away from the terrestrial planet zone and
additionally extremely high planetesimal formation efficiencies that convert
hundreds of Earth masses of CC material to planetesimals. This raises the
concern that a large fraction of those planetesimals would subsequently become
implanted in the asteroid belt due to the migration of Jupiter (?).

Here we instead propose that the viscous expansion of the gas naturally
separates the two disk reservoirs (Figure 2c). As such, there is no need to
invoke very early Jupiter formation in order to explain the NC-CC separation.
In this case, the core of proto-Jupiter forms in a wide orbit and then
undergoes inward migration. Adopting a state-of-the-art pebble accretion and
planet migration model (?), we provide an illustration of Jupiter’s growth and
migration track in Figure 2c. In our model, Jupiter finishes the growth of its
$10\ \rm M_{\oplus}$ core at a time of $t{\simeq}3$ Myr in the $10$ AU disk
region and reaches its full mass (by gas accretion) and the current position
at $t{\simeq}4.2$ Myr.

These two contrasting models for the NC/CC dichotomy in the context of the
known constraints on parent-body formation timings are illustrated in Figure 3
(see also Materials and Methods). The expansion of the protoplanetary disk is
a key physical mechanism needed to understand the asteroid and planetary
record of the Solar System. As shown in Figure 2c, with viscous disk
spreading, the outer CC material arrives in the inner disk region only after
$2{-}3$ Myr. This expansion allows the parent bodies of NC differentiated
meteorites, achondrites, enstatite, and ordinary chondrites to form
sequentially during the first $2{-}3$ Myr in the inner disk region (Figure
3c). The later formed ordinary chondrites reveal a modest compositional
deviation from earliest formed Ureilites towards CC chondrites, implying a
continuous admixing between the original inner NC disk reservoir and outer CI-
like material (e.g., ureilites and CI chondrites as end members for the
mixing) (?). This is a natural consequence of the planetesimals forming at the
time where the NC and CC reservoirs partly mix, as illustrated by the shaded
region in Figure 3c. The small (${<}100$ km) NC asteroids are protected
against further accretion of substantial amounts of drifting CC pebbles. The
pebble accretion efficiency depends strongly on the inclinations and
eccentricities of the planetesimals. The internal gravitational stirring of
the primitive NC asteroids will quickly raise their eccentricities to a degree
where subsequent accretion of CC material is insignificant (see Materials and
Methods). A similar early excitation of the orbit of Mars has been proposed to
explain how Mars terminated its accretion earlier than Earth and thus
incorporated a lower fraction of CC material (?).

The differentiated meteorite parent-bodies of CC composition form at
relatively large orbital distances ($r{>}r_{\rm CC}$) within $1$ Myr, whereas
the parent-bodies of CC chondrites form at $t{\simeq}3{-}4$ Myr exterior to
the orbit of Jupiter and are later scattered into the asteroid belt by the
combined gravity of Saturn and Jupiter (?). The terrestrial planets – proto-
Earth and Mars – accrete considerable CC material and complete their main
accretion within $4{-}5$ Myr (?, ?, ?). Our model allows Jupiter’core to
finalize its growth at a relatively late time (e.g., $t{\sim}2{-}3$ Myr) and
at large orbital distances, such as $r{>}10$ AU, in agreement with planet
formation models that include inwards migration due to gravitational torques
exerted by the protoplanetary gas disk on the planet (?, ?). Large-scale
inward migration of proto-Jupiter plays a decisive role in sweeping through
the formation zone of CC differentiated planetesimals. The orbital crossings
and close-encounters between Jupiter and those planetesimals result in the CC
iron meteorite parent-bodies to be frequently scattered into the inner
asteroid belt region (?).

Discussion

We first discuss the role of Jupiter on dust filtration. Hydrodynamic studies
find that while large mm-to-cm size particles are stopped at the pressure
maximum exterior to the planet, small grains that are well coupled to the disk
gas can pass the gap (?, ?, ?, ?). How the above size-dependent filtering
affects the isotope anomalies is not consistently accounted for in the early
Jupiter scenario (?). We show in Figure 4 the efficiency of Jupiter as a dust
filter, taking into account a size distribution of solids. Two different
viscosity prescriptions are considered: a constant $\alpha$ everywhere in the
disk (blue) and a higher value exterior to the gap (red). The latter assumes
an elevated viscosity due to the generation of vortices by the Rossby wave
instability associated with gap-opening in the inner regions of the disk where
the background turbulence is inherently low (?). The critical Stokes number of
particles to cross the gap decreases as Jupiter’s mass increases, whereas the
maximum Stokes number of particles in the fragmentation regime remains
constant at the planet location. When the mass of Jupiter is low, all
particles are below the critical Stokes number and can drift through the gap.
Jupiter begins to block particles only when the critical Stokes number is
lower than the fragmentation-limited particle size. Although the maximum size
particles dominate the total mass of the whole dust population, a non-
negligible fraction of them would unavoidably fragment into smaller particles.
We assume that the dust size distribution follows a power law, $n(a)\propto
a^{p}$, where $p{=}{-}3.5$, $a_{\rm min}{=}0.1\rm\ \mu m$ and $a_{\rm max}$ is
the maximum size in the fragmentation-limit. We find that a dust fraction of
$(a_{\rm crit}/a_{\rm max})^{4-p}$ still diffuse through the gap region when
$a_{\rm max}{>}a_{\rm crit}$. As shown in Figure 4e when Jupiter grows
rapidly, $24\%$ and $65\%$ of CC dust would drift through Jupiter’s gap after
$5$ Myr at $\alpha_{\rm gap}{=}0.001$ and $0.003$. This dust filtering works
much less effectively for a $3$ Myr gas accretion timescale in Figure 4f.

We note that larger cm-sized CAIs, condensed in the inner regions of the
protoplanetary disk and likely transported to the outer disk by winds and
outflows, will be efficiently trapped by the growing Jupiter and their igneous
nature will prevent the formation of CAI fragments that could pass the gap and
hence this can explain how the ordinary chondrites avoided accreting CAIs (?,
?).

Our scenario supports that the disk angular momentum is transported outwards
by the turbulent viscosity, for instance, via the magnetorotational
instability (MRI) (?). Although MRI turbulence is likely to be suppressed due
to non-ideal MHD effects in the inner ten AU disk region where disk winds
could drive a laminar accretion flow, it can operate at large disk radii where
the CC dust particles reside initially. In the outer disk region where
ambipolar diffusion dominates, the turbulent level depends on the strength of
the net magnetic field, dust abundance and XUV/cosmic ray penetration depth
(?). The disk turbulence can be inferred from the spectral energy distribution
or molecular line velocity measurements. Such observations show that some
disks display a relatively high $\alpha{\gtrsim}10^{-3}{-}0.1$ (?, ?),
compatible with measurements from non-ideal MHD disk simulations (?).
Moreover, a recent study found that even in the magnetized wind-driven disk
where the MRI is fully quenched, the magnetic torque can also drive the
expansion of the gas in the outer disk region, analogous to the viscous
spreading effect (?). This implies that our expanding disk scenario can be
applicable for a broad range of circumstances. Despite uncertainties in
measuring disk size, the observed sizes of Class II disks are plausibly larger
than those of Class I disks, in alignment with the above theoretical
predictions (?).

Summary

In summary, we propose that a generic outward-then-inward radial motion of
pebbles, driven first by the viscous expansion of the disk gas and thereafter
by the radial drift of pebbles, is the key ingredient that explains the
isotopic compositions of the entire known meteoritic and planetary record. In
this view, the observed NC/CC dichotomy is set by a secular change to the
composition of the disk and not by an early spatial separation by an in-situ
formed Jupiter (?, ?). We show that the gap opening by Jupiter will not stop
all inward-drifting solid particles and that Jupiter lets considerable amounts
of CC dust and pebble fragments enter the inner disk to contribute to the mass
budgets of the terrestrial planets as well as to the parent bodies of the
ordinary chondrites.

Our work implicitly supports a distant origin of Jupiter, which is consistent
with orbital migration theory (?, ?, ?). Indeed, many heavy elements including
ultra-volatile noble gases are found to be enriched in Jupiter’s atmosphere
compared to the solar photospheric values. This feature cannot be explained by
in-situ accretion and therefore strongly favors that Jupiter’s core formed in
a distant, cold orbit (Figure 3c) (?). In this view, heavy elements can be
accreted as solids that would later become diluted into the envelope,
resulting in the observed enrichment pattern.

Main Figures

Figure 1: Radial drift of dust particles in a viscously spreading disk with
(a) uniform turbulent model, (b) dead zone model, and (c) dead zone + infall
model. The black curves trace the radial drift of dust particles whose size is
limited in the fragmentation regime, while the grey dashed lines indicate the
gas trajectories. The background color refers to the size of the particle at
the corresponding time and disk location, with the magenta dash-dot lines
marking pebbles of $0.1$, $1$ and $10$ mm in size. The disk parameters are:
$v_{\rm F}{=}1\rm\ m/s$, $r_{\rm d}{=}10$ AU, $C_{\rm d}{=}0.14\ M_{\odot}$,
$T_{\rm irr0}{=}130$ K and $\alpha{=}5{\times}10^{-3}$ in the uniform
turbulent model, $\alpha_{\rm A}{=}10^{-2}$ and $\alpha_{\rm
D}{=}5{\times}10^{-4}$ in the dead zone model (fiducial run). In the dead zone
+ infall model, the early infall period where gas only accretes inward lasts
for $0.3$ Myr, and after that, the disk gas transports angular momentum
through viscous spreading. The outwards flow of gas in the expanding disk
brings along with it the solids from the outer regions of the disk. As the
solids detach from the gas flow, they drift into the inner disk regions with
the arrival time determined by their initial distance from the star.  Figure
2: Radial drift of pebbles (top) and the pebble flux entering the inner
terrestrial planet formation region (bottom). The trajectories of the NC and
CC dust particles are shown in red and blue, while the growth of Jupiter is
traced by the black curve with $1\ \rm M_{\oplus}$, $3\ \rm M_{\oplus}$, $10\
\rm M_{\oplus}$, $100\ \rm M_{\oplus}$ and full mass in circles. The gas moves
only inwards in panels (a) and (b) whereas the gas outward viscous spreading
is included in panel (c). In panel (a) we assume that Jupiter formed early and
that it entirely blocked the inward drifting solid particles after $0.5$ Myr,
while in panel (b) Jupiter is assumed to let all dust particles pass through.
In the first case, the inner (NC) reservoir is quickly depleted, while in the
second case the CC reservoir quickly takes over the inner disk. In panel (c)
we show that viscous expansion can keep the two disk reservoirs separated
within $2{-3}$ Myr. Jupiter is here assumed to form at a wide orbit after $1$
Myr. In top panels the black dashed lines represent Jupiter is not an
efficient dust filter while the black solid line assumes that solid particles
are all blocked by Jupiter. The vertical dashed lines in the bottom panels
mark when the NC and CC reservoirs get truncated. For example, in the left
bottom panel the NC reservoir is depleted at $t{=}0.45$ Myr while Jupiter
blocks the CC reservoir at $t{=}0.5$ Myr. The disk parameters are: $v_{\rm
F}{=}1.5\rm\ m/s$, $r_{\rm d}{=}8$ AU, $C_{\rm d}{=}0.14\ M_{\odot}$, $T_{\rm
irr0}{=}130$ K, $\kappa_{0}{=}10^{-2}$ and $\alpha_{\rm A}{=}8\times 10^{-3}$
and $\alpha_{\rm D}{=}10^{-3}$. Figure 3: Meteorite dating records and a
schematic illustration of the two Solar System formation scenarios. Panel (a):
symbols give the peak accretion ages of meteorites with uncertainties shown in
horizontal lines. Panel (b,c): The magenta and blue lines show example
trajectories of NC and CC dust particles, while the dashed line in the viscous
spreading scenario is the separation radius for gas inflow/outflow. For the
early Jupiter scenario, the core grows in-situ within $1$ Myr and blocks all
particles drifting from the outer part of the disk. The inner disk drains out
solids quickly, which conflicts with the later formation of ordinary/enstatite
chondrites and the incorporation of CC material into the terrestrial planets.
For the viscous spreading scenario, Jupiter’s core forms beyond (and slightly
later than) the CC differentiated parent bodies. The parent bodies of
primitive NC meteorites (ordinary chondrites and enstatite chondrites) form at
$2{-}3$ Myr while the inner disk region is changing composition. The primitive
CC parent bodies form at $t{\sim}3{-}4$ Myr and are scattered into the
asteroid belt together with the CC differentiated parent bodies by Jupiter’s
migration. Figure 4: Evolution of Jupiter’s mass, particle Stokes numbers and
CC dust mass interior to Jupiter after its core formation. Panel (a+b): the
mass of Jupiter starting from $20\ \rm M_{\oplus}$ at $t{=}1$ Myr, with a
characteristic growth timescale of $0.5$ Myr (left) and $3$ Myr (right). Panel
(c+d): critical (maximum) Stokes number $\tau_{\rm s,crit}$ that particles can
diffuse through the gap vs disk particles’ Stokes number in the fragmentation
regime for $\alpha_{\rm gap}{=}0.001$ (blue) and $0.003$ (red). The blue dots
and red triangles refer to $\tau_{\rm s,crit}(M_{\rm p})$ at two different
$\alpha_{\rm gap}$, where the data from the low-masses and Jupiter’s full mass
are adopted from Ref (?) and setup I of Ref (?), respectively. From Ref (?) we
calculate the critical sizes when the dust-to-gas ratio interior of the planet
drops below 50% of the exterior value and convert those into Stokes numbers.
Panel (e+f): The mass of CC dust particles that drift across the Jupiter’s gap
for for $\alpha_{\rm gap}{=}0.001$ (blue) and $0.003$ (red). The grey line
shows the CC dust drift exterior to the orbit of Jupiter for comparison. The
other disk parameters are $\alpha{=}10^{-3}$, $T_{\rm irr0}{=}130$ K, $C_{\rm
d}{=}0.14\ M_{\odot}$, $v_{\rm F}{=}1\ \rm m/s$ and $r_{\rm d}{=}10$ AU. The
growing Jupiter lets a significant fraction of the dust mass past its orbit,
unless it grows very fast and has a low turbulent diffusion outside the gap.

Materials and Methods

Chronology of meteorites

We summarize the literature chronological meteorite studies in Figure 3a. The
ages of chondritic parent-bodies can be deduced from the time when chondrules
were last melted and crystallized. This age is determined from either the
long-lived, U-corrected Pb-Pb dating or short-lived Al-Mg and Hf-W datings. In
the Pb-Pb dating approach, recent studies proposed that chondrules form
simultaneously with CAIs and last for $3{-}4$ Myr, whereas the age peaks
around $0$ to $1$ Myr (?, ?). In the Al-Mg dating approach, chondrules
typically record a lower initial Al isotopic ratio compared to that of CAIs,
the latter of which is treated as the canonical value for the onset of the
Solar System (26Al/27Al ${=}5.5\times 10^{-5}$). This is interpreted as a
$1{-}4$ Myr protracted formation of chondrules after CAIs, with a major age
peak at $t{=}2{-}3$ Myr (?, ?, ?, ?). Note that Pb-Pb dating of single
chondrules shows different ages from that of the pooled chondrule separates,
the latter of which seems more aligned with the results obtained from the Al-
Mg dating (?). On the other hand, the short-lived Al-Mg dating relies on the
assumption of an initial homogeneous distribution of 26 Al throughout the
inner Solar System, the validity of which is subject to debate in the
community (?, ?).

On the other hand, the age of meteorites can instead be inferred from the
isotope dating of specific physical/chemical events occurred in their history.
For iron meteorites (differentiated planetesimals), Hf-W dating probes the
ages when their parent-bodies’ cores became segregated from mantles. Then the
formation time of these bodies can be traced back via thermal modeling of
their internal evolution, based on the assumption that 26Al is the predominant
heat source. Similar approaches have been used for achondrites, ordinary and
carbonaceous chondrites, with different dating events such as magma
crystallization, formation of minerals like carbonates, and secondary
fatalities. In such a way, Vesta and the parent-bodies of ureilites, angrites
are deduced to form within $0.5$ Myr after the CAI formation (?). The parent-
bodies of iron meteorites are found to form earlier than $0.5$ Myr for the NC
population and earlier than $1$ Myr for the CC population (?). The parent-body
accretion of ordinary chondrites takes place at $t{\sim}1.6{-}1.8$ Myr (?),
while those of carbonaceous chondrites occur at ${\sim}2.5$ to $4$ Myr after
CAIs, spanning from CO, CV and CM types (?, ?).

Furthermore, the accretion history of terrestrial planets can be constrained
from their isotopic compositions. Many isotopes such as 48Ca, 50Ti and 54Cr of
Earth, Mars and Moon are found to be different from either NC or CC asteroid
populations (?, ?). The terrestrial planets originally grew in the NC
environment. One expects that these NC/CC disk reservoirs are temporarily
separated early on and mixed together later through drifting of CC dust
particles. As such, in order to account for their isotopic compositions, Earth
and Mars need to accrete a non-negligible fraction of CC material (an upper
limit of $30{-}40\%$ depending on different estimated isotopes) during the
protoplanetary disk phase. Inferred from Hf-W chronometry, Mars attains its
half mass within $3$ Myr (?), while the formation of Earth takes much longer,
approximately at $30{-}100$ Myr (?). The above Earth age derivation, however,
could be substantially affected by the moon-forming impact. It is plausible
that proto-Earth had completed the main accretion within the gas disk lifetime
and the last Moon-forming giant impact happened a few tens of Myr later (?).
The early and rapid formation of proto-Earth is more consistent with iron
isotope evidence (?). Ref (?) proposed a pebble accretion terrestrial planet
formation model that explains their architecture, bulk compositions as well as
isotopic variability. They suggested that Mars and proto-Earth accrete CC
materials $3{-}4$ Myr after the CAI formation.

Spreading disk model

We want to track the $1$D evolution of the gas and solid components in the
protoplanetary disk. A conventional disk model is constructed, including both
the gas spreading and dust radial drift. We focus on the stage when the star
completed its main accretion and disk has already formed. Thus, the disk-cloud
interaction and infall of envelope material onto the disk are neglected for
our fiducial setup. The influence of the early infall is further addressed in
Section Early disk formation. Although the disk formation and evolution are
found to be much more complicated in zoom-in $3$D hydrodynamic simulations
(?), the advantage of our simplified $1$D treatment is that it captures the
essential physical processes, at the same time with the feasibility of testing
the influence of model parameters at a considerable detail. Such explorations
can inspire and guide future sophisticated numerical modelings.

We recapitulate the classical theory of viscous spreading disk (?, ?). Based
on the assumption that viscosity increases linearly with radial distance
$\nu{=}\nu_{\rm d}(r/r_{\rm d})$, the characteristic timescale of viscous
accretion/spreading is expressed as

$t_{\nu}=\frac{r_{\rm d}^{2}}{3\nu_{\rm d}},$ (4)

where $r_{\rm d}$ is the characteristic disk radius at $t{=}0$ yr, and
$\nu_{\rm d}$ is the corresponding viscosity at $r_{\rm d}$. Here we define
the dimensionless time as $\tau_{\nu}{=}1+t/t_{\nu}$. The gas surface density
and disk accretion rate from the self-similarity solution are written as (?)

$\Sigma_{\rm g}=\frac{M_{\rm d0}}{2\pi r_{\rm d}^{2}}\left(\frac{r}{r_{\rm
d}}\right)^{-1}\tau_{\nu}^{-1.5}\exp\left[-\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right],$ (5)

and

$\dot{M}_{\rm g}=-3\pi\Sigma_{\rm g}\nu\left[1-2\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]=-\frac{M_{\rm
d0}}{2t_{\nu}}\tau_{\nu}^{-1.5}\exp\left[-\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]\left[1-2\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right],$ (6)

where $M_{\rm d0}$ is the initial gas disk mass and $r$ is the radial distance
from the central star. The gas velocity is given by

$u_{\rm g}=\frac{\dot{M}_{\rm g}}{2\pi r\Sigma_{\rm
g}}=-\frac{3\nu}{2r}\left[1-2\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right].$ (7)

Hereafter we consider a realistic two-component disk structure based on
different heating sources. The inner disk region is heated by viscous
dissipation while the outer disk region is dominated by stellar irradiation
(?). The standard $\alpha$ prescription is adopted for viscosity $\nu{=}\alpha
R_{\rm g}T/(\mu\Omega_{\rm K})$ (?) where $R_{\rm g}$ is the gas constant, $T$
is the disk temperature, $\mu$ is the gas mean molecular weight and
$\Omega_{\rm K}{=}\sqrt{GM_{\star}/r^{3}}$ is the angular velocity and
$M_{\star}$ is the stellar mass. The self-similarity solution is applicable
for the outer stellar irradiated disk region. The gas temperature and surface
density are given by

$T_{\rm irr}=\left(\frac{\phi L_{\star}}{4\pi\sigma_{\rm SB}\ \rm
AU^{2}}\right)^{1/4}\left(\frac{r}{1\ \rm AU}\right)^{-1/2}=T_{\rm
irr0}\left(\frac{r}{1\ \rm AU}\right)^{-1/2}.$ (8) $\Sigma_{\rm
g,irr}=\frac{C_{\rm d}}{2\pi r_{\rm d}\ \rm
AU}\tau_{\nu}^{-1.5}\exp\left[-\left(\frac{r}{r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]\left(\frac{r}{\ \rm AU}\right)^{-1},$ (9)

where $\phi$ is the grazing angle where the stellar light hits the disk
surface, $L_{\star}$ is the stellar luminosity, $C_{\rm d}$ is the equivalent
disk mass when the disk is purely stellar irradiated from Eq. (5),
$\sigma_{\rm SB}$ is the Stefan-Boltzmann constant and AU is the mean distance
between Earth and Sun. It is worth noting that $\phi$ is much lower than the
unit and should principally depend on $r$ in a sophisticated disk model
(leading to $T_{\rm irr}{\propto}r^{-3/7}$) (?). Here we simply adopt it to be
a constant. For the fiducial disk model when $\phi{=}0.01$ and $L_{\star}{=}1\
L_{\odot}$, the irradiation temperature $T_{\rm irr0}$ at $1$ AU is $130$ K.
Due to the variation of $\phi$ (related to disk opacity) and more luminous
young star with a higher $L_{\star}$ during its pre-main sequence, $T_{\rm
irr0}$ could be varied from $130$ K to $300$ K.

For the inner viscously heated disk region, the viscosity depends on both the
disk surface density and radial distance, and hence, the self-similarity
solution is not exactly applicable. We assume that in the inner disk region
the gas flow is maintained by Equation (6) such that $(\Sigma_{\rm g}\nu)_{\rm
vis}{=}(\Sigma_{\rm g}\nu)_{\rm irr}$. In the inner viscously heated region,
the energy balance between heating and cooling provides that

$\frac{9}{4}\Sigma_{\rm g}\nu\Omega_{\rm K}^{2}=2\sigma T_{\rm e}^{4},$ (10)

where $T_{\rm e}$ is the effective gas temperature at the disk surface. The
heat that is generated from viscous dissipation in the midplane radiates
vertically towards to the surface layer. For an optically thick radiative
disk, the midplane temperature gives $T^{4}{\simeq}3\tau/8T_{\rm e}^{4}$ and
the optical depth is $\tau{=}\kappa\Sigma_{\rm g}$. We follow a simple disk
opacity law $\kappa{=}\kappa_{0}T$ (?), and then the energy equation reads

$\frac{9}{4}\Sigma_{\rm g}\nu\Omega_{\rm K}^{2}=\frac{16\sigma
T^{3}}{3\kappa_{0}\Sigma_{\rm g}}.$ (11)

We obtain

$T_{\rm vis}=T_{\rm vis0}\tau_{\nu}^{-3/4}\exp\left[-\left(\frac{r}{2r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]\left(\frac{r}{1\ \rm AU}\right)^{-9/8},$ (12)

and

$\Sigma_{\rm vis}=\left(\frac{C_{\rm d}}{2\pi r_{\rm d}\ \rm AU}\frac{T_{\rm
irr0}}{T_{\rm vis0}}\right)\tau_{\nu}^{-3/4}\exp\left[-\left(\frac{r}{2r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]\left(\frac{r}{1\ \rm AU}\right)^{-3/8},$ (13)

where

$T_{\rm vis0}=\left(\frac{27\alpha\kappa_{0}R_{\rm g}\Omega_{\rm
K0}}{64\mu\sigma_{\rm SB}}\right)^{1/4}\left(\frac{C_{\rm d}}{2\pi r_{\rm d}\
\rm AU}\right)^{1/2}T_{\rm irr0}^{1/2},$ (14)

and $\Omega_{\rm K0}$ is the Keplerian angular velocity at $1$ AU.

The transition radius $r_{\rm tran}$ between these two disk regions can be
solved by equating $T_{\rm vis}$ and $T_{\rm irr}$,

$r_{\rm tran}^{-5/8}\exp\left[-\left(\frac{r_{\rm tran}}{2r_{\rm
d}}\right)\tau_{\nu}^{-1}\right]=\left(\frac{T_{\rm irr0}}{T_{\rm
vis0}}\right)\tau_{\nu}^{3/4}$ (15)

The initial disk mass is

$M_{\rm d0}=2\pi\int_{\rm in}^{\rm out}r\Sigma_{\rm g}(t=0)dr,$ (16)

and the outer edge of the disk (or disk size, $r_{\rm out}$) defines that
includes $95\%$ of its mass. The above two quantities can be solved
numerically. For the fiducial setup in Table 1, the initial disk mass $M_{\rm
d0}$ is $0.065\ M_{\odot}$ for the uniform turbulence model and $0.06\
M_{\odot}$ for the dead zone model. Figure 5 illustrates the evolution of gas
temperature and surface density for our fiducial setup.

We note that the disk angular momentum is transported from the inner region of
the disk to the outer region through the viscous stress. As such, most of the
disk gas moves inward while the remaining gas expanse to larger radii to
conserve angular momentum. Based on Eq. (7), the radius that separates mass
inflow and outflow is expressed as

$r_{\rm exp}=\frac{r_{\rm d}\tau_{\nu}}{2}=\frac{r_{\rm
d}}{2}\left(1+\frac{t}{t_{\nu}}\right).$ (17)

At the disk radius larger than $r_{\rm exp}$, the mass flux moves outward and
$u_{\rm g}{>}0$; at the disk radius smaller than $r_{\rm exp}$, the mass flux
moves inward and $u_{\rm g}{<}0$. Inserting Eq. (17) into Eq. (7), we rewrite
the gas velocity as

$u_{\rm g}=-\frac{3\nu}{2r}\left(1-\frac{r}{r_{\rm exp}}\right).$ (18)

The radial drift velocity of solid particle is given by (?)

$v_{\rm drift}=-\frac{2\tau_{\rm s}}{1+\tau_{\rm s}^{2}}\eta v_{\rm
K}+\frac{1}{1+\tau_{\rm s}^{2}}u_{\rm g},$ (19)

where $v_{\rm K}{\equiv}\Omega_{\rm K}r$ is the Keplerian velocity. The Stokes
number of the particle $\tau_{\rm s}{\equiv}\Omega_{\rm K}t_{\rm stop}$, and
$t_{\rm stop}$ is the stopping time that measures how fast the particle
adjusts its momentum to the surrounding gas. The headwind prefactor that
measures the disk gas pressure gradient is given by

$\eta=-\frac{1}{2}\frac{\partial\ln P}{\partial\ln r}\left(\frac{H_{\rm
g}}{r}\right)^{2}=C_{\eta}h_{\rm g}^{2},$ (20)

where $h_{\rm g}{=}H_{\rm g}/r$ is the disk aspect ratio, $H_{\rm g}$ is the
gas scale height, $C_{\eta}$ is an order unit factor related to the disk
structure. Combing Eq. (18) and Eq. (19), the outward movement of dust
particles requires

$\frac{\alpha}{\tau_{\rm s}}>\frac{4C_{\eta}}{3}\frac{r_{\rm exp}}{r-r_{\rm
exp}}.$ (21)

The final size that particles can reach depends on the growth pattern. We
consider two circumstances: particles are either in fragmentation-limited or
bouncing-limited regimes (?). In the fragmentation-limited growth regime the
highest Stokes number of particles is given by $\tau_{\rm s,F}{\equiv}v_{\rm
F}^{2}/\alpha c_{\rm s}^{2}$, where $c_{\rm s}{=}H_{\rm g}\Omega_{\rm K}$ is
the gas sound speed and $v_{\rm F}{=}1\ \rm m/s$ is the adopted fragmentation
threshold speed, related to the composition and size of the monomer grains (?,
?). The turbulent speed of the dust particles is given by $\Delta v_{\rm
t}{=}\sqrt{\alpha\tau_{\rm s}}c_{\rm s}$. Inserting it into Eq. (21), we
obtain the criterion of particles’ outward drift in the fragmentation regime:

$\alpha>\frac{v_{\rm F}}{c_{\rm s}}\sqrt{\frac{4C_{\eta}}{3}\frac{r_{\rm
exp}}{r-r_{\rm exp}}}.$ (22)

As can be seen from the above equation, a high $\alpha$ and hot disk promotes
the outward movement of dust particles.

On the other hand, some laboratory experiments find that the growth of the
particles is stalled at around $0.1{-}1$mm when the bouncing effect becomes
dominant (?, ?). In this work we assume that particles in the bouncing-limited
growth regime reach a radius of $0.3\rm\ mm$. The particles’ Stokes number
reads

$\tau_{\rm s,B}=\sqrt{2\pi}\frac{\rho_{\bullet}R_{0}}{\Sigma_{\rm g}},$ (23)

where $\rho_{\bullet}$ is the internal density of the dust particles, and
$R_{0}$ is the radius of the bouncing particles. Inserting Eq. (23) into Eq.
(21), we obtain the criterion of particles’ outward drift in the bouncing
regime:

$\alpha>\frac{4\sqrt{2\pi}C_{\eta}}{3}\frac{\rho_{\bullet}R_{0}}{\Sigma_{\rm
g}}\frac{r_{\rm exp}}{r-r_{\rm exp}}.$ (24)

We assume that dust particles have already reached the maximum sizes at the
beginning of numerical integration for the above two cases. This is valid as
long as the growth timescale is shorter than the viscous spreading timescale.
The growth timescale from micron-sized grains to mm-sized particles that
initiate substantial radial drift is given by $t_{\rm
grow}{\sim}\xi(Z\Omega_{\rm K})^{-1}$, where $\xi{\sim}10$ and $Z{=}10^{-2}$
is the canonical dust-to-gas ratio, while the viscous timescale is
$t_{\nu}{\sim}(\alpha h_{\rm g}^{2}\Omega_{\rm K})^{-1}$. It is easy to
calculate that $t_{\rm grow}{\ll}t_{\nu}$ for standard disk parameters.

The turbulent velocity is isotropic when the MRI is the predominant disk
turbulence source (?). The MRI is not able to operate when the ionization
level is low Hence, a layered accretion model has also been proposed, where
the turbulence in the midplane is strongly damped and accretion is mainly
through an ionized, active surface layer (?). We account for both these two
models: one assumes a uniform disk turbulence (constant $\alpha$ everywhere),
and the other incorporates the effect of the dead zone. In the latter model,
$\alpha$ – which sets the global viscous evolution of the disk – is much
higher than the turbulent level $\alpha_{\rm t}$ in the quiescent midplane
layer. The ionization relates to the column density of the disk gas, which
increases with orbital distance. We adopt a simple prescription for
$\alpha_{\rm t}$ in the dead zone model,

$\alpha_{\rm t}=\alpha_{\rm D}+(\alpha_{\rm A}-\alpha_{\rm
D})\times\frac{r-r_{\rm in}}{r_{\rm out}-r_{\rm in}},$ (25)

where $\alpha_{\rm D}$ and $\alpha_{A}{=}\alpha$ are the low and high boundary
of turbulent viscosity parameter and $r_{\rm in}$ is the inner disk radius
where the disk temperature is $1000$ K.

Parameter study

We investigate how the radial drift of dust particles influenced by different
parameters in both the uniform turbulence and dead zone models. The particles
are either in the fragmentation-limited or in the bouncing-limited growth
regimes. The parameter setups are provided in Tables $1$, while the simulation
results are depicted in Figures 6 and 7. For the size of the particles limited
by the fragmentation threshold, we find in Figure 6 that particles maintain
their outward movement for a less time when the disk is less turbulent, the
temperature is lower, the initial disk size is larger, and the fragmentation
threshold velocity is higher. Meanwhile, the outward-then-inward drifting
particles that eventually enter the asteroid belt have a larger size when the
disk is less turbulent, the temperature is lower, the initial disk size is
larger, the disk is more massive, the fragmentation threshold velocity is
higher and/or the particles are more porous with a lower internal density.

We note that the fragmentation velocity is a crucial factor in determining the
duration of the NC-CC separation, since the Stokes number of the particles
directly correlates to $v_{\rm F}^{2}$. We find that very large pebbles form
when the fragmentation threshold is high, (e.g., $v_{\rm F}{=}3\rm\ m/s$ in
Figure 6h). Such particles quickly decouple from the diffusing gas and drift
into the inner disk within less than $1$ Myr. It is therefore difficult to
generate the NC-CC separation if the fragmentation velocity is very high.

For the size of the particles limited by the bouncing barrier, we find in
Figure 7 that particles beyond certain disk radii (${\gtrsim}20$ AU) drift in
a convergent manner. Further out particles have high Stoker numbers that
already decouple from the disk gas to move inward, while particles with
slightly inner orbits still couple with the outwardly spreading gas. The
bouncing-limited size is also varied at $0.1$, $0.3$, and $1$ mm. When the
dust particles reach a smaller size, they can couple to the expanding gas for
a longer time, resulting in a longer NC-CC separation. Similarly, porous dust
particles with a lower internal density have a lower Stokes number compared to
those with a compact structure. Therefore, the porous dust particles can
maintain the outward expansion for a longer time. To summarize, in order to
separate the two disk reservoirs for more than $2{-3}$ Myr, a lower bouncing-
limited size or particles with a lower internal density are preferred.

In order to analyze the results statistically, we also perform a high number
of Monte Carlo sampling simulations by varying two disk parameters at each
time. The CC material injection radius $r_{\rm CC}$ is uniformly chosen from
$5$ to $35$ AU, the initial disk size $r_{\rm d}$ is uniformly chosen from
$10$ to $40$ AU, the disk temperature at $1$ AU ranges from $120$ to $400\rm\
K$, and $\alpha$ is log-uniformly selected from $10^{-3}$ to $10^{-2}$ in the
uniform turbulent model or $\alpha_{\rm D}$ from $10^{-4}$ to $10^{-3}$ in the
dead zone model. The particles in the bouncing-limited regime are assumed to
be $0.6$ mm in diameter. The results are given in Figure 8.

The duration of the NC-CC separation in the expanding disk model is strongly
influenced by the properties of the protoplanetary disk. Figure 8 shows
parameter dependencies of the arrival time of CC particles in the inner disk.
For the explored range of parameters, the time for CC particles reaching the
inner $1$ AU disk region spans from $10^{5}$ to $10^{7}$ yr, with a later
entering time for a larger $r_{\rm CC}$. Overall, the inner boundary of CC
material larger than $20$ AU is needed for the corresponding drift time of
${>}$ a few Myr. The drift timescale of particles beyond $r_{\rm CC}$
sensitively depends on the initial disk size, since it directly correlates to
places where disk gas can diffuse inward/outward ($r_{\rm exp}$).

Early disk evolution

Our study separates the disk-forming stage from the disk-evolution stage. We
define the former phase as the period in which gas from the parent molecular
cloud accretes onto the central protostellar core and the surrounding
circumstellar disk, while in the latter phase this process has come to an end.
The disk is then approximated as a closed and isolated system where gas
accretion onto the host star is paired with an expanding outer disk, in line
with conventional disk theory and observations (?). In the main paper, we show
that this disk evolution phase is sufficient to explain the observed isotopic
dichotomy. There is, however, a significant degree of uncertainty on the
earlier disk building phase (?, ?, ?).

Here we show that our assumptions on the disk formation stage are in line with
theoretical and observational constraints. Firstly, disks likely form rapidly:
a conventional Bonnor-Ebert sphere collapses within a free-fall timescale of
$T_{\rm ff}{=}\sqrt{r_{\rm BE}^{3}/GM_{\star}}{\sim}0.4$ Myr, where we take
the core mass to be a solar mass and a critical radial extend of $r_{\rm BE}$
to be ${\approx}0.1$ pc. Although some early condensates like CAIs may form
within this early period, most solids form in the disk evolution phase (Figure
3). This estimated free-fall timescale is compatible with the range of disk
formation timescales measured in the state-of-the-art magneto-hydrodynamical
simulations (?, ?, ?).

It is less clear how exactly mass is deposited onto the disk during this disk-
forming phase. In the classical picture, not considering magnetic fields,
material will collapse inside-out when the infalling material conserves its
angular momentum. The cloud-collapsing front expands outward, with high
angular momentum material deposited onto the disk at a later time, e.g. (?).

Current magneto-hydrodynamic (MHD) simulations similarly argue for inside-out
formation without a strong imprint of magnetic braking: ideal MHD simulations
typically show a strong and changing misalignment between the total angular
momentum and magnetic field vectors of young disks, when taking into account
the turbulent nature of the giant molecular cloud (?). Therefore, disks
undergo relatively isotropic accretion – with strong local and temporal
variation – of gas at $\sim$ $50$ AU that results in a radial mass flow in the
disk midplane (?, ?). We therefore assume that, within the infall phase, the
gas only accretes inward. After $t_{\rm infall}$, the disk evolution switches
to the nominal situation described in Section Spreading disk model. Recently,
it has been argued that magnetic braking and vertical accretion play a larger
role (?, ?). However, a statistical analysis of these results, starting from
the molecular cloud level, shows that disk properties for ideal and non-ideal
MHD (including ambipolar diffusion) simulations appear to be largely similar,
but in the latter case disks tend to be less massive ($<0.1\ $ M⋆) (?).
Regardless, the current mass and radial extent of the Solar System points
towards it not forming from such a potential population of small, low-mass
disks, but instead from an intermediate-sized disk of size $>50$ AU. Moreover,
outer Solar System solids with a low degree of thermal processing, like comets
and CI chondrites, were more likely delivered radially from the outer regions
of the collapsing cloud. This is in line with the model presented here with an
outer – more pristine – CC component. Delivering the bulk material closer to
the host star would give a potentially excessive degree of thermal processing.

Jupiter’s growth and migration

We adopt the state-of-the-art pebble-driven planet formation model (?). The
key equations for Jupiter’s growth and migration are recapitulated as follows.

The pebble accretion efficiency in the $2$D and $3$D limits read

$\begin{split}\varepsilon_{\mathrm{PA,2D}}&=\frac{0.64(1+\tau_{\rm
s}^{2})}{2\tau_{\rm s}\eta+(1+\tau_{\rm s}^{2})u_{\rm g}/v_{\rm
K}}\sqrt{\frac{M_{\rm p}}{M_{\star}}\frac{\Delta v}{v_{\rm K}}\tau_{\rm
s}}\end{split}$ (26)

and

$\begin{split}\varepsilon_{\mathrm{PA,3D}}&=\frac{0.78}{\left[2\eta+(1+\tau_{\rm
s}^{2})(u_{\rm g}/v_{\rm K})/\tau_{\rm s}\right]h_{\mathrm{peb}}}\frac{M_{\rm
p}}{M_{\star}}.\end{split}$ (27)

We take into account of the gas flow velocity in the above equations, $M_{\rm
p}$ and $M_{\star}$ are the masses of the planet and the star, and $\Delta v$
is the relative velocity between the planet and the pebble.

The pebble mass flux ($\dot{M}_{\rm peb}$) is assumed to be attached to the
gas flow with a constant mass flux ratio such that $\xi{\equiv}\dot{M}_{\rm
peb}/\dot{M}_{\rm g}$. Particles of very low Stokes number are well-coupled to
the disk gas. The gas and dust particles drift together with the same
velocities, and therefore $\xi$ remains equal to the initial disk metallicity.
When the Stokes number is high, pebbles drift faster than the disk gas. In
this case, in order to maintain a constant flux ratio, $\Sigma_{\rm
peb}/\Sigma_{\rm g}$ becomes lower than the nominal disk metallicity and the
pebble surface density gets reduced. The mass accretion onto the planetary
core can be written as

$\frac{dM_{\rm p,PA}}{dt}=\varepsilon_{\mathrm{PA}}\dot{M}_{\rm
peb}=\varepsilon_{\mathrm{PA}}\xi\dot{M}_{\rm g}.$ (28)

The core mass growth is terminated when the planet reaches the pebble
isolation mass,

$\begin{split}M_{\rm iso}=&25\left(\frac{h_{\rm
g}}{0.05}\right)^{3}\left(\frac{M_{\star}}{\ \rm
M_{\odot}}\right)\left[0.34\left(\frac{-3}{{\rm log}\alpha_{\rm
t}}\right)^{4}+0.66\right]\\\ &\left[1-\frac{\partial{\rm ln}P/\partial{\rm
ln}r+2.5}{6}\right]\ \rm M_{\oplus}.\end{split}$ (29)

After reaching the pebble isolation mass, the planet starts rapid gas
accretion

$\dot{M}_{\rm p,g}=\min\left[\left(\frac{dM_{\rm p,g}}{dt}\right)_{\rm
KH},\left(\frac{dM_{\rm p,g}}{dt}\right)_{\rm Hill},\dot{M}_{\rm g}\right],$
(30)

The first term in Equation (30) corresponds to the Kelvin-Helmholtz
contraction

$\begin{split}\left(\frac{dM_{\rm p,g}}{dt}\right)_{\rm
KH}=10^{-5}\left(\frac{M_{\rm p}}{10\ \rm
M_{\oplus}}\right)^{4}\left(\frac{\kappa_{\rm env}}{1\rm\
cm^{2}g^{-1}}\right)^{-1}\rm\ M_{\oplus}\,yr^{-1},\end{split}$ (31)

where $\kappa_{\rm env}$ is the opacity in of the planetary envelope.

The second term in Equation (30) indicates how much gas the in the planet Hill
is able to get accreted by the planet,

$\left(\frac{dM_{\rm p,g}}{dt}\right)_{\rm Hill}=\frac{f_{\rm
acc}}{3\pi}\left(\frac{R_{\rm H}}{H_{\rm g}}\right)^{2}\frac{\dot{M}_{\rm
g}}{\alpha}\frac{\Sigma_{\rm gap}}{\Sigma_{\rm g}},$ (32)

where $R_{\rm H}{=}(M_{\rm p}/3M_{\star})^{1/3}r$ is the Hill radius of the
planet, and $\Sigma_{\rm g}$ and $\Sigma_{\rm gap}$ are the unperturbed gas
surface density and the surface density at the bottom of the gap. The relative
depth of the gap can be given by

$\frac{\Sigma_{\rm g}}{\Sigma_{\rm gap}}=1+0.04\left(\frac{M_{\rm
p}}{M_{\star}}\right)^{2}\left(\frac{1}{h_{\rm
g}}\right)^{5}\left(\frac{1}{\alpha}\right).$ (33)

We also define the gap opening mass $M_{\rm gap}$ when $\Sigma_{\rm gap}$ is
reduced by $50\%$ compared to the unperturbed value.

Briefly, the Kelvin-Helmholtz contraction is the dominant gas accretion
channel when the planet mass is low. When $(dM_{\rm pg}/dt)_{\rm
KH}{>}(dM_{\rm pg}/dt)_{\rm Hill}$, the gas accretion is limited by the amount
of gas within the planet’s Hill sphere. Once the planet becomes sufficiently
massive, the available gas is eventually determined by the global disk
accretion flow. In this work, $\kappa_{\rm env}$ and $f_{\rm acc}$ are the two
variables that determine the timescale of gas accretion.

The planet interacts with sounding disk gas, inducing orbital migration. Such
a planetary migration rate is given by

$\dot{r}=\dot{r}_{\rm I}/\left[1+\left(\frac{M_{\rm p}}{M_{\rm
gap}}\right)^{2}\right].$ (34)

This formula summarizes for both the low-mass planet in the type I and massive
planet in the type II torque regimes. The type I migration rate reads

$\dot{r}_{\rm I}=-f_{\rm I}\left(\frac{M_{\rm
p}}{M_{\star}}\right)\left(\frac{\Sigma_{\rm g}r^{2}}{M_{\star}}\right)h_{\rm
g}^{-2}v_{\rm K},$ (35)

where $f_{\rm I}{=}2$ is adopted from Eq.(8) of Ref. (?) approximately for the
migration prefactor in a local isothermal disk.

We start the growth track of a protoplanet with $1\ \rm M_{\oplus}$ and
$t_{0}$,$r_{0}$, $\xi$, $f_{\rm acc}$ and $\kappa_{\rm env}$ are the
parameters that can be varied to generate the current Jupiter’s mass and
location. One example is illustrated in Figure 2c, where $t_{0}{=}10^{5}$ yr,
$r_{0}{=}45$ AU, $\xi{=}0.04$, $\kappa_{\rm env}{=}0.05\rm\ g/cm^{2}$ and
$f_{\rm acc}{=}0.9$. The other parameters are: $v_{\rm F}{=}1.5\rm\ m/s$,
$r_{\rm d}{=}8$ AU, $C_{\rm d}{=}0.14\ M_{\odot}$, $T_{\rm irr0}{=}130$ K,
$\alpha_{\rm A}{=}8\times 10^{-3}$ and $\alpha_{\rm D}{=}10^{-3}$ (dead zone
model).

Pebble accretion for asteroids on eccentric and inclined orbits

We explain how the inclination and eccentricity affect an asteroid’s growth by
pebble accretion. The orbits of the planetesimals are excited through their
mutual gravitational interactions. The random velocities of massive embryos
remain low, whereas the eccentricities/inclinations of small asteroids become
high. There are two consequences. First, the orbits of asteroids are lifted
away from the dust midplane layer due to increased inclinations, resulting in
fewer pebbles able to be accreted. Second, the pebble-asteroid encounter time
becomes short owing to their high relative velocity. When the encounter time
is shorter than the stopping time of pebbles, the pebble accretion efficiency
drops greatly. Both effects are more profound for lower-mass planetesimals.

Figure 9 shows the mass growth of asteroids varying with their eccentricities
and inclinations. The parent-bodies of ordinary and enstatite chondrites form
in the NC reservoir at $t{=}2$ Myr. However, even though $100\ \ \rm
M_{\oplus}$ CC dust particles drift across their orbits over the next $1$ Myr,
these asteroids accrete an insignificant amount to reverse their isotopic
signatures as long as their eccentricities/inclinations are beyond a few times
$10^{-3}$.

Supplementary Figures and Tables

Table 1:  Model setups for the parameter study. The radius of particles is either chosen to be $0.1$ (fiduical), $0.3$ or $1$ mm in the bouncing-limited growth regime or calculated from their Stokes number of $\tau_{\rm s,F}{=}v_{\rm F}^{2}/(\alpha_{\rm t}c_{\rm s}^{2})$ in the fragmentation-limited growth regime where $v_{\rm F}{=}1$ (fiducial), $0.3$ or $3\rm\ m/s$.  $\rm Run$ | $\rm dead\ zone$ | $\rm uniform$ | $T_{\rm irr0}$ | $C_{\rm d}$ | $v_{\rm F}$ | $R_{\rm peb}$ | $r_{\rm d}$ | $\rho_{\bullet}$
---|---|---|---|---|---|---|---|---
| $\alpha({=}\alpha_{\rm A})$ | $\alpha_{\rm D}$ | $\rm turbulence\ \alpha$ | $[\rm K]$ | $[\ \rm M_{\odot}]$ | $\rm[m/s]$ | $\rm[mm]$ | $\rm[AU]$ | $\rm[gcm^{-3}]$
$\rm fiducial$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $1$ | $0.3$ | $10$ | $1.5$
$\rm low\ \alpha$ | $10^{-3}$ | $5\times 10^{-4}$ | $10^{-3}$ | $130$ | $0.14$ | $1$ | $0.3$ | $10$ | $1.5$
$\rm hot\ disk$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $280$ | $0.14$ | $1$ | $0.3$ | $10$ | $1.5$
$\rm low\ mass\ disk$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.014$ | $1$ | $0.3$ | $10$ | $1.5$
$\rm large\ size\ disk$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $1$ | $0.3$ | $20$ | $1.5$
$\rm porous\ particle$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $1$ | $0.3$ | $10$ | $0.15$
$\rm high\ v_{\rm F}$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $3$ | $\setminus$ | $10$ | $1.5$
$\rm low\ v_{\rm F}$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $0.3$ | $\setminus$ | $10$ | $1.5$
$\rm large\ particle$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $\setminus$ | $1$ | $10$ | $1.5$
$\rm small\ particle$ | $10^{-2}$ | $5\times 10^{-4}$ | $5\times 10^{-3}$ | $130$ | $0.14$ | $\setminus$ | $0.1$ | $10$ | $1.5$

Figure 5: Two-component structure for a spreading disk. Panel (a): disk
temperature as a function of radial distance. Panel (b): gas surface density
as a function of radial distance. The thickness of line represents different
time of $t{=}0$ yr, $0.5$ Myr, $1$ Myr and $2$ Myr, respectively. Panel (c):
time evolution of the transition radius. The parameters are adopted from the
fiducial run with a uniform turbulence model in Table $1$.

Figure 6:  Parameter study on the radial drift of dust particles with
fragmentation-limited sizes in the uniform turbulence (left) and dead zone
(right) disk model. The black curves trace the radial drift of dust particles,
while the grey dashed lines indicate the gas spreading at the same initial
disk locations. The color refers to the size of the particle at the
corresponding time and disk location. The model setups are presented in Table
$1$.

Figure 7:  Parameter study on the radial drift of dust particles with
bouncing-limited sizes in the uniform turbulence (left) and dead zone (right)
disk model. The black curves trace the radial drift of dust particles, while
the grey dashed lines indicate the gas spreading at the same initial disk
locations. The color refers to the Stokes number of the particle at the
corresponding time and disk location. The model setups are presented in Table
$1$.

Figure 8: Monte Carlo simulation outcomes as a function of disk parameters.
The dead zone and uniform turbulence models with particles in the
fragmentation and bouncing-limited regimes are illustrated in from top to
bottom. The color corresponds to the time of CC dust particles entering the
inner disk region of $r{<}1$ AU. The circles with black edgecolor refer to the
systems that CC particles entering the terrestrial planet formation region
during $3{-}4$ Myr after the CAI formation. We fix the other model parameters
and vary the initial CC material injection radius vs midplane $\alpha$ (left),
initial disk size (middle) and $T_{\rm irr0}$ (right), respectively. The
default parameters are $\alpha_{\rm D}{=}5\times 10^{-4}$,
$\alpha_{A}{=}10^{-2}$ (for the dead zone model) or $\alpha{=}5\times 10^{-3}$
(for the uniform turbulence model), $r_{\rm d}{=}10$ AU, $v_{\rm F}{=}1\rm\
m/s$, $T_{\rm irr0}{=}130$ K and $C_{\rm d}{=}0.14\ M_{\odot}$, respectively.

Figure 9: Planetesimal mass growth as a function of inclination. The initial
sizes of planetesimals are $100$ km (thick) and $20$ km (thin), whereas their
orbits are either circular (solid) or with eccentricities two times of
inclinations (dashed). The disk is adopted to be the fiducial setup of the
uniform turbulence model and the size of particles is bouncing-limited to be
one millimeter. The planetesimals are assumed to form at $t{=}2$ Myr and the
following mass growth by pebble accretion is accounted for when $100\ \ \rm
M_{\oplus}$ pebbles drift bypass their orbits. The pebble accretion
prescription is adopted from Refs (?, ?).

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Acknowledgments

Funding
B. L. is supported by the start-up grant of the Bairen program from Zhejiang
University, National Natural Science Foundation of China (No.12173035 and
12111530175), and the Swedish Walter Gyllenberg Foundation. A. J. acknowledges
funding from the European Research Foundation (ERC Consolidator Grant
724687-PLANETESYS), the Knut and Alice Wallenberg Foundation (Wallenberg
Scholar Grant 2019.0442), the Swedish Research Council (Project Grant
2018-04867), the Danish National Research Foundation (DNRF Chair Grant
DNRF159) and the Göran Gustafsson Foundation. M. B. acknowledges funding from
the Carlsberg Foundation (CF18 1105) and the European Research Council (ERC
Advanced Grant 833275-DEEPTIME). T. H. acknowledges funding from the
Independent Research Fund Denmark (DFF8021-00350B).

Author contributions
A. J., B. L. and M. L. contribute equally to initiating the idea and drafting
the manuscript. B. L. performed numerical simulations. All authors contributed
to analyzing and discussing the results, their implications and in finalizing
the manuscript.

Competing interests
The authors declare that they have no competing interests.

Data and materials availability
All data needed to evaluate the conclusions in the paper are present in the
paper and the Supplementary Materials.