# Formation of Rocky Super-Earths From A Narrow Ring of Planetesimals

Konstantin Batygin1 & Alessandro Morbidelli2 1Division of Geological and
Planetary Sciences California Institute of Technology, Pasadena, CA 91125, USA
—<EMAIL_ADDRESS>
2Laboratoire Lagrange, Université Cote d’Azur, CNRS, Observatoire de la Cote
d’Azur, Nice, France —<EMAIL_ADDRESS>
The formation of super-Earths, the most abundant planets in the Galaxy,
remains elusive. These planets have masses that typically exceed that of the
Earth by a factor of a few; appear to be predominantly rocky, although often
surrounded by H/He atmospheres; and frequently occur in multiples. Moreover,
planets that encircle the same star tend to have similar masses and radii,
whereas those belonging to different systems exhibit remarkable overall
diversity. Here, we advance a theoretical picture for rocky planet formation
that satisfies the aforementioned constraints: building upon recent work —
which demonstrates that planetesimals can form rapidly at discrete locations
in the disk — we propose that super-Earths originate inside rings of silicate-
rich planetesimals at approximately $\sim$1AU. Within the context of this
picture, we show that planets grow primarily through pairwise collisions among
rocky planetesimals, until they achieve terminal masses that are regulated by
isolation and orbital migration. We quantify our model with numerical
simulations and demonstrate that our synthetic planetary systems bear a close
resemblance to compact, multi-resonant progenitors of the observed population
of short-period extrasolar planets. Our results thus indicate that the absence
of short-period super-Earths within the solar system can simply be attributed
to the comparatively low mass of the primordial planetesimal ring within the
protosolar nebula.

### Introduction.

It has long been known that the genesis of planets begins through the
coalescence of solids within protoplanetary nebulae, and models of planet
formation have traditionally assumed that dust within circumstellar disks is
smoothly distributed. Despite being common, this simplifying assumption may be
unfounded. Several lines of evidence have recently been marshaled in support
of the notion that rather than arising from a smooth gradient of solids,
planetesimal formation unfolds in a small number of discrete rings [1, 2, 3,
4, 5, 6]. In this vein, the work reported in ref. [3] proposes that
protoplanetary nebulae generally originate as decretion disks that spread
radially from tenths of an AU, facilitating the condensation of outward-
diffusing silicate vapor into rocky dust grains at the disk’s primordial
silicate sublimation-line. Importantly, this process naturally leads to the
formation of rocky planetesimals at a stellocentric distance comparable to the
Earth’s orbital radius (as well as the generation of more distant icy bodies
close to Jupiter’s present-day orbit) through gravito-hydrodynamic
instabilities [7, 8]. Such a model further yields a self-consistent
explanation for the isotopic dichotomy of carbonaceous and non-carbonaceous
iron meteorites, as well as the physical origins of the solar system’s broader
architecture [3, 6].

Figure 1: Schematic diagram of the planet formation scenario considered in
this work. Strong magnetic braking during the infall phase implies that the
protoplanetary nebula originates as a decretion disk that viscously spreads
outward from a few tenths of an AU, carrying minute dust grains to large
stellocentric distances (top panel). Beyond the condensation front, grains
grow and begin to drift inwards. Accordingly, a balance between the radial
outflow and sub-Keplerian azimuthal rotation of the gas leads to the
accumulation of rocky grains at the silicate sublimation line, at an orbital
distance of $r\sim 1\,$AU (middle panel). The envisioned picture requires the
satisfaction of three criteria [3]: a small (sub-AU) centrifugal radius for
the infalling material, phase-transitions of silicate species that facilitate
significant changes in characteristic particle radii across one or more
sublimation front(s), and a sufficiently quiescent disk for radial particle
pile-up and vertical settling to occur. While the cumulative mass of the
accrued dust ring is regulated by both metallicity as well as the vigor of
disk turbulence, given nominal parameters, the silicate annulus can readily
reach a mass on the order of tens of Earth masses. Gravito-hydrodynamic
instabilities facilitate the conversion of $\sim 1\,$mm dust into $\sim
100\,$km planetesimals, which subsequently merge to generate
multi-$M_{\oplus}$ objects. As planets deplete their local supply of solids
and grow massive enough to experience substantial disk-driven migration, they
exit the planetesimal feeding zone and undergo orbital decay (bottom panel),
which terminates when they reach the inner edge of the protoplanetary disk.
Importantly, when planets exit the planetesimal ring, their accretion stalls.
Thus, the terminal mass of super-Earth type planets is approximately set by
the balance between accretion and migration timescales, yielding a natural
propensity towards intra-system uniformity.

Within the framework of the aforementioned disk model, the mass budget of
silicate material that forms at the rock-line is distinctively variable
(Extended Data Figure). That is, depending on the specific combination of disk
viscosity and metallicity, the cumulative mass of rocky planetesimals
entrained within the $r\sim 1\,$AU silicate annulus can readily reach tens of
Earth masses (although we note that it can also be null if the threshold for
planetesimal formation through gravitational collapse is not met). Moreover,
numerical modeling indicates that planetesimal formation is expected to occur
over a relatively short temporal burst, such that dust is incorporated into
planetary building blocks over a timescale of $\sim 10^{5}$ years.

Adopting the ringed planetesimal formation paradigm as a platform, a key goal
of our work is to consider the possibility that a typical system of extrasolar
super-Earths originates within such a radially confined annulus of rocky
material. As we describe below, the process of planetary conglomeration within
a narrow ring of silicate-rich planetesimals naturally yields a characteristic
multi-$M_{\oplus}$ mass scale of the resulting planets, and the simultaneous
operation of accretion and orbital migration regulates the emergence of
uniformity among the growing planetary embryos (Fig. 1).

### Results.

The starting point of our calculation corresponds to the epoch of large-scale
planetesimal formation within a protoplanetary disk. For definitiveness, here
we adopt disk conditions derived from the simulations reported in ref. [3],
although we note that for the purposes of our calculations, any ringed
planetesimal formation scenario is likely to lead to similar results. Our
fiducial disk model is initialized with a gas surface density of
$\Sigma_{0}=2500\,$g/cm2 at $1\,$AU, a corresponding peak dust surface density
of $\Sigma_{\bullet}=500\,$g/cm2, and a dust grain radius of
$s_{\bullet}=1\,$mm, consistent with fragmentation-limited growth [9]. Owing
primarily to viscous energy dissipation, the disk maintains an appreciable
aspect ratio of $h/r\sim 0.05$ throughout the planetesimal formation epoch.
While the gas surface density is taken to dissipate exponentially with a time-
constant of $\tau_{\rm{disk}}=1.5\,$Myr, the dust surface density decays much
more rapidly, owing to the fact that pebbles get incorporated into a
$M_{\rm{ring}}\sim 20\,M_{\oplus}$ planetesimal swarm over a $\sim 10^{5}$
year timescale. The specific functional parameterizations of these quantities
are delineated in the Methods section.

Figure 2: The formation sequence of a mass-uniform exoplanetary system. Over
the course of the first $100{,}000$ years (panels A-E), $\mathcal{D}=100\,$km
super-planetesimals (gray, orange, red, green, and blue points, labeled
according to their inclinations) and lunar-mass planetary embryos (purple
circles) – comprising $M_{\rm{ring}}\approx 20\,M_{\oplus}$ in total – are
gradually introduced into the simulation domain. These objects originate with
eccentricities and inclinations of $\langle e\rangle\sim\langle i\rangle\sim
0.01$, across a radial range spanned by the horizontal line shown in panel A.
Growth of planetary embryos is driven primarily by accretion of planetesimals,
with aerodynamic drag and collisional damping facilitating enhanced
gravitational focusing (panels C-E). Injection of new material into the system
terminates at the $t=10^{5}$ year mark (panel E), and over the course of the
following few hundred thousand years, multi-Earth-mass planets emerge, with
the conglomeration process largely completed within the first $0.5\,$Myr
(panel F, G). Over the course of the remaining lifetime of the disk, the
formed planets migrate inwards, locking into a mass-uniform multi-resonant
chain (panels H). Recent work [23, 10] has shown that tightly packed multi-
resonant planetary configurations serve as ideal initial conditions for
reproducing both the period ratio distribution of observed extrasolar planets
as well as their inferred degree of mass-uniformity.

As clouds of dust within the $r\sim 1\,$AU silicate ring consolidate into
planetary building blocks by means of gravitational collapse, their continued
growth can proceed through two distinct channels: pairwise mergers among
planetesimals and pebble accretion. The efficiency of planetesimal accretion
is controlled by the extent to which gravitational focusing can increase the
collisional cross-section of protoplanetary embryos. Pebble accretion, on the
other hand, depends critically on whether the capture of dust proceeds in the
2D or 3D regimes — a determination that is sensitive to the characteristic
size of dust particles. Generically speaking, the process of collisional
fragmentation inhibits the growth of silicate grains beyond the millimeter-
scale within protoplanetary disks, ensuring that even in relatively quiescent
nebulae, turbulent stirring can maintain the dust sub-disk’s aspect ratio at
an inflated level [11]. Correspondingly, pebble accretion proceeds in the
comparatively inefficient 3D regime, contributing very little to the planetary
conglomeration process during the planetesimal formation epoch. We further
find that leftover dust that is not incorporated into planetary building
blocks through gravito-hydrodynamic instabilities, rapidly flows away from the
planetesimal ring as the nebula matures into an accretion disk, and our
estimates (see Methods section 6) indicate that any auxiliary exterior flux of
pebbles plays a negligible role in driving the formation of rocky super-Earths
(we confirm these analytic expectations with numerical simulations below).

#### Analytical Estimates.

In contrast with the relative inefficiency of pebble accretion in the inner
regions of a protoplanetary disk, the efficacy of planetesimal accretion
within a narrow annulus of rocky planetesimals is strongly enhanced. The
reasons for this are two-fold: first, by concentrating tens of Earth masses of
solids into a radially confined ring of planetesimals, the rate of collisions
among the constituent bodies is strongly amplified. Second, the combined
action of aerodynamic drag and inelastic collisions among planetesimals
constitutes a fast-acting damping mechanism for the planetesimal velocity
dispersion, magnifying the effect of gravitational focusing. In this regime,
the associated mass-accretion rate of a planetary embryo can be deduced from a
$n-\sigma-v$ relation, and the result is well-known [12, 14]:
$\dot{M}\sim\Sigma_{\rm{pl}}\,\pi\,R^{2}\,\Omega\,(1+\Theta)$, where
$\Sigma_{\rm{pl}}\sim\Sigma_{\bullet}$ is the planetesimal surface density,
$\Omega$ is the orbital frequency, and $\Theta=(v_{\rm{esc}}/\langle
v\rangle_{\rm{pl}})^{2}$ is the Safronov number (i.e., the ratio of the square
of the escape velocity to the square of the planetesimal velocity dispersion
[13]). Moreover, under the simplifying assumption of strong and time-invariant
gravitational focusing, it is straightforward to show that a crude estimate
for the timescale for an Earth-mass body to emerge within the ring of rocky
planetesimals is given by
$\mathcal{T}_{\oplus}\sim\bar{\rho}\,R_{\oplus}/(\Sigma_{\rm{pl}}\,\Omega\,\Theta)$
[15], where $\bar{\rho}\sim 3\,$g/cc is the embryo’s density. If we adopt the
fiducial parameters of our model and assume that the escape velocity of the
planetary embryo exceeds the planetesimal velocity dispersion by a factor of a
few (corresponding to $\Theta\sim 10$), $\mathcal{T}_{\oplus}$ can be as short
as $\sim 10^{5}\,$years.

No matter the dominant growth mode, planetary accretion cannot proceed without
bounds. For the problem at hand, two distinct processes constitute natural
termination mechanisms for planetary conglomeration, the first being
isolation. Isolation occurs due to the depletion of planetesimals from the
local feeding zone of the embryo. The expression for the isolation scale is
easily obtained by equating the cumulative mass of planetesimals within the
feeding zone (approximately two Hill radii within a heavily dissipated disk)
and the planetary mass itself, to yield $M\sim
8\,\pi^{3/2}\,\Sigma_{\rm{pl}}^{3/2}\,r^{3}/\sqrt{3\,M_{\star}}$. Given our
fiducial parameters, the isolation mass within the planetesimal ring evaluates
to $M\sim 3\,M_{\oplus}$.

A second growth-limiting process that ensues in our model is gas-driven
orbital migration. As a growing planet becomes massive enough to raise a
substantial wake within the gaseous nebula, the gravitational back-reaction of
the wake upon the planet drives an exchange of energy and angular momentum
between the planet and the disk, which in turn expels the planet from the
planetesimal ring altogether. Thus, within the framework of our theoretical
picture, an approximate equivalence between the mass doubling timescale
$\mathcal{T}_{\rm{mass}}\sim
3\,M^{1/3}\,\bar{\rho}^{2/3}/(\Sigma_{\rm{pl}}\,\Omega\,\Theta)$ and the
migration timescale [16]
$\mathcal{T}_{\rm{mig}}\sim(4/\Omega)(M_{\star}/M)(M_{\star}/\Sigma_{0}\,r^{2})(h/r)^{2}$
yields an estimate for the mass of planets that are expected to emerge from
the rocky annulus of planetesimals. Auspiciously, for the aforementioned
nominal parameters, the planetary mass scale that comes out from this relation
also evaluates to $M\sim 3\,M_{\oplus}$.

Figure 3: Architectures of exoplanetary systems at time of disk-dispersal,
generated within the framework of our model. The formation and evolution of
the system highlighted with a light-blue-green rectangle is depicted in Figure
2. As the mass of the planetesimal ring is increased from
$M_{\rm{ring}}=5\,M_{\oplus}$ to $40\,M_{\oplus}$, both the number of embryos
that achieve the planetary mass-scale, as well as their average mass itself
increase. Within the numerical model, radial migration is taken to smoothly
terminate at $r=0.5\,$AU across a characteristic length-scale of $\pm 0.1\,$AU
(see Methods section 5). This stellocentric distance marked by a dotted line
on the Figure, and also represents the boundary between short-period
progenitors to the observed population of Super-Earths and low-mass objects
that remain stranded close to their formation site. By and large, these
($r<0.5\,$AU) synthetic planetary systems adhere to a pattern of intra-system
mass uniformity with the normalized mass dispersion, $\mathcal{D}_{M}/\langle
M\rangle$, that systematically reaches values comparable to, or smaller than,
the observed value of $(\mathcal{D}_{M}/\langle M\rangle)_{\rm{data}}\approx
0.48.$ Cumulatively, these results explain how planetary systems can emerge
with a broad diversity of masses while retaining an unexpectedly high degree
of self-uniformity.

Because the isolation and accretion-migration terminal mass scales are
similar, the process of planetary conglomeration is unlikely to depend
sensitively on the detailed character of type-I torques, and the envisioned
fiducial picture is expected to hold even in a scenario where inward migration
is initially suppressed or even directed outward. And while these mass-
limiting mechanisms operate simultaneously, it is nevertheless important to
note that they scale differently with the planetesimal surface density.
Crucially, this scaling is super-linear ($M\propto\Sigma_{\rm{pl}}^{3/2}$) and
sub-linear ($M\propto\Sigma_{\rm{pl}}^{3/4}$) for isolation and migration-
regulated growth respectively, meaning that migration is expected to act as
the primary accretion-quenching mechanism for massive systems (that yield
$M\gtrsim 3\,M_{\oplus}$ planets), while isolation regulates the formation of
planets in lower-mass rings.

Beyond yielding a mass-scale that is broadly consistent with the observational
sample of extrasolar super-Earths [17, 18], the scenario described above
entails the emergence of a pattern of intra-system uniformity among the
forming bodies [19, 20, 21]. That is, while parameters such as the surface
density, disk aspect ratio, etc., may differ from system to system, the masses
of planets that form within a given disk are likely to be similar, since they
are largely determined by the isolation and accretion-migration relations. We
further remark that this paradigm is not specific to circumstellar nebulae:
the standard model for the formation of giant planet satellites [22, 15]
follows an analogous narrative, suggesting that architectural similarities
between extrasolar multi-planet systems and the Galilean moons are not
coincidental.

#### Numerical Simulations.

The analytical estimates quoted above provide a useful reference point for the
planetary growth sequence that is expected to ensue within a ringed disk.
Nevertheless, the chaotic intricacy of planet formation cannot be captured
with such considerations alone. Accordingly, we have simulated the formation
of Super-Earths within an annulus of rocky planetesimals employing a full-
fledged numerical model. Our simulations build upon an N-body framework and
augment the self-consistent treatment of gravitational dynamics of
planetesimals and planetary embryos with the effects of aerodynamic drag
exerted on planetesimals by the gas, collisional damping, disk-driven (type-I)
migration, as well as pebble accretion, in a parameterized manner. The details
of our implementation are provided in the Methods section.

In our numerical experiments, the ring of rocky planetesimals is modeled as a
population of $N_{\rm{pl}}=1000$ super-particles that form a Gaussian
distribution of solid material centered on $1\,$AU, with an initial radial
spread of $\Delta_{r}=0.1\,$AU. Along with $N_{\rm{emb}}=10$ lunar-mass
planetary embryos, the planetesimals were introduced into the simulation
gradually over a course of $10^{5}$ years, while the ambient surface density
of solid dust is reduced in concert. Our numerical experiments followed the
conventional “big-small” categorization of bodies, wherein planetesimals could
accrete onto the embryos but not onto each other. We further note that
although the total number of particles in our simulations is limited by
computational cost, we have also found that increasing the number of planetary
seeds beyond ten does not yield materially different results, and can even
diminish the realism of our simulations by over-exaggerating the effects of
dynamical heating (see the Methods section for a discussion). Cumulatively, we
carried out our calculations over a timespan of $3\,$Myr, in agreement with
typical lifetimes of protoplanetary disks.

All in all, our numerical experiments follow a similar narrative to that
outlined by the analytic estimates, and a typical formation sequence observed
in our fiducial simulation suite is depicted in Fig. 2. As dust is converted
into planetary building blocks over the course of the first one hundred
thousand years, planetary embryos experience rapid growth. Simultaneously, the
initially narrow ring of solid material spreads radially due to gravitational
stirring within the system. As such, a small number of separated massive
objects emerge even before the epoch of large-scale planetesimal formation is
complete. Over the following $\sim 10^{5}$ years, a chaotic phase of impacts
ensues, generating super-Earth-class objects. As collisional accretion wanes
and the embryos dynamically isolate themselves within the planetesimal swarm,
the planets enter a prolonged period of inward migration. In due course, the
planetary orbits lock into a multi-resonant chain and stabilize near the
disk’s assumed inner edge at radii spanning tenths of an AU, where they
eventually become observable by photometric surveys.

### Discussion.

The degree of intra-system mass uniformity within our modelized planetary
systems is keenly reminiscent of that observed in the data. As a specific
example, the final results of the simulation shown in Fig. 2 are characterized
by a normalized mass dispersion of $\mathcal{D}_{M}/\langle M\rangle\approx
0.13$. We have also carried out a variant of the same numerical experiment
where the phase of long-range inward migration is delayed until after the
principal phase of planetary growth is complete and obtained comparable
average mass of $\langle M\rangle\approx 2\,M_{\oplus}$ and a normalized
dispersion of $\mathcal{D}_{M}/\langle M\rangle\approx 0.42$. While these
numbers are low, they not anomalous: a normalized mass dispersion of
$\mathcal{D}_{M}/\langle M\rangle\lesssim 0.5$ is an expected outcome of our
proposed formation scenario. To this end, we have run a series of twelve
numerical experiments akin to that depicted in Fig. 2, each time randomizing
the initial conditions and varying the mass of the planetesimal ring from $5$
to $40\,M_{\oplus}$.

The census of the generated systems is depicted in Fig. 3. Overall, these
numerical experiments confirm that in $M_{\rm{ring}}\geqslant 20\,M_{\oplus}$
systems, the typical planetary mass scales as the $\sim$3/4th power of the
initial planetesimal surface density, as is expected from the accretion-
migration timescale balance. Conversely, for planets generated within
$M_{\rm{ring}}\leqslant 20\,M_{\oplus}$ rings, this dependence is slightly
super-linear ($\sim 8/7$ index), signaling the increasingly important role of
isolation for lower-mass systems. Regardless of the relevant scaling, the
generated sample of synthetic systems conforms to a clear pattern of mass (and
presumably radius) homogeneity. In agreement with recent observational
determinations [24], however, our model also predicts that this “peas-in-a-
pod” pattern is limited to short-period orbits and does not extend beyond
$r\sim 0.5\,$AU. Conversely, larger stellocentric distances are primarily
occupied by stranded low-mass planetary objects. Finally, beyond reproducing
the rocky composition of the planets themselves, the orbital architectures of
our synthetic planetary systems are markedly resonant, and require post-
nebular dynamical instabilities to generate the observed period-ratio
distribution of the Galactic Planetary Census [23, 10].

We have not simulated the onset of such post-nebular instabilities here
because the terminal point of our model corresponds to the epoch of nebular
dissipation. Nevertheless, a broad array of dynamical pathways through which
short-period resonant chains of planets can become unstable — including mass-
loss through photo-evaporation [25] as well as interactions with a fading
quadrupolar moment of the host star [26] — is well-documented in the
literature, and large-scale operation of such instabilities has already been
shown to be likely [27]. It is further worth noting that destabilization of
resonant chains of planets yields a period ratio distribution that is
indistinguishable from the data [23] and recent work has demonstrated that the
degradation in the planetary mass-homogeneity is generally very mild during a
dynamical instability, such that systems that experience transient epochs of
scattering remain fully consistent with the “peas-in-a-pod” pattern of
uniformity observed in the population of extrasolar super-Earths [10].

We conclude this work by remarking upon the connection between our exoplanet
formation model and the formation of the Earth itself [6], as the two are
indeed related within the context of our picture. Despite the innate
complexity inherent to planetary accretion, the terminal outcome of our
envisioned scenario is determined chiefly by the mass of the planetesimal
ring, which in turn depends on a variety of the disk’s physical properties,
with turbulent viscosity playing a central role [3]. This dependence is driven
by the fact that the viscosity (and overall metallicity) controls the
cumulative mass entrained within the population of rocky planetesimals (see
Extended Data Figure). To this end, we note that beyond the masses of the
planets themselves, their terminal orbital architecture is also sensitive to
this parameter, since planets that do not accrete rapidly enough do not
experience long-range inward migration, and remain close to their original
formation site. Thus, within the framework of our model, one of the key
reasons that the sun is encircled by the Earth — and not a group of more
massive short-period planets — is simply that the protosolar nebula was
sufficiently turbulent to inhibit the agglomeration of a more massive ring of
rocky planetesimals at 1AU, which prevented the terrestrial planets from
growing massive enough to migrate inwards before the nebular clock had run
out. If the general predictions of our model endure, a unifying model for the
origins of the Earth, the moons of Jupiter and Saturn, as well as extrasolar
planets may finally lie within reach.

### Data availability.

Ascii output files summarizing the time-series of our reference simulation
(with an output interval of 1000 years, totaling 1010 files) are provided at
https://www.konstantinbatygin.com/setimeseries.

### Code availability.

This work utilizes the publically available mercury6 code. The subroutine
detailing user-defined forces is available on request from the corresponding
author (K.B.).

### Acknowledgments.

K.B. and A.M. thank the referees for providing thorough and insightful
reports. K. B. is grateful to Caltech, Observatoire de la Côte d’Azur, the
David and Lucile Packard Foundation, and the National Science Foundation
(grant number: AST 2109276) for their generous support. A. M. is grateful for
support from the ERC advanced grant HolyEarth N. 101019380.

### Author contributions.

K.B. and A.M. jointly conceived the project and collaborated on the
interpretation of the results. K.B. carried out the N-body simulations and led
the writing of the paper. A.M. ran particle-in-a-box simulations and
contributed to writing of the manuscript.

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Extended Data Figure

Figure 4: Output of the disk evolution model delineated in ref. [3]. Owing to
a hydrodynamic balance between the radial outflow and drag exerted on dust by
gas, solids concentrate in the vicinity of their respective sublimation lines,
forming planetesimals within discrete rocky and icy rings. Depending on the
minimal level of turbulent viscosity that the disk can attain,
$\alpha_{\rm{min}}$, the mass entrained within the $r\sim 1\,$AU annulus of
silicate-rich material varies drastically. Importantly, for relatively
quiescent nebulae, the rock ring can reach masses on the order of tens of
$M_{\oplus}$, readily serving as the birthplace of super-Earths.

Methods

To quantify the concurrent processes of planetary accretion and orbital
migration, we have employed the mercury6 $N$-body simulation code [2], and
have augmented it to account for a series of effects that are expected to
arise within protoplanetary disks. Some of these effects were modeled self-
consistently, while others were implemented through ad-hoc parameterizations
for the sake of computational efficiency. Below, we describe the individual
elements of our model and their physical rationale.

##  1 Protoplanetary Embryos and Super-Planetesimals: Resolution and Initial
Conditions

Following conventional practice of $N$-body calculations of planet formation,
we break up our simulations into two classes of particles: fully self-
gravitating planetary “embryos” and semi-active “planetesimals”, which
interact with embryos but not one-another (strictly speaking, only direct
coupling between planetesimals is suppressed; indirect interactions among
these particles – that are transmitted through the reflex motion of the
central body – remain, and drive a minor but non-physical excitation of the
planetesimals’ velocity dispersion [3]). The computational cost of a given
numerical experiment scales quadratically with the number of embryos ($\propto
N_{\rm{emb}}^{2}$) and linearly with the number of planetesimals ($\propto
N_{\rm{emb}}\,N_{\rm{pl}}$). Because $N_{\rm{pl}}$ is taken to exceed
$N_{\rm{emb}}$ by a large margin in typical planet formation calculations, the
computational cost is primarily controlled by the product of $N_{\rm{emb}}$
and $N_{\rm{pl}}$. For this reason, we capped the planetesimal count in our
simulations at $N_{\rm{pl}}=10^{3}$.

The initial embryo masses used in our simulations were informed by particle-
in-a-box calculations of collisional growth within a $20\,M_{\oplus}$
planetesimal ring. To this end, we employed the Boulder code [4] to simulate
the growth of protoplanetary embryos originating from $100\,$km objects,
accounting for self-stirring of the planetesimal velocity dispersion, as well
as collisional damping, gas drag and dynamical friction. This calculation
showed the emergence of Lunar-mass planetary embryos after $\sim 10{,}000$
years of evolution.

It is important to keep in mind that even in the most numerically heavy
calculations, $N_{\rm{pl}}$ is much smaller than the actual number of
planetesimals that exist within protoplanetary disks. This means that each of
our model planetesimals represents a “super-particle” of mass
$M_{\rm{pl}}=M_{\rm{ring}}/N_{\rm{pl}}$, and its dynamical evolution should be
interpreted as a tracer of a large consortium of small bodies. Further, to
simulate the inherent time-dependence of the planetesimal formation process,
we injected the particles into the simulation domain at a constant rate over a
span of 100,000 years, consistent with the duration of planetesimal formation
epoch in the model outlined in ref. [3]. Both the embryos and planetesimals
were introduced following a Gaussian profile in the semi-major axis, centered
at $\langle a\rangle=1\,$AU and a standard deviation of $\Delta_{r}=0.1\,$AU.
Choosing a smaller value of $\Delta_{r}$ yielded similar results because the
gravitational scattering within the planetesimal swarm facilitates a
relatively rapid radial spreading of the system. The initial eccentricities
and inclinations were drawn from the Rayleigh distribution with a scale
parameter of $\Delta_{e}=\Delta_{i}=0.01$, while all orbital angles were
uniformly randomized.

In principle, the ratio $N_{\rm{pl}}/N_{\rm{emb}}$ is an adjustable (and
somewhat artibraty) parameter of the numerical model. Due to chaotic self-
regulation that ensues within the planet-forming region, however, we found
that the results of our calculations are only weakly dependent on the number
of embryos that are injected into the annulus of rocky material during the
planetesimal formation epoch. To quantify this relative insensitivity, we
carried out a series of 12 simulations, where the total mass of the super-
planetesimal disk was fixed at $20$ Earth masses, but the number of lunar-mass
embryos was varied from $N_{\rm{emb}}=5$ to 10 to 20, running four
realizations of each case. From these numerical experiments, we found that at
the 300,000-year mark, the median number of proto-Super-Earths that attain a
mass greater than $1M_{\oplus}$ clocked in at $3.5$, $4.5$ and $5$, for
simulations with 5, 10 and 20 embryos, respectively. Furthermore, we found a
broad consistency in the properties of the emergent planets with average
masses of $M=2.5\pm 0.8\,M_{\oplus}$, $M=2.3\pm 1.3\,M_{\oplus}$, and
$M=2.4\pm 1.4\,M_{\oplus}$ for the three simulation subsets.

Cumulatively, these results indicate that the dependence on the number of
accreting particles saturates around $N_{\rm{emb}}\sim 10$. We did, however,
find that unlike simulations with 10 embryos, those with $N_{\rm{emb}}=20$
consistently demonstrated a pronounced (and almost certainly unphysical)
difference in the velocity dispersion of the super-planetesimals and low-mass
proto-planets, with the former having factor of $\sim 2$ lower eccentricities
on average. Consequently, we chose to adopt $N_{\rm{emb}}=10$ as a fiducial
parameter in our model. With these parameters, the completion of a single run
of the model on a 2.3-GHz machine required $60-80$ CPU hours.

##  2 Gravitational Interactions: Integration Scheme and Time-Step

The gravitational dynamics of our system of planetary embryos and
planetesimals were solved using the hybrid Wisdom-Holman/Bulirsch-Stoer
integration algorithm [5, 6], implemented within the mercury6 software package
[2]. Because our particle swarm was initialized in the vicinity of $r\sim
1\,$AU, we adopted an initial timestep of $\Delta t=10$ days. However, as
disk-driven orbital evolution caused planetary orbits to decay towards their
host star, we found it necessary to reduce the timestep to $\Delta t=1\,$ day
at the $t=300{,}000$ year mark (and in some cases to an even lower value at
later stages), to ensure that the symplectic timestep remained smaller than
$\sim 1/20$th of the shortest orbital period of any particle within the
simulations. An adaptive time-step was used to resolve close encounters, with
an inter-particle separation of $\Delta r=3\,R_{\rm{Hill}}$ marking the
change-over radius for the symplectic-to-conventional integration scheme. The
Bulirsch-Stoer accuracy parameter was set to $\hat{\epsilon}=10^{-8}$.
Finally, the radii of embryos were computed assuming a bulk density of
$3\,$g/cc, and all collisions were treated as perfect mergers. Notably, we
found that in our simulations, the collision cross-section was entirely
dominated by that of the embryos, meaning that our results are insensitive to
the choice of the mean density of the super-planetesimals. To demonstrate
this, we carried out a simulation where the physical collision radii of super-
planetesimals were effectively suppressed (by choosing a corresponding density
of $10^{4}\,$g/cc) while keeping the rest of the calculation – including
damping prescriptions – unchanged. With such a setup, we obtained essentially
identical results to our nominal simulation with planetesimal densities of
$3\,$g/cc. Consequently, we adopted equal bulk densities of planetesimals and
embryos, for definitiveness.

##  3 Protoplanetary Nebula: Gas and Dust Profiles

The presence of the gaseous component of the protoplanetary disk plays a key
role in driving the early evolution of a forming planetary system. For
definitiveness, in our work, we adopted various parameters directly derived
from the ringed disk model delineated in ref. [3] as a guide for our
functional parameterizations. More specifically, we first assumed that the gas
surface density followed a Mestel-like profile [7] that decays exponentially
in time:

$\displaystyle\Sigma=\Sigma_{0}\,\bigg{(}\frac{r_{0}}{r}\bigg{)}\,\exp\big{(}-t/\tau_{\rm{disk}}\big{)},$
(1)

where the initial value of the surface density at $r=1\,$AU is equal to
$\Sigma_{0}=2500\,$g/cm2, and the disk decay constant is set to
$\tau_{\rm{disk}}=1.5\,$Myr. Second, we adopted a constant disk aspect ratio
of $h/r=0.05$ throughout the simulation. To this end, we note that while $h/r$
does in principle change in time and is in general dependent on the disk
viscosity itself [8], this variation is not central to our numerical
experiments, and is expected to only influence the results on a detailed
level.

The dust component of the disk is assumed to be composed of
$s_{\bullet}=1\,$mm particles, in agreement with experimentally-derived
fragmentation threshold of silicate grains [9] as well as theoretical
computation of the Stokes number within the inner nebula [11]. The dust
surface density itself is envisioned to be comprised of a “local” component –
which stems from an aerodynamically-assisted buildup of solids at the silicate
sublimation front – as well as an externally supplied flux of pebbles,
$\mathcal{F}_{\rm{\bullet}}=10^{-4}\,M_{\oplus}/$yr [9, 10], which is
facilitated by the radial drift of solids [11]:

$\displaystyle\Sigma_{\bullet}$
$\displaystyle=\Sigma_{\bullet\,0}\,\exp\bigg{[}-\bigg{(}\frac{r-r_{0}}{\Delta_{r}}\,\bigg{)}^{2}\,\bigg{]}\,\exp\bigg{[}-\bigg{(}\frac{t}{\tau_{\bullet}}\bigg{)}^{2}\,\bigg{]}$
$\displaystyle+\frac{\mathcal{F}_{\rm{\bullet}}}{4\,\pi\,r\,v_{\rm{kep}}\,\eta\,\rm{St}}\,\exp\bigg{[}-\frac{t}{\tau_{\rm{disk}}}\bigg{]}.$
(2)

In the above expression, $\Sigma_{\bullet\,0}=500\,$g/cm2,
$\tau_{\bullet}=10^{5}\,$yr is the dust depletion timescale and the functional
form of the leading term was chosen to adequately approximate the spatial and
temporal profiles of the $r\sim 1\,$AU dust ring modeled in ref. [3].

Strictly speaking, $\mathcal{F}_{\bullet}$ represents the maximal pebble flux
of the nebula, and it would have been appropriate to reduce this value in
accordance with sequestration of material at the rock and ice sublimation
lines, to account for the finite supply of solids in the disk. As our
calculations show, however, the process of pebble accretion is highly
inefficient in the inner disk, so lowering the pebble flux would merely
diminish an already-negligible effect. For the adopted surface density
profile, the sub-Keplerian factor $\eta\approx(3/2)(h/r)^{2}=3.75\times
10^{-3}$. Additionally, in the Epstein drag regime – applicable for the
problem at hand – the Stokes number takes the form:

$\displaystyle\mathrm{St}=\sqrt{\frac{\pi}{8}}\frac{\rho_{\bullet}}{\rho}\frac{s_{\bullet}}{c_{s}}\,\Omega_{\mathrm{kep}}=\frac{\pi\,s_{\bullet}\,\rho_{\bullet}}{2\,\Sigma}.$
(3)

##  4 Planetesimal Evolution: Aerodynamic Drag and Collisional Damping

Interactions between planetesimals and the considerably more massive gaseous
component of the disk ensue primarily through aerodynamic drag [11]. In the
high-Reynolds number regime (appropriate for planetesimals with
$\mathcal{R}\gtrsim 1\,$km; [12]), the relevant drag acceleration has the form
[11, 13]:

$\displaystyle\vec{a}\approx(1+\xi)\,\frac{3\,\rho}{16\,\bar{\rho}\,\mathcal{R}}\,v_{\rm{rel}}\,\vec{v}_{\rm{rel}}.$
(4)

where $\vec{v}_{\rm{rel}}\approx\vec{v}-\vec{v}_{\rm{kep}}$ refers to the
relative velocity between a planetesimal and gas, and $\xi\geqslant 0$ is a
numerical factor that can be used to mimic the effects of non-aerodynamic
damping (see below). This acceleration was implemented into our model assuming
that rocky planetesimals born at $r\sim 1\,$AU have diameters of
$\mathcal{D}=2\,\mathcal{R}=100\,$km and bulk densities of
$\bar{\rho}=3\,$g/cc.

A second important dissipative effect that affects planetesimals within the
context of our calculations is collisional damping. Fundamentally, this
process is driven by inelastic collisions among planetesimals, which occur
with a characteristic frequency [14]:

$\displaystyle\frac{1}{\tau_{\rm{coll}}}\sim\frac{3\,\Sigma_{\rm{pl}}\,\Omega}{8\,\bar{\rho}\,\mathcal{R}}.$
(5)

Although it is impossible to resolve this effect self-consistently in our
super-particle calculations, we can crudely account for it by assuming that
its effective functional form is similar to that of aerodynamic drag. Under
this assumption, we can envision modeling the consequences of collisions (at
least during the first few hundred thousand years, when the planetesimals’
velocity dispersion matters most) through the enhancement factor $\xi$
introduced in equation (4).

To quantify the relative importance of collisional damping to aerodynamic drag
– and thereby determine the value of $\xi$ – we begin by noting that the rate
of aerodynamic damping of eccentricity and inclination is given by [13]:

$\displaystyle\frac{1}{\tau_{\rm{aero}}}=-\frac{1}{e}\frac{de}{dt}=-\frac{2}{i}\frac{di}{dt}=\frac{3\,\rho\,v_{\rm{kep}}}{16\,\bar{\rho}\,\mathcal{R}}$
$\displaystyle\times\bigg{(}\frac{5}{8}\,e^{2}+\frac{1}{2}\,i^{2}+\eta^{2}\bigg{)}^{1/2}\sim\frac{3\,\eta\,\rho\,v_{\rm{kep}}}{16\,\bar{\rho}\,\mathcal{R}}.$
(6)

The value of $\xi$ can then be approximated as:

$\displaystyle\xi=\frac{\tau_{\rm{aero}}}{\tau_{\rm{coll}}}\sim\frac{\sqrt{8\,\pi}}{\eta}\frac{\Sigma_{\rm{pl}}}{\Sigma}\frac{h}{r}\sim\sqrt{\frac{1}{2\,\pi}}\frac{1}{\eta}\frac{M_{\rm{ring}}}{\Sigma\,r_{0}\,\Delta_{r}}\frac{h}{r}.$
(7)

For nominal parameters quoted above, this factor evaluates to $\xi\sim 10$.

While we adopt this value as an upper limit in our calculations, in order to
approximately capture the dependence of the collisional damping process on the
planetesimal surface density, we scale $\xi$ by the ratio
$(N_{\rm{pl}}(t)/N_{\rm{pl}}^{\rm{tot}})$ in our implementation, where
$N_{\rm{pl}}(t)$ is the number of super-planetesimals present in the
simulation at a given timestep and $N_{\rm{pl}}^{\rm{tot}}=10^{3}$ is the
total number of super-planetesimals injected into the simulation. Importantly,
this prescription is only designed to represent the relevant physics during
the first $\sim 2-3\times 10^{5}$ years of the simulation – a period when the
planetesimal population remains radially concentrated and collisional damping
aids the accretion process. At later times, planetesimal dynamics play a
negligible role in dictating the architecture of the emergent planetary
system.

##  5 Planet-Disk Interactions: Type-I Orbital Migration

Generally speaking, the interactions between the gaseous nebula and solid
material are not limited to planetesimals and aerodynamic drag. As planetary
embryos accrete a sufficient amount of mass to raise significant wakes in the
gaseous disk, the gravitational back-reaction of the spiral density waves upon
the planet drives an exchange of energy and angular momentum [15, 16]. In a
chemically inhomogeneous disk model, both the rate and the direction of the
resulting planetary migration can depend sensitively on a variety of local
disk properties [17, 18, 16], as well as the planetary mass itself. In other
words, even at the qualitative level, the picture of planetary migration can
be rather complex in a detailed model of the nebula.

In a simpler, power-law, locally isothermal disk model (of the type we adopt
in this work) however, the sense of migration is strictly inward, and the
semi-major axis, eccentricity, as well as inclination damping timescales are
well defined [16, 29]:

$\displaystyle\mathcal{T}_{\rm{mig}}=\frac{\gamma}{\Omega_{\rm{kep}}}\frac{M_{\star}}{M}\frac{M_{\star}}{\Sigma\,r^{2}}\bigg{(}\frac{h}{r}\bigg{)}^{2}$
$\displaystyle\mathcal{T}_{\rm{damp}}=\frac{\mathcal{T}_{\rm{mig}}}{2}\bigg{(}\frac{h}{r}\bigg{)}^{2},$
(8)

where the dimensionless constant $\gamma\approx 4$. The effects of migration
were implemented into our $N$-body scheme through auxiliary accelerations of
the form [19]:

$\displaystyle\vec{a}=-\frac{\zeta}{\mathcal{T}_{\rm{mig}}}\,\vec{v}-\frac{2}{\mathcal{T}_{\rm{damp}}}\bigg{(}\frac{(\vec{v}\cdot\vec{r})\,\vec{r}}{r^{2}}+(\vec{v}\cdot\hat{z})\,\hat{z}\bigg{)},$
(9)

that were only applied to the planetary embryos.

A practically important and well-known attribute of protoplanetary disks is
that they do not extend all the way down to stellar surfaces, but are instead
truncated by their host stars’ magnetospheres [20]. Due to a dramatic
enhancement of the corotation torque associated with a sharp surface density
gradient, the migration directions reverses at the disk’s inner edge, meaning
that disk cavities act as effective planet traps [21, 22]. The resulting
stalling of inward migration at $r\sim 10-20\,\mathcal{R}_{\odot}$ facilitates
the formation of resonant chains that span the $a\sim 0.1-0.5\,$AU range [23,
23, 10].

A number of distinct approaches have been employed to model this
magnetospheric cavity-driven torque reversal within the framework of $N$-body
simulations, including semi-major axis re-normalization [24, 15], as well as
ad-hoc (e.g., sinusoidal) modifications of the disk-driven accelerations [9,
25]. While our numerical experiments are not sufficiently idealized for semi-
major axis rescaling to be applicable, a drawback of the latter approach is
that it requires careful tuning of the damping effects, including the
introduction of non-linear eccentricity dependence in
$\mathcal{T}_{\rm{damp}}$, etc., to suppress unphysical excitation of the
orbits.

To avoid the usual difficulties of modeling the disk’s inner edge, here we
opted for a simpler approach of smoothly diminishing the migration torque
interior to $r_{\rm{mig}}=0.5\,$AU such that our resonant chains stabilized
with their outermost member at $a\lesssim r_{\rm{mig}}$. We implemented this
by choosing the following functional form form for the multiplicative constant
$\zeta$ in equation (9):

$\displaystyle\zeta=\frac{1}{2}\,\bigg{(}1+\operatorname{erf}\bigg{[}\frac{a-r_{\rm{mig}}}{r_{\rm{mig}}/10}\bigg{]}\bigg{)}.$
(10)

In practice, we found that lowering the threshold semi-major axis to
$r_{\rm{mig}}=0.25\,$AU did not alter the structure of the emergent resonant
chains in any appreciable manner.

##  6 Planetary Growth: Pebble Accretion

In addition to growth facilitated by pairwise collisions between planetary
embryos and planetesimals, in our simulations we also modeled direct capture
of solid dust by the growing protoplanets. In general, this process –
routinely referred to as pebble accretion [27, 28] – can proceed in a number
of distinct physical regimes, with their relative efficiency determined by the
dust particle size, disk turbulence, planetary mass, etc.

Recent work [11] has argued that in typical disks, pebble accretion is
expected to unfold in the comparatively inefficient 3D regime interior to the
water ice sublimation line. Our assumption of a fixed $s_{\bullet}=1\,$mm
particle radius yields results that are consistent with this presumption. By
and large, this is due to the fact that in our model, the Stokes number (given
by equation 3) evaluates to very small values – e.g., $\mathrm{St}\sim 2\times
10^{-4}$ at $r\sim 1\,$AU. In turn, this implies that even for relatively low
values of the turbulence parameter, $\alpha$, vertical settling is
prohibitively inefficient, and the dust layer’s thickness remains comparable
to that of the scale height of the gaseous nebula [26]:

$\displaystyle\frac{h_{\bullet}}{h}=\frac{1}{\sqrt{1+\mathrm{Sc}\,\mathrm{St}/\alpha}}\sim
1.$ (11)

We remark that the constant particle radius assumption can be lifted in favor
of a more self-consistent theory for dust-gas coupling that accounts for
turbulent stirring of dust grains and collisional fragmentation. Importantly,
such a model also predicts that dust should be well-mixed with the gas in the
inner regions of the disk, in agreement with the above estimate [11].

Because the dust component of the nebula never forms a thin sub-disk, the 2D
regime of pebble accretion never ensues. However, for consistency with the
model of ref. [3], where the value of the turbulent Schmidt number is taken to
be on the order of $\mathrm{Sc}\sim 10$, we take $h_{\bullet}/h=2/5$ in our
model, such that $h_{\bullet}/r=0.02$. As we discuss below, our results are
not sensitive to this choice.

The rate of 3D pebble accretion is given by [30]:

$\displaystyle\dot{M}_{\rm{3D}}$
$\displaystyle=6\,\pi\,\mathrm{St}\,R_{\rm{Hill}}^{3}\,\Omega\,\bigg{(}\frac{\Sigma_{\bullet}}{\sqrt{2\,\pi}\,h_{\bullet}}\bigg{)}$
$\displaystyle=\sqrt{2\,\pi}\,\mathrm{St}\,\Sigma_{\bullet}\,r^{2}\,\Omega\,\frac{M}{M_{\star}}\,\frac{r}{h_{\bullet}}.$
(12)

Although this growth mode was implemented in our simulations, we found pebble
accretion to be largely inconsequential in our calculations. This can be
understood as follows.

As the most favorable scenario for dust capture, let us consider an isolated
protoplanetary embryo of mass $m_{0}$ that accretes pebbles at $r=1\,$AU,
where the primordial solid surface density is maximized. Substituting the
functional form (2) for $\Sigma_{\bullet}$, it is straightforward to show that
the accreted mass is bounded from above by:

$\displaystyle\Delta{M}<\int_{0}^{\infty}\dot{M}_{\rm{3D}}\,dt=\frac{\pi}{\sqrt{2}}\,\mathrm{St}\,\Sigma_{\bullet\,0}\,r^{2}\,\Omega\,\tau_{\bullet}\,\frac{M_{0}}{M_{\star}}\frac{r}{h_{\bullet}}$
$\displaystyle+\frac{\mathcal{F}_{\bullet}\,\tau_{\rm{disk}}}{2\,\sqrt{2\,\pi}\,\eta}\frac{M_{0}}{M_{\star}}\,\frac{r}{h_{\bullet}}\approx
5.2\,M_{0},$ (13)

with the dominant contribution arising from the leading term (the externally
supplied pebble flux accounts for less than a quarter of the accreted mass).

This simple estimate indicates that even under the most optimistic conditions,
pebble accretion can only boost the mass of an embryo by a factor of a few.
Moreover, it is important to keep in mind that the efficiency of this process
diminishes rapidly over the course of the first $\sim 10^{5}$ years, as the
majority of pebbles are converted into planetesimals (because of the smallness
of this effect, we did not implement filtering of pebble flux in our model).
Accordingly – while we include the pebble accretion process in our simulations
for completeness – our numerical experiments indicate that a far more dominant
role in facilitating planetary growth is played by the accretion of
planetesimals and mergers among protoplanetary embryos.

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# The zero divisor conjecture and Mealy automata

Ievgen Bondarenko and Kate Juschenko

###### Abstract.

The zero divisor conjecture is sufficient to prove for certain class of
finitely presented groups where the relations are given by a pairing of
generators. We associate Mealy automata to such pairings, and prove that the
zero divisor conjecture holds for groups corresponding to invertible automata
with three states. In particular, there cannot be zero divisors of support
three corresponding to invertible pairings.

In 1956 Irving Kaplansky proposed several problems on group rings. We are
interested in two of them:

######  Conjecture 1 (Zero Divisor Conjecture).

Let $K$ be a finite field and $G$ be a torsion-free group. Then the group ring
$K[G]$ has no zero divisors.

######  Conjecture 2 (Direct Finiteness Conjecture).

Let $K$ be a finite field and $G$ be a group. Then the group ring $K[G]$ is
directly finite, that is, for every $a,b\in K[G]$ the equation $ab=1$ implies
$ba=1$.

One of the other famous Kaplansky’s conjectures on group rings — the unit
conjecture — was recently disproved by Giles Gardam in [Gardam].

Both listed conjectures are open even for the field ${\mathbb{F}}_{2}$, which
we will be considering through the paper. The conjectures were confirmed for
many classes of groups. In particular, the Direct Finiteness conjecture is
true for sofic groups (see [ES04]). This class of groups is very large, and at
the moment there are no examples of non-sofic groups. We refer the reader to a
classical book of Passman on group rings [Pa77] for more discussion on
Kaplansky’s conjectures and their relation to each other.

In [DHJ, DJ] and [Schw], the authors proposed an algorithmic approach to both
conjectures (the idea circulated among mathematicians for some time, see for
example [Thom]). They introduced a class of finitely presented groups
$\Gamma_{C}$ that are universal among possible counterexamples: if $G$ is a
counterexample, then $G$ contains a subgroup $H$ that is a quotient of some
$\Gamma_{C}$ that is also a counterexample. Here $C$ is a pairing matrix, that
is, an integer $n\times m$ matrix $C=(c_{ij})$, where $nm$ is even and
$c_{i,j}\in\\{1,2,\ldots,nm/2\\}$, with the following property: no row or
column contain a repeated value, and for every pair $(i,j)$ there exists
exactly one pair $(i^{\prime},j^{\prime})\neq(i,j)$ such that
$c_{ij}=c_{i^{\prime}j^{\prime}}$. To every pairing matrix $C$, we can
associate a finitely presented group:

$\Gamma_{C}=\langle
a_{1},\ldots,a_{n},b_{1},\ldots,b_{m}\,|\,a_{i}b_{j}=a_{i^{\prime}}b_{j^{\prime}}\mbox{
whenever $c_{ij}=c_{i^{\prime}j^{\prime}}$}\rangle.$

Let us recall the connection with the group ring.

Let $G$ be a group and consider two elements $a,b\in{\mathbb{F}}_{2}[G]$
satisfying $ab=0$. We may write

$a=a_{1}+\cdots+a_{n}\ \text{ and }\ b=b_{1}+\cdots+b_{m}$

for distinct elements $a_{1},\ldots,a_{n}$ and distinct elements
$b_{1},\ldots,b_{m}$ of $G$. In this case we say $a$ has rank $n$ and the
support of $a$ is $\\{a_{1},\ldots,a_{n}\\}$. By distributing the identity
$ab=0$, we get $a_{i}b_{j}=a_{i^{\prime}}b_{j^{\prime}}$ for certain pairing
$(i,j)\sim_{\pi}(i^{\prime},j^{\prime})$ of the indices. Then the subgroup of
$G$ generated by $a_{1},\ldots,a_{n}$ and $b_{1},\ldots,b_{m}$ is a quotient
of the group $\Gamma_{C}$, where the matrix $C$ is given by the pairing $\pi$.
Therefore, the Zero Divisor Conjecture over ${\mathbb{F}}_{2}$ is equivalent
to:

######  Conjecture 3.

Let $C$ be a pairing matrix and the generators $a_{1},\ldots,a_{n}$ and
$b_{1},\ldots,b_{m}$ of $\Gamma_{C}$ are distinct. Then $\Gamma_{C}$ contains
a nontrivial element of finite order.

The groups $\Gamma_{C}$ were analyzed with respect to these conjectures in
[DHJ, Schw]. It was shown that there are no counterexamples with an element of
rank two, the groups $\Gamma_{C}$ for $2\times n$ pairings satisfy the
conjectures. With the computer assistance, all groups $\Gamma_{C}$ up to sizes
$3\times 11$ and $5\times 5$ were determined in [DHJ], and they satisfy the
Direct Finiteness Conjecture. In [Schw], it is shown that there are no zero
divisors, where the ranks of elements are $(3,m)$ for $m\leq 15$ and $(4,m)$
for $m\leq 6$. In [Ali, Ali2], this result was extended to the ranks $(3,m)$
for $m\leq 19$ and $(4,m)$ for $m\leq 8$.

In this paper, we associate Mealy automata to pairings matrices and relate
certain cycles in these automata to elements of finite order. We prove that
the group $\Gamma_{C}$ corresponding to invertible automata with three states
contains nontrivial elements of finite order (Theorem 4). In particular, there
cannot be zero divisors of support three corresponding to invertible pairings.

##  1\. Mealy automata and pairing matrices

A Mealy automaton over an alphabet $X$ is a quadruple
$\mathsf{A}=(S,X,\mu,\nu)$, where $S$ is the set of states; $X$ is an
alphabet; $\mu:S\times X\rightarrow S$ is the transition map; and $\nu:S\times
X\rightarrow X$ is the output map. An automaton $\mathsf{A}$ can be identified
with a directed labeled graph whose vertices are identified with the states of
$\mathsf{A}$, and for every state $s\in S$ and every letter $x\in X$, the
graph has an arrow from $s$ to $\mu(s,x)$ labeled by $x|\nu(s,x)$.

Let $C$ be a $n\times m$ pairing matrix. We associate to $C$ a Mealy automaton
$\mathsf{A}_{C}$ with the set of states $S=\\{a_{1},\ldots,a_{n}\\}$ over the
alphabet $X=\\{b_{1},\ldots,b_{m}\\}$ and the following arrows:

$a_{i}\xrightarrow{b_{j}|b_{j^{\prime}}}a_{i^{\prime}}\ \mbox{ whenever
$c_{ij}=c_{i^{\prime}j^{\prime}}$},$

see example in Figure 1.

$C=\left(\begin{array}[]{cccc}1&2&3&4\\\ 5&6&2&1\\\ 3&4&5&6\\\
\end{array}\right)$

$a_{1}$$a_{2}$$a_{3}$$b_{1}|b_{4}$ $b_{2}|b_{3}$$b_{4}|b_{1}$
$b_{3}|b_{1}$$b_{1}|b_{3}$, $b_{2}|b_{4}$$b_{3}|b_{1}$,
$b_{4}|b_{2}$$b_{1}|b_{3}$ $b_{2}|b_{4}$$b_{3}|b_{1}$ $b_{4}|b_{2}$

Figure 1. Example of a pairing matrix $C$ and the associated automaton
$\mathsf{A}_{C}$

An automaton $\mathsf{A}=(S,X,\mu,\nu)$ is called invertible if, for all $s\in
S$, the transformation $\nu(s,\cdot):X\rightarrow X$ is a permutation of $X$.
For invertible automata, one can naturally extend the functions $\mu,\nu$ to
$S^{-1}\times X$ as $\mu(s^{-1},y)=t^{-1}$ and $\nu(s^{-1},y)=x$, where
$y=\nu(s,x)$ and $t=\mu(s,x)$. An invertible automaton $\mathsf{A}$ is called
bireversible if the map

$(s,x)\mapsto(\mu(s,x),\nu(s,x))$

is a bijection on $S\times X$ and $S^{-1}\times X$. For more information about
Mealy automata and related groups, see [BK, GNS].

######  Proposition 1.

Let $C$ be a pairing matrix and $\mathsf{A}_{C}$ the associated automaton. The
following statements are equivalent:

  1. (1)

every row and every column of $C$ have exactly one equal coefficient besides
the common coefficient in their intersection;

  2. (2)

$\mathsf{A}_{C}$ is invertible;

  3. (3)

$\mathsf{A}_{C}$ is bireversible.

###### Proof.

The statement in the first item is equivalent to saying that for every row $i$
the map $j\mapsto j^{\prime}$ defined on $\\{1,2,\ldots,m\\}$ by the relation
$c_{ij}=c_{i^{\prime}j^{\prime}}$ is a permutation. This exactly means that
the automaton $\mathsf{A}_{C}$ is invertible. Another reformulation of
invertability: for every pair $(i,j^{\prime})$ there exists a unique pair
$(i^{\prime},j)$ such that $c_{ij}=c_{i^{\prime}j^{\prime}}$. It follows that
for every $(i^{\prime},j^{\prime})$ there exists a unique $(i,j)$ such that
$c_{i^{\prime}j^{\prime}}=c_{ij}$, and for every $(i^{\prime},j)$ there exists
a unique $(i,j^{\prime})$ with $c_{i^{\prime}j^{\prime}}=c_{ij}$. This means
that $\mathsf{A}_{C}$ is bireversible. ∎

######  Definition 1.

We say that a matrix $C$ defines an invertible pairing if it satisfies the
conditions of Proposition 1.

We can characterize invertible automata corresponding to pairings in term of
their Helix graph. The Helix graph of an automaton $\mathsf{A}$ is the
directed graph $\mathcal{H}_{\mathsf{A}}$ on the vertex set $S\times X$ with
arrows

$(s,x)\rightarrow(t,y)\mbox{ in $\mathcal{H}_{\mathsf{A}}$ \quad whenever
\quad}s\xrightarrow{x|y}t\mbox{ in $\mathsf{A}$}.$

######  Proposition 2.

A bireversible automaton $\mathsf{A}$ corresponds to a certain pairing matrix
if and only if $\mathsf{A}$ does not contain loops and arrows of the form
$s\xrightarrow{x|x}t$, and the Helix graph of $\mathsf{A}$ consists of cycles
of length two.

###### Proof.

The Helix graph of a bireversible automaton consists of cycles. If
$\mathsf{A}$ corresponds to a paring matrix $C$, then the pairing
$c_{ij}=c_{i^{\prime}j^{\prime}}$ implies a cycle in the Helix graph
$\mathcal{H}_{\mathsf{A}}$:

$(a_{i},b_{j})\rightarrow(a_{i^{\prime}},b_{j^{\prime}})\rightarrow(a_{i},b_{j}).$

The absence of loops and arrows $s\xrightarrow{x|x}t$ is equivalent to absence
of repeated values in rows and columns of $C$. Conversely, if all cycles have
length two, then we can construct a matrix $C=(c_{sx})$ with rows labeled by
the states and columns labeled by the letters such that

$c_{sx}=c_{ty}\ \mbox{ if and only if \
$(s,x)\rightarrow(t,y)\rightarrow(s,x)$ in $\mathcal{H}_{\mathsf{A}}$}.$

The matrix $C$ is a pairing matrix, because $(t,y)\neq(s,x)$ and $(t,y)$ is
uniquely determined by $(s,x)$. ∎

An automaton $\mathsf{A}=(S,X,\mu,\nu)$ can be uniquely extended to an
automaton $\mathsf{A}^{*}$, where the states and letters are the words over
$S$ and $X$ respectively:

$\displaystyle s\xrightarrow{x_{1}|y_{1}}q\mbox{ \ and \
}q\xrightarrow{x_{2}|y_{2}}t\quad$ $\displaystyle\Rightarrow\quad
s\xrightarrow{x_{1}x_{2}|y_{1}y_{2}}t,$ $\displaystyle
s_{1}\xrightarrow{x|z}t_{1}\mbox{ \ and \ }s_{2}\xrightarrow{z|y}t_{2}\quad$
$\displaystyle\Rightarrow\quad s_{1}s_{2}\xrightarrow{x|y}t_{1}t_{2}.$

Certain cycles in the automaton $\mathsf{A}_{C}^{*}$ produce elements of
finite order in the group $\Gamma_{C}$.

######  Proposition 3.

Let $C$ be a pairing matrix. Then

$\displaystyle s\xrightarrow{xx\ldots x|yy\ldots y}s\mbox{\quad in \
$\mathsf{A}_{C}^{*}$}\ $ $\displaystyle\Rightarrow$ $\displaystyle\
(yx^{-1})^{k}=e\mbox{\quad in \ $\Gamma_{C}$},$ $\displaystyle ss\ldots
s\xrightarrow{x|x}tt\ldots t\mbox{\quad in \ $\mathsf{A}_{C}^{*}$}\ $
$\displaystyle\Rightarrow$ $\displaystyle\ (t^{-1}s)^{k}=e\mbox{\quad in \
$\Gamma_{C}$}.$

###### Proof.

The arrow $s\xrightarrow{xx\ldots x|yy\ldots y}s$ in $\mathsf{A}_{C}^{*}$
means that we have the following sequence of arrows in $\mathsf{A}_{C}$:

$s\xrightarrow{x|y}s_{2},\ s_{2}\xrightarrow{x|y}s_{3},\ \ldots,\
s_{k}\xrightarrow{x|y}s.$

These arrows correspond to the following relations in $\Gamma_{C}$:

$sx=s_{2}y,\ s_{2}x=s_{3}y,\ \ldots\ ,s_{k}x=sy.$

Then

$\displaystyle
sx=s_{2}y=s_{3}yx^{-1}y=s_{4}yx^{-1}yx^{-1}y=\ldots=s(yx^{-1})^{k-1}y$
$\displaystyle\Rightarrow\quad(yx^{-1})^{k}=e.$

∎

The Kaplansky’s zero divisor conjecture can be equivalently formulated as
$a^{2}=0$ implies $a=0$ instead of $ab=0$ implies $a=0$ or $b=0$. This allows
to reduce the generators of the associated finitely presented group to
$a_{1},\ldots,a_{n}$. Let $C$ be an $n\times n$ pairing matrix (here $n$ is
even), and define a finitely presented group

$G_{C}=\langle a_{1},\ldots,a_{n}\ |\
a_{i}a_{j}=a_{i^{\prime}}a_{j^{\prime}}\mbox{ whenever
}c_{ij}=c_{i^{\prime}j^{\prime}}\rangle.$

Then the Kaplansky’s conjecture is equivalent to the existence of non-trivial
elements of finite order in the group $G_{C}$. We get a different way to
associate an automaton to a pairing matrix: define an automaton
$\mathsf{B}_{C}$ with the set of states $S=\\{a_{1},\ldots,a_{n}\\}$, alphabet
$X=\\{a_{1},\ldots,a_{n}\\}$, and arrows

$a_{i}\xrightarrow{a_{j}|a_{i^{\prime}}}a_{j^{\prime}}\ \mbox{ whenever
$c_{ij}=c_{i^{\prime}j^{\prime}}$}.$

Then any transition $s\xrightarrow{v|u}t$ in $\mathsf{B}_{C}^{*}$ implies the
relation $sv=ut$ in $G_{C}$. Unfortunately, the automaton $\mathsf{B}_{C}$ is
never invertible. Indeed, if the first row of $C$ is $[1,2,\ldots,n]$, then
some other row $k$ contains at least two numbers $1\leq i<j\leq n$. This means
that there are two arrows in $\mathsf{B}_{C}$:

$a_{1}\xrightarrow{a_{i}|a_{k}}b\qquad\mbox{ and }\qquad
a_{1}\xrightarrow{a_{j}|a_{k}}b^{\prime},$

what contradicts invertibility.

##  2\. The zero divisor conjecture for invertible pairings

We show that the zero divisor conjecture holds for $3\times 2n$ invertible
pairings.

######  Theorem 4.

Let $C$ be a $3\times 2n$ invertible pairing matrix. Let us assume that the
generators $a_{1}$, $a_{2}$, $a_{3}$ and $b_{1}$, …, $b_{2n}$ of the group
$\Gamma_{C}$ are distinct. Then $\Gamma_{C}$ contains a nontrivial element of
finite order.

###### Proof.

Each row and column of $C$ do not contain two equal coefficients, since
otherwise $a_{i}=a_{j}$ or $b_{i}=b_{j}$ for $i\neq j$. We want to show that
by permuting the columns/rows and relabeling equal coefficients, the matrix
$C$ can be brought to the following form:

$C=\left(\begin{array}[]{cccccccc}1&2&\ldots&n&n+1&n+2&\ldots&2n\\\
2n+1&2n+2&\ldots&3n&c_{2,n+1}&c_{2,n+2}&\ldots&c_{2,2n}\\\
c_{3,1}&c_{3,2}&\ldots&c_{3,n}&c_{3,n+1}&c_{3,n+2}&\ldots&c_{3,2n}\\\
\end{array}\right),$

where $(c_{2,n+1},c_{2,n+2},\ldots,c_{2,2n})$ is a permutation of
$1,2,\ldots,n$, $(c_{3,1},c_{3,2},\ldots,c_{3,n})$ is a permutation of
$n+1,n+2,\ldots,2n$, and $(c_{3,n+1},c_{3,n+2},\ldots,c_{3,2n})$ is a
permutation of $2n+1,2n+2,\ldots,3n$. This gives certain “bipartite” structure
on $C$.

First, half of the numbers $1,2,\ldots,2n$ should be in the second row and the
other half in the third one, since otherwise there would be two equal elements
in one row. By permuting the rows if necessary, we can assume that
$1,2,\ldots,n$ are in the second row.

Second, the numbers $1,2,\ldots,n$ in the second row cannot belong to columns
$1$, $2$,…, $n$. Indeed, if $c_{2,i}\in\\{1,2,\ldots,n\\}$ for $1\leq i\leq
n$, then the $i$th column and $3$rd row do not contain common coefficient
except for their intersection; this contradicts invertibility. Therefore,
$(c_{2,n+1},c_{2,n+2},\ldots,c_{2,2n})$ is a permutation of $1,2,\ldots,n$. By
the same reason, $(c_{3,1},c_{3,2},\ldots,c_{3,n})$ is a permutation of
$n+1,n+2,\ldots,2n$ and $(c_{3,n+1},c_{3,n+2},\ldots,c_{3,2n})$ is a
permutation of $2n+1,2n+2,\ldots,3n$.

Let us denote $x_{i}=b_{i}$ and $y_{i}=b_{n+i}$ for $i=1,\ldots,n$ in order to
reflect the bipartite structure of $C$. Then we get the following relations in
$\Gamma_{C}$:

$\displaystyle t_{1}=a_{1}^{-1}a_{2}$
$\displaystyle=x_{1}y_{i_{1}}^{-1}=x_{2}y_{i_{2}}^{-1}=\ldots=x_{n}y_{i_{n}}^{-1},$
$\displaystyle t_{2}=a_{2}^{-1}a_{3}$
$\displaystyle=x_{1}y_{j_{1}}^{-1}=x_{2}y_{j_{2}}^{-1}=\ldots=x_{n}y_{j_{n}}^{-1},$
$\displaystyle t_{3}=a_{3}^{-1}a_{1}$
$\displaystyle=x_{1}y_{k_{1}}^{-1}=x_{2}y_{k_{2}}^{-1}=\ldots=x_{n}y_{k_{n}}^{-1},$

where $(i_{1},i_{2},\ldots,i_{n})$, $(j_{1},j_{2},\ldots,j_{n})$, and
$(k_{1},k_{2},\ldots,k_{n})$ are permutations of $1,2,\ldots,n$. Notice that
if we write the corresponding relations for each
$t_{1}^{-1},t_{2}^{-1},t_{3}^{-1}$, we will see
$y_{1}x_{l_{1}}^{-1},y_{2}x_{l_{2}}^{-1},\ldots,y_{n}x_{l_{n}}^{-1}$, where
$(l_{1},l_{2},\ldots,l_{n})$ is a permutation of $1,2,\ldots,n$.

We want to show that $t_{1}t_{2}^{-1}$ has finite order. Let us construct a
directed $(n,n)$ bipartite graph: the left part consists of vertices
$x_{1}y_{i_{1}}^{-1}$, …, $x_{n}y_{i_{n}}^{-1}$, which represent $t_{1}$, the
right part consists of vertices
$y_{1}x_{l_{1}}^{-1},y_{2}x_{l_{2}}^{-1},\ldots,y_{n}x_{l_{n}}^{-1}$, which
represent $t_{2}^{-1}$. We put a directed edge from $x_{s}y_{i_{s}}^{-1}$ to
$y_{i_{s}}x_{l_{is}}^{-1}$ for $s=1,\ldots,n$, and a directed edge from
$y_{s}x_{l_{s}}^{-1}$ to $x_{l_{s}}y^{-1}_{i_{l_{s}}}$ for $s=1,\ldots,n$. Now
there is a directed edge passing from any vertex. Hence, there is a directed
cycle of some length $1\leq 2m_{12}\leq 2n$. By multiplying consequently the
elements along this cycle, we get $(t_{1}t_{2}^{-1})^{m_{12}}=e$.

The same observation holds for $t_{2}t_{3}^{-1}$ and $t_{3}t_{1}^{-1}$. Hence

$(t_{1}t_{2}^{-1})^{m_{12}}=(t_{2}t_{3}^{-1})^{m_{23}}=(t_{3}t_{1}^{-1})^{m_{31}}=e$

for some $1\leq m_{ij}\leq n$. If $t_{i}\neq t_{j}$ for some $i\neq j$, we get
a nontrivial element $t_{i}t_{j}^{-1}$ of finite order. If
$t_{1}=t_{2}=t_{3}$, then

$t_{1}^{3}=t_{1}t_{2}t_{3}=a_{1}^{-1}a_{2}\cdot a_{2}^{-1}a_{3}\cdot
a_{3}^{-1}a_{1}=e$

and $t_{1}=a_{1}^{-1}a_{2}\neq e$ by assumption. The theorem is proved. ∎

######  Remark 1.

The previous proof also works for larger pairing matrices $C$/automata
$\mathsf{A}_{C}$ as soon as they admit the following cyclic-bipartite
structure: there is an order on the states $s_{1},s_{2},\ldots,s_{n}$ and a
partition of the alphabet $X=A\sqcup B$, $|A|=|B|$ so that the arrows in
$\mathsf{A}_{C}$ are of the form

$s_{i}\xrightarrow{A|B}s_{i+1}\ \mbox{ and }\ s_{i}\xrightarrow{B|A}s_{i-1}.$

Invertible pairing $3\times 2n$ matrices always admit such a structure.

######  Remark 2.

The automata $\mathsf{A}_{C}$ associated to invertible pairings are
bireversible. There is a strong connection between bireversible automata and
lattices in the product of the automorphism groups of two regular trees that
are given by presentation

$G_{\mathsf{A}}=\\{S,X\,|\,sx=yt\mbox{ whenever $s\xrightarrow{x|y}t$ in
$\mathsf{A}$ }\\}.$

The groups $G_{\mathsf{A}}$ are always torsion-free; however, some of them are
non-residually finite, even virtually simple, and not elementary sofic (see
[BK] are the references therein). It is unknown whether the groups
$G_{\mathsf{A}}$ are sofic or satisfy the Kaplansky’s conjectures.

## References