through the vanishing of $L/\xi_{trans}$ seen in Fig. 33(b), which would
correspond to a quantized thermal conductance at the $Q=0$ boundary Akhmerov
_et al._ (2011). The topological phase $Q<0$ (seen in blue in panel a) is seen
in Fig. 33(b) to potentially support localized MZMs based on the low
transmission that can be inferred from $L/\xi_{trans}>2$.

Figure 35: Conductance tensor $G_{ab}$ of a three-terminal Majorana device
where $a,b$ label leads $L,R$ at a temperature of $T=40\,mK$. The local
conductances $G_{LL}$ and $G_{RR}$ show zero-bias conductance peak signatures
of the topological phase. The non-local conductance $G_{LR},G_{RL}$ can
indicate a topological gap closure. The conductance $G_{ab}$ are plotted as
magnetic field $B$ and chemical potential $\mu$ is varied around the reference
point $\mu=0.29$ meV and $B=0.42$ T in Fig. 33. (a) Shows a waterfall of
$G_{ab}$ where $B$ varies from $B=0.378$ T to $B=0.462$ T. (b) Shows a
waterfall of $G_{ab}$ where $\mu$ varies from $\mu=0.24$ meV to $\mu=0.34$
meV. The conductance curves in both panels are shifted vertically
symmetrically about the reference point to make the curves visible.  Figure
36: (a) Same as panel Fig. 35(a) repeated here for reference. (b) Presents the
zero-temperature results from which panel (a) is generated at $T=40$ mK. We
see that small gaps at $T=0$ mK often disappear at $T=40$ mK. While this is
not apparent because of the vertical shift of the conductances in the wter-
fall plots, we note that the $T=0$ local conductance peak heights near zero
bias are nearly (i.e. within $5\%$) quantized with $G\sim 2G_{0}$ both from
the left (i.e. $G_{LL}$)and right (i.e. $G_{RR}$).

To verify this conjecture, we plot the LDOS as a function of energy $E$ and
position $x$ at a reference point $\mu=0.29$ meV and $B=0.42$ T in Fig. 33(c).
This reference point, which is marked by a star in Fig. 33(a), is at a
parameter value where the topological charge is $Q=-0.995$. Furthermore, the
reference point is deep in the topological region, away from the fluctuating
phase boundary for the long wire phase diagram shown in Fig. 34 and is in this
sense unambiguously in the topological superconducting phase. This, however,
is not obvious just by looking at the finite size effective phase diagram in
Fig. 33. Note that to enhance visibility in panel (c), the LDOS plot (in this
and following figures) is raised to power $1/3$ to prevent localized states
from dominating the color scale. This has no impact on the qualitative
interpretation of the figure and only implies that the color bar cannot be
interpreted quantitatively for these figures. We see that the reference point
LDOS at $E=0$ shows strong intensity that is localized near the ends of the
wire. Thus, the system has end MZMs here. The quantitative form of this
localization becomes clear in the sectional plot of the logarithm of the LDOS
at $E=0,40$ mK shown in Fig. 33(d). The peaks of the $\textrm{Log}(LDOS)$
shows a linear decrease from the ends of the wire to the middle consistent
with a the transmission estimate $L/\xi_{trans}>2$ from transmission.

Figure 37: (a) Topological charge $(Q=Detr)$ as a function of magnetic field
$B$ and chemical potential $\mu$ in the superconductor for a $L=3\,\mu m$ for
same parameters as Fig. 33 with disorder amplitude increased $V_{dis}=1.5$
meV. shows topological patches (b) $(L/\xi)_{trans}=1/2\textrm{Log}(trans)$
corresponding to the topological charge shows the regimes where the
transmission is low enough (i.e. $(L/\xi)_{trans}\gtrsim 2$) to support a
well-defined topological phase. (c) LDOS at $\mu=0.28$ meV and $B=0.56$ T
(with $Q=-0.846$, marked by star in panel (a)) shows delocalized zero-energy
state (d) Plot of $1/2\textrm{Log}(LDOS)$ i.e. cross-sections of LDOS from (c)
at two energies $E=0,40$ mK shows delocalized state consistent with
$(L/\xi)_{trans}=0.8$ at that point.  Figure 38: (a,b) Similar to Fig. 37
except for an $L=40\,\mu m$ long wire. This figure shows that the topological
patch reference point marked by a star in Fig. 37(a) when extended to a longer
wire ends up in the trivial phase.

The topological charge $Q$ is not measurable and even the thermal conductance
which experimentally marks the boundary $Q=0$ Akhmerov  _et al._ (2011) of the
topological phase is difficult to measure (and we know of no attempt to
measure the thermal conductance experimentally in this context). However, the
topological phases have been proposed to be electrically detectable based on
zero-bias peaks in local conductance Sengupta  _et al._ (2001) as well as gaps
in non-local conductance Rosdahl  _et al._ (2018); Pan  _et al._ (2021).

Figure 39: Conductance tensor $G_{ab}$ of a three-terminal Majorana device
where $a,b$ label leads $L,R$ at a temperature of $T=40\,mK$. The local
conductances $G_{LL}$ and $G_{RR}$ show zero-bias conductance peak signatures
of the topological phase. The non-local conductance $G_{LR},G_{RL}$ can
indicate a topological gap closure. The conductance $G_{ab}$ are plotted as
magnetic field $B$ and chemical potential $\mu$ is varied around the reference
point $\mu=0.29$ meV and $B=0.42$ T in Fig. 33. (a) Shows a waterfall of
$G_{ab}$ where $B$ varies from $B=0.378$ T to $B=0.462$ T. (b) Shows a
waterfall of $G_{ab}$ where $\mu$ varies from $\mu=0.24$ meV to $\mu=0.34$
meV. The conductance curves in both panels are shifted vertically
symmetrically about the reference point to make the curves visible.  Figure
40: (a) Same as panel Fig. 35(a) repeated here for reference. (b) Presents the
zero-temperature results from which panel (a) is generated at $T=40$ mK. We
see that small gaps at $T=0$ mK often disappear at $T=40$ mK. The local zero-
bias conductance peaks on the right are seen to be strongly split and
substantially below quantization. This is also true to a lesser extent for
most of the ZBCPs on the left. This can be thought to be a result of strong
hybridization of the zero energy state with other states.  Figure 41: (a) Zoom
in of Fig. 37(a) shows a reference point $\mu=0.29$ meV and $B=0.61$ T (with
$Q=-0.983$) marked by star. (b) Shows that this point has weak transmission
i.e. $(L/\xi)_{trans}\sim 1.9$) and may support a well-defined topological
phase. (c) LDOS at shows a somewhat localized state where the DOS dips
significantly in the middle (d) This is confirmed by the section of the LDOS
plot on a log scale.

Fig. 35 shows the three-terminal conductance tensor $G_{ab}$ (with $a,b$
labeling leads $L,R$) at temperature $T=40\,mK$, which is consistent with the
apparent electron temperature in the recent experiments Aghaee  _et al._
(2022). The middle curve in each of these panels shows the conductance of the
reference topological point $\mu=0.29$ meV and $B=0.42$ T marked by a star in
Fig. 33(a). The other curves in Fig. 35(a) shows the variation of the
conductance with changing magnetic field $B$ while Fig. 35(b) shows variation
with varying chemical potential $\mu$. All parameters plotted in this figure
are in the topological phase according to the topological invariant $Q<0$ as
is suggested by the presence of a robust ZBCP in all local conductance curves
$G_{LL}$ and $G_{RR}$. The non-local conductances $G_{LR},G_{RL}$ are more
complicated to interpret. While the reference point appears gapped with a
nearly flat non-local conductance, changing the magnetic field $B$ or chemical
potential $\mu$ appears to rapidly close the gap suggesting a very small patch
in for the topological phase. This contradicts the phase diagram Fig. 33(a),
which shows a rather contiguous region of topological phase with a relatively
low transmission according to Fig. 33(b). The apparent contradiction is
resolved by comparing the above plots with the zero-temperature plot shown in
Fig. 36, which shows a small gap in the non-local conductance. This suggests
that while a robust topological phase might be possible for a $L=3\,\mu m$
wire, non-local conductance at $T=40\,mK$ might not be able to distinguish the
relatively well-defined topological phase discussed in this section with the
regime of small topological patches in the next subsection. Thus, not
observing a finite temperature gap in the nonlocal conductance does not
necessarily rule out the existence of a topological phase!

###  V.3 Topological patch regime

Fig. 37(a) shows the topological charge $Q$ for a $L=3\,\mu m$ long Majorana
nanowire model Lutchyn  _et al._ (2010); Oreg  _et al._ (2010) with parameters
chosen to match InAs wires from the recent Microsoft experiment Aghaee  _et
al._ (2022). The topological charge is plotted in this figure versus the
chemical potential $\mu$ of the Majorana nanowire (controlled by the plunger
gate) and the applied magnetic field $B$. The disorder potential for this
calculation is chosen to have an RMS amplitude $V_{dis}=1.5$ meV (larger than
the value $0.8\mu eV$ used in the examples of Figs. 33\- 36), which supports
only small topological patches. The putative topological phase boundary $Q=0$
seen from the figure shows substantial mesoscopic fluctuations, including
islands of topological phase in the non-topological region, which is
consistent with previous results on short wires Adagideli  _et al._ (2014).
These fluctuations are seen to be substantially reduced in the longer wire
results shown in Fig. 38 and reveal an unambiguously trivial superconducting
phase for this system. These topological patches are also eliminated by
increasing the strength of the disorder by $30\%$ as seen in Fig. 42. The
topological patch $Q<0$ (seen in blue in panel a) is seen in Fig. 38(b) might
be expected to support MZMs based on the non-trivial topological invariant.
Thus, finding topological patches in finite wires with small gaps may not
necessarily imply the robust existence of topology even when these patches
satisfy the $Q<0$ criterion, as emphasized already in sec. IV.

Figure 42: (a,b) Similar to Fig. 37 except for increased disorder
$V_{dis}=2.0$ meV. This figure shows that the topological patches in Fig. 37
are eliminated with increasing disorder.

Despite the suggestion of MZMs, the LDOS as a function of energy $E$ and
position $x$ at the reference point $\mu=0.28$ meV and $B=0.56$ T (star in
Fig. 37(a)) shown in Fig. 37(c) indicates a zero-energy state that is
delocalized across the wire. This delocalization is confirmed by the sectional
plot of the logarithm of the LDOS at $E=0,40$ mK shown in Fig. 37(d). The
peaks of the $\textrm{Log}(LDOS)$ appear to be relatively uniform from the
ends of the wire to the middle of the wire, which is consistent with the
relatively high transmission ($(L/\xi)_{trans}=0.8$). Even in the absence of
localized zero-bias peaks, we find that the end conductance at the reference
point, shown as the middle curve in each of these panels of Fig. 39 displays a
zero-bias conductance peak at both ends. The other curves in Fig. 39(a) shows
the variation of the conductance with changing magnetic field $B$ while Fig.
35(b) shows variation with varying chemical potential $\mu$. Variation of the
magnetic field suggests a robust ZBCP in all local conductance curves $G_{LL}$
and $G_{RR}$ despite the fact that only a few of these curves are in the
topological phase. This is quite different from the variation of the chemical
potential, which seems to split the ZBCP rather rapdily. The non-local
conductances $G_{LR},G_{RL}$ shows that reference point is gapped with a
nearly flat non-local conductance. However, changing the magnetic field $B$ or
chemical potential $\mu$ appears to rapidly close this gap, which would be
interpreted as a very small topological patch according to the criteria used
to analyze the experimental data Aghaee  _et al._ (2022). While this is
qualitatively consistent with the phase diagram Fig. 37(a) surrounding the
reference point, the patch from the topological gap appears rather smaller
than the theoretical one for this example. The absence of end localized MZMs
could be diagnosed from the zero temperature local conductance (Fig. 40) which
show ZBCPs to be strongly split and substantially below quantization.

The delocalized zero-energy state in the LDOS in Fig. 37(c,d) was accompanied
by a low transmission. However, a closer examination of the same phase diagram
in the vicinity of the topological patch, shown in Fig. 41(a) shows that a
slightly different reference point ($\mu=0.29$ meV $B=0.61$ T) shows a much
stronger topological charge $Q=-0.983$. The corresponding transmission shown
in Fig. 41(b) is also found to be much smaller corresponding to an effective
length $(L/\xi)_{trans}\sim 1.9$. Indeed, this less that $10\%$ change of the
reference point along either direction relative to Fig. 37 introduces a dip in
the the middle of the profile of the LDOS shown in Fig. 41(c). An examination
of the log plot Fig. 41(d), shows that this dip is consistent with the
transmission coherence length $\xi_{trans}$. Thus, topology in this more
disordered case for the finite system has no end MZMs even in the patches
where the $Q<0$ condition is unambiguously satisfied. Going to long wires in
this case establish decisively that the short wire topological patches were
simply finite size fluctuations and cannot be construed to be ’topological’ in
any practical sense since there are no end MZMs in the system. We emphasize
that transport experiments cannot detect the existence or not of the end MZMs
unless the local conductance is robustly quantized at $2e^{2}/h$ – just seeing
a ZBCP does not ensure the existence of MZMs. Pan and Das Sarma (2020)

###  V.4 Summary of analysis of topological patches

To summarize the findings of this section, an intermediate strength of
disorder ($\sim 1.5\,meV$) for a finite length nanowire leads to a topological
”phase diagram” with small topological patches (Fig. 37). This is consistent
with previous analysis of topological superconductors Adagideli  _et al._
(2014); Aghaee  _et al._ (2022). Stronger disorder ($\sim 2.0\,meV$)
eliminates even the patches at $L=3\,\mu m$ (Fig. 42). Here we use quotation
marks to refer to phase diagram because a finite size system does not
technically represent a phase unless it can be shown to survive extension to
the so-called thermodynamic limit. Indeed, the parameters that demonstrate
such topological patches at $L=3\,\mu m$ are found to be topologically trivial
over the broad vicinity of the patches for longer wires (see Fig. 38), making
it clear that the wires are non-topological in a thermodynamic sense. However,
the finite wire still shows somewhat robust ZBCPs and a gap at a reference
point for the $L=3\,\mu m$ long wire (see Fig. 39), which are characteristics
that could be associated with MZMs Aghaee  _et al._ (2022). However an
examination of the LDOS shown in Fig. 37 shows something essentially
delocalized making this interpretation difficult to justify. On the other
hand, the wire with slightly lower disorder (Fig. 33) is found to show
exponentially localized MZMs in the LDOS as well as a relatively large
topological regime. In this case the topological regime falls within the range
of the topological phase of a longer wire (Fig. 34) that is essentially in the
thermodynamic limit with weak fluctuations. However, even the intermediate
disorder topological patch regime does show a singular point (Fig. 41) with a
potentially localized MZM and low transmission. The challenge is that this
point cannot be separated from the delocalized MZM state by transport
measurements at $T=40$ mK, making the intermediate disorder topological patch
regime problematic to interpret. Unless the topological patch is stable over a
reasonably large parameter regime (e.g. magnetic field) for the finite wire,
we cannot be sure that the system is topological in any operational sense even
if the nonlocal conductance manifests a small gap in this patch.

##  VI Conclusion

Motivated by the recent Majorana nanowire experiment from Microsoft Quantum
Aghaee  _et al._ (2022), we have carried out an extensive analysis of the
“topological” properties of finite length disordered semiconductor nanowires
(e.g., InAs) proximitized by a superconductor (e.g., Al) in the presence of
spin-orbit coupling and Zeeman splitting Sau  _et al._ (2010a, b); Lutchyn
_et al._ (2010); Oreg  _et al._ (2010); Lutchyn  _et al._ (2011); Stanescu
_et al._ (2011). The key issues are the extent to which topology can be
inferred based on simultaneous local and nonlocal tunneling conductance
spectroscopy, and the precise meaning of such an inference. In particular, we
investigate the density of states, the local density of states, the Majorana
localization length, the relevant topological index (in the scattering
geometry), and the conductance matrix (i.e. both local and nonlocal
components) to conclude that disordered finite length nanowires may produce
mixed signals with respect to the existence or not of topological properties
depending on system parameters, disorder strength, and the wire length. For
example, just seeing a gap opening in the nonlocal conductance over some small
regions of parameter values, which superficially implies the existence of a
topological gap, may be misleading because the corresponding local density of
states may not manifest any Majorana zero modes at the wire ends. Such
superficial signatures of an apparent topological gap may simply reflect the
existence of disorder induced low-energy states with localization lengths
comparable to the wire length. The only way to ensure “topology” is to find a
topological gap region which is stable over a large range of parameter values
(e.g., magnetic field and gate voltage). The typical magnetic field range over
which the topological signatures should remain stable is expected to be
comparable to the typical magnetic field where the zero bias conductance peaks
show up. The key problem we identify is that the existence of gap closing and
opening signatures in nonlocal conductance along with the existence of zero
bias conductance peaks in the local conductance, by themselves, are not
definitive in ascertaining “topology” in finite disordered wires, unless such
features persist over a large variation in the tuning parameters, such as the
Zeeman splitting and chemical potential. To make things even more complicated,
there is the possibility that occasionally real MZMs may be localized somewhat
away from the ends of the system, thus producing no signatures in the local
tunneling and leading to an erroneous conclusion that the system is trivial
(since no ZBCP shows up from tunneling at the end). A similar false negative
may also arise in the nonlocal tunneling where the gap reopening signal may
occasionally be far too weak for it to effectively manifest itself in the
experiment even when a real topological gap actually exists.

One key aspect of the current SM-SC nanowire samples playing a crucial role in
the Majorana physics is the fact that the relative magnitudes of the three
important length scales in the problem are comparable: mean free path (or
localization length), SC coherence length, and the wire length. Of these three
length scales, only the wire length is approximately ($\sim$ 3 microns) known
in the Microsoft experiment. Our best estimates of the other two length scales
indicate that the conditions necessary for the realization of isolated end
MZMs, namely, that the wire length is much longer than the coherence length
($\sim$ 1-3 microns) which in turn is much less than the mean free path
($\sim$ 1 micron), may not be decisively satisfied because of the relatively
small disorder to gap ratios in the system. These problems are soluble by
using longer wires and larger gap to disorder ratios in the future
experiments. This approximate one micron estimate for the mean free path is
also consistent with the interface random charged impurity density of $\sim
3.10^{12}/{\rm cm}^{2}$, as estimated for the samples used in the Microsoft
experiment. Aghaee  _et al._ (2022) We note that this experiment is already a
huge improvement over all earlier nanowire experiments in this context, since
the earlier experiments used much shorter wires (of lengths $\sim 1$ micron)
having larger disorder.

A key finding of ours, which was not realized in earlier works on Majorana
zero modes (including our own work), is that it is insufficient to observe
simultaneous unquantized ZBCPs from both ends in the local tunneling
conductance along with weak gap closing/opening features in the nonlocal
conductance over small variations of the system parameters (most particularly,
the Zeeman field) in order for a decisive claim of topology. To have stable
Majorana zero modes, the “topological phase” should persist over a reasonably
large parameter region, instead of consisting of small patches surrounded by
trivial regions, with repeated field-tuned transitions between topological and
trivial regimes indicating the presence of considerable disorder-induced low
energy ’Griffiths’ states in the bulk of the wire. The problem is that such
low energy states throughout the bulk, if present, completely suppress the end
MZM properties, destroying the bulk-boundary correspondence, and in such
situations, having a small bulk gap may not necessarily imply the existence of
anything resembling a pair of isolated non-Abelian end MZMs, since the bulk of
the wire is filled with “effective” Majorana modes created by the random
disorder. Our theory explicitly shows that such a disturbing situation is
generic in the intermediate disorder regime, and experimental observations of
fragile gap closing/opening along with weak ZBCPs is insufficient for the
manifestation of any topology with non-Abelian properties. A pair of Majorana
modes (strongly) overlapping with disorder-induced low-energy states may
exhibit some stability within a small patch in the control parameter space,
but this does not imply that it possesses non-Abelian properties and does not
guarantee that its stability will be enhanced by increasing the system size.
Of course, it is possible, by sheer luck, that some of these fragile
“topological” patches inferred on the basis of nonlocal gap closing/opening
are in fact truly topological, with non-Abelian end MZMs, but this is unlikely
if the stability of the effective “topology” persists only over small magnetic
field regions. In any case, local and nonlocal transport is incapable of
decisively settling this point in finite disordered wires, unless the
effective topology (as obtained by the existence of gap closing/opening along
with ZBCPs) is stable over a large parameter space. Disorder-induced low
energy bulk states may very well lead to artificial ZBCPs at finite
temperatures, mimicking MZMs, but these ZBCPs are neither quantized nor
conclusive signatures for isolated end MZMs. In the same vein, some of these
bulk localized states may have localization lengths comparable to or larger
than the wire length, thus leading to artificial gap opening features which
are not the topological bulk gaps. Such disorder-induced gap opening features
in the nonlocal conductance are misleading and would disappear in longer
wires. Since the current experiments cannot directly measure any topological
invariants, there are limited options in deciding about the topological nature
of the system from tunneling transport spectroscopy, including (1) efforts to
measure the thermal conductance which should be quantized at the TQPT; (2)
trying to find quantized ZBCPs stable to large variations in system
parameters, including tunnel barrier conductance Lai  _et al._ (2022); (3)
observing stable topological features in transport over large regimes of the
Zeeman field and the gate voltage; (4) trying experiments to look for non-
Abelian properties (e.g. braiding). We point out that although the measurement
of nonlocal conductance is necessary to establish the closing and the opening
of the bulk gap at the TQPT, such nonlocal measurements will be unsuccessful
in long wires if the disorder is not sufficiently reduced so that the mean
free path is longer than the wire length, and therefore, both longer wires and
lower disorder are necessary to demonstrate the exponential protection
characterizing Majorana zero modes which underlie topological quantum
computation. We believe, and our current work establishes that the gap to
disorder ratio in the SM-SC hybrid systems must increase considerably before
one can be certain that decisive topology has been established. In this
context, it may be worthwhile to consider some additional diagnostic
measurements for Majorana signatures such as the fractional Josephson effect,
Kwon  _et al._ (2004) noise correlation spectroscopies Bolech and Demler
(2007); Akhmerov  _et al._ (2011), finite -frequency spectroscopy around the
TQPT Tewari  _et al._ (2012), and ZBCP fidelity studies Lai  _et al._ (2022)
which could meaningfully complement the local and nonlocal transport
spectroscopies of Ref. Aghaee  _et al._ , 2022. It may also be worthwhile to
carry out noninvasive local tunneling measurements along the wire to ensure
that the bulk is free of abundant low energy bound states suppressing Majorana
physics. The breakthrough Microsoft experiment provides clear guidance for
future efforts and tells us precisely how the experiments should improve in
our search for Majorana. Although some of the current Microsoft samples are
likely to have manifested topological patches with small gaps, longer wires
and more robust stability to parameter variations as well as larger values of
gap to disorder ratios seem to be necessary for developing Majorana-based
topological qubits in future experiments. The current work indicates that a
factor of 2-4 increase both in the gap to disorder ratio and the wire length
compared with the current Microsoft samples in Ref. Aghaee  _et al._ , 2022
may be adequate to reach the decisively topological regime with stable
Majorana zero modes at the wire ends. An encouraging recent materials
development in this context is the just-appeared report of the fabrication of
InAs-Al hybrid SM-SC structures with mobilities exceeding $10^{5}cm^{2}/V.s$
Zhang  _et al._ (2023).

We believe that the Microsoft experiment is an essential breakthrough in the
search for topological superconducting phase, but unless the topological gap
regime is stable over a large parameter regime, it is unclear that the system
would manifest non-Abelian topological properties. Our best guess based on our
work is that the current systems, although much improved from the earlier
nanowires used in the older experiments, may still be in the intermediate
disorder regime with considerable low energy subgap Andreev bound states,
which may produce some signatures (e.g., gap opening/closing over small
patches of parameter values) of topological gap. However, these signatures may
disappear if longer wires are used, as shown in our concrete examples. We
cannot rule out that some of the studied nanowires are indeed topological with
small gaps and weak stability, but any definitive conclusion would necessitate
wires with much reduced disorder to gap ratios, which can be achieved either
by reducing the disorder or by increasing the induced gap (or both). We remain
optimistic that future experiments on cleaner samples would provide definitive
evidence for a topological superconducting phase with localized non-Abelian
Majorana zero modes.

###### Acknowledgements.

We are grateful to Roman Lutchyn, Chetan Nayak, Haining Pan and Anton Akhmerov
for valuable discussions and advice. JS acknowledges using KWANT code from Bas
Nijoholt’s gitbuh (https://github.com/basnijholt/majorana-nanowire-
conductance) for Sec. V. This work is supported by the Laboratory for Physical
Sciences and by NSF-2014156.

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# (523599) 2003 RM: The asteroid that wanted to be a comet

Davide Farnocchia Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Darryl Z. Seligman
Department of Astronomy and Carl Sagan Institute, Cornell University, 122
Sciences Drive, Ithaca, NY, 14853, USA  Mikael Granvik Asteroid Engineering
Laboratory, Luleå University of Technology, Box 848, SE-98128 Kiruna, Sweden
Department of Physics, University of Helsinki, PO Box 64, FI-00014 Helsinki,
Finland  Olivier Hainaut ESO, Karl-Schwarzschild-Straße 2, 85748 Garching-bei-
München, Germany  Karen J. Meech Institute for Astronomy, University of
Hawaii, 2680 Woodlawn Dr., Honolulu, HI 96822, USA  Marco Micheli ESA NEO
Coordination Centre, Largo Galileo Galilei, 1, 00044 Frascati (RM), Italy
Robert Weryk Physics and Astronomy, The University of Western Ontario, 1151
Richmond St., London, ON N6A 3K7, Canada  Steven R. Chesley Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena,
CA 91109, USA  Eric J. Christensen  Lunar and Planetary Laboratory, University
of Arizona, 1629 E University Blvd., Tucson, AZ 85721, USA  Detlef Koschny ESA
NEO Coordination Centre, Via Galileo Galilei, 1, 00044 Frascati (RM), Italy
ESA ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands  LRT, TU Munich,
Boltzmannstraße 15, 85748 Garching bei München, Germany  Jan T. Kleyna
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr., Honolulu, HI
96822, USA  Daniela Lazzaro Observatório Nacional, R. Gen. José Cristino, 77,
20921-400 Rio de Janeiro, RJ, Brazil  Michael Mommert School of Computer
Science, University of St. Gallen, Rosenbergstrasse 30, CH-9000 St. Gallen,
Switzerland  Richard Wainscoat Institute for Astronomy, University of Hawaii,
2680 Woodlawn Dr., Honolulu, HI 96822, USA

###### Abstract

We report a statistically significant detection of nongravitational
acceleration on the sub-kilometer near-Earth asteroid (523599) 2003 RM. Due to
its orbit, 2003 RM experiences favorable observing apparitions every 5 years.
Thus, since its discovery, 2003 RM has been extensively tracked with ground-
based optical facilities in 2003, 2008, 2013, and 2018. We find that the
observed plane-of-sky positions cannot be explained with a purely gravity-
driven trajectory. Including a transverse nongravitational acceleration allows
us to match all observational data, but its magnitude is inconsistent with
perturbations typical of asteroids such as the Yarkovsky effect or solar
radiation pressure. After ruling out that the orbital deviations are due to a
close approach or collision with another asteroid, we hypothesize that this
anomalous acceleration is caused by unseen cometary outgassing. A detailed
search for cometary activity with archival and deep observations from Pan-
STARRS and the VLT does not reveal any detectable dust production. However,
the best-fitting H2O sublimation model allows for brightening due to activity
consistent with the scatter of the data. We estimate the production rate
required for H2O outgassing to power the acceleration, and find that, assuming
a diameter of 300 m, 2003 RM would require Q(H2O)$\sim 10^{23}$ molec s-1 at
perihelion. We investigate the recent dynamical history of 2003 RM and find
that the object most likely originated in the mid-to-outer main belt ($\sim
86\%$) as opposed to from the Jupiter-family comet region ($\sim 11\%$).
Further observations, especially in the infrared, could shed light on the
nature of this anomalous acceleration.

Asteroid dynamics — Comet dynamics — Near-Earth Objects

##  1 Introduction

Modeling the trajectory of small bodies is a non-trivial problem. As the data
quality improves and observational arcs get extended, nongravitational
perturbations can become a significant consideration. Comets are especially
affected by nongravitational forces because of sublimation and outgassing of
volatiles (Meech & Svoren, 2004). Therefore, orbit determination centers that
estimate cometary orbits such as the Jet Propulsion
Laboratory111https://ssd.jpl.nasa.gov and the Minor Planet Center
222https://minorplanetcenter.net account for nongravitational perturbations in
the orbit model. Typically, these perturbations are incorporated based on the
standard model of Marsden et al. (1973), although more sophisticated models
(e.g., Yeomans & Chodas, 1989; Królikowska, 2004; Yeomans et al., 2004;
Chesley & Yeomans, 2005) are sometimes employed.

As opposed to cometary orbits, the trajectories of volatile-depleted asteroids
are generally better approximated with a motion purely driven by gravitational
forces. While asteroids are subject to nongravitational perturbations such as
solar radiation pressure (Vokrouhlický & Milani, 2000) and the Yarkovsky
effect (Vokrouhlický et al., 2015), these are orders of magnitude weaker than
forces induced via outgassing. These weaker forces can still be detectable
with astrometric data, especially with radar measurements (Ostro et al., 2002)
and/or long observational arcs. Solar radiation pressure has been measured for
a handful of small near-Earth asteroids (MPEC
2008-D12,333https://www.minorplanetcenter.org/mpec/K08/K08D12.html Micheli et
al., 2012, 2013, 2014; Mommert et al., 2014a, b; Farnocchia et al., 2017;
Fedorets et al., 2020), while searches for detections of the Yarkovsky effect
are performed regularly (e.g., Farnocchia et al., 2013; Del Vigna et al.,
2018; Greenberg et al., 2020) and have led to hundreds of detections.

Asteroid (523599) 2003 RM was discovered by the Near Earth Asteroid Tracking
Program (Pravdo et al., 1999) on 2003 September 2 (MPEC
2003-R17).444https://www.minorplanetcenter.net/mpec/K03/K03R17.html 2003 RM
has a semimajor axis $a=2.92$ au and eccentricity $e=0.60$, which yields
apsidal distances of $q\times Q=1.17$ au $\times$ 4.68 au. The inclination is
10.9∘ relative to the ecliptic plane. Since the initial discovery, there have
been over 300 optical observations of 2003 RM over the course of four
apparitions in 2003, 2008, 2013, and 2018. While searching for the Yarkovsky
effect among near-Earth asteroids, Chesley et al. (2016) detected a clear
signal of a transverse acceleration in the motion of 2003 RM. They noted that
the observed acceleration was far too large to be caused by the Yarkovsky
effect, and argued that cometary activity was the most likely explanation.
However, Chesley et al. (2016) also reported that closer inspection of
observational images of 2003 RM by a number of near-Earth object search
programs (Christensen et al., 2016; McMillan et al., 2016; Wainscoat et al.,
2016) did not reveal any clear evidence of cometary activity. To date, no
detection of cometary activity has been reported to the Minor Planet Center
and therefore 2003 RM currently remains classified as an asteroid.

A similar puzzle affects the understanding of the cometary nature of
1I/‘Oumuamua, the first interstellar object to be discovered passing through
the inner Solar System (MPECs 2017-U181 and
2017-V17).555https://minorplanetcenter.net//mpec/K17/K17UI1.html,666https://minorplanetcenter.net/mpec/K17/K17V17.html
While no cometary activity was detected around ‘Oumuamua (Meech et al., 2017;
Trilling et al., 2018), its astrometric positions could only be explained with
the addition of nongravitational perturbations, most plausibly due to
outgassing activity (Micheli et al., 2018). Additionally, ‘Oumuamua had an
extreme shape, a reddened reflection spectrum (Meech et al., 2017) and a low
incoming velocity indicating a young $<40$ Myr age (Feng & Jones, 2018).
Reconciling the lack of activity and the observed nongravitational
perturbations remains a challenge (Jewitt & Seligman, 2022), even though
models have been proposed that could explain all the observed properties of
‘Oumuamua, e.g., by assuming a significant presence of molecular hydrogen ice
(Seligman & Laughlin, 2020). Alternative explanations have invoked the
presence of N2 and CO driven activity (Desch & Jackson, 2021; Jackson & Desch,
2021; Seligman et al., 2021) or radiation pressure (Micheli et al., 2018).
Spin-up was detected in the object (Taylor et al., 2022) consistent with
outgassing torques (Rafikov, 2018).

##  2 Astrometry

2003 RM is currently in a 5:1 mean motion resonance with the Earth.
Subsequently, the object has apparitions visible from the Earth every five
years. Since its discovery in 2003, the object has been extensively tracked
with ground-based optical facilities over the course of the following four
apparitions:

  * •

From 2003 September 2 to 2003 October 19: 85 observations;

  * •

From 2008 June 24 to 2008 October 29: 73 observations;

  * •

From 2013 May 16 to 2013 October 24: 45 observations;

  * •

From 2018 March 18 to 2018 November 14: 98 observations.

The full observational dataset is available from the Minor Planet
Center.777https://minorplanetcenter.net/db_search/show_object?object_id=523599
For the majority of the existing astrometric data, observations were weighted
using the scheme presented by Vereš et al. (2017). Positional uncertainties
were estimated for our own measurements:

  * •

Siding Spring Survey (station code
E12)888https://minorplanetcenter.net/iau/lists/ObsCodesF.html observations on
2008 June 24 and 25, which we remeasured and weighted at 0.5′′;

  * •

Catalina Sky Survey (station code 703) observations on 2008 September 7
(weights at 0.2′′) and 21 (0.6′′), 2013 August 28 (0.8′′), 2013 September 15
(0.5′′) and 23 (0.8′′), which we remeasured;

  * •

Pan-STARRS 1 (station code F51) observations on 2013 July 7 (0.2′′), 2013
August 17 (0.1′′) and 28 (0.2′′), 2013 September 14 (0.1′′), and 2013 October
24 (0.1′′);

  * •

OASI, Nova Itacuruba (station code Y28) observations on 2018 March 18 (0.7′′),
2018 April 19 (0.1′′) and 20 (0.2′′), 2018 May 17 (0.2′′), 2018 July 09
(0.2′′), 15 (0.1′′), 17 (0.1′′), and 18 (0.1′′), 2018 August 7 (0.1′′), 2018
September 6 (0.1′′) and 11 (0.2′′);

  * •

Las Campanas Observatory (station code 304), with the Magellan Baade
telescope, on 2018 June 22 (0.1′′);

  * •

Lowell Discovery Telescope (station code G37) on 2018 September 4 (0.15′′);

  * •

Very Large Telescope (station code 309) with the Unit Telescope 1,
observations on 2018 September 19 (0.1′′).

We assumed a 1 s uncertainty in the reported time of observation and corrected
for star catalog systematic errors using the Eggl et al. (2020) debiasing
scheme. To reject outliers, we employed the Carpino et al. (2003) algorithm.

##  3 Detection of nongravitational perturbations

Our default gravitational force model configuration (e.g., Farnocchia et al.,
2015) is based on JPL planetary ephemeris DE441 (Park et al., 2021) and the 16
most massive small-body perturbers in the main belt (Farnocchia,
2021).999ftp://ssd.jpl.nasa.gov/pub/eph/small_bodies/asteroids_de441/SB441_IOM392R-21-005_perturbers.pdf
As was initially pointed out by Chesley et al. (2016), this gravity-only model
configuration fails to satisfactorily model the 2003 RM astrometric data.
While any given set of two consecutive apparitions can be fit, the addition of
a third apparition results in unacceptably high residuals.

To illustrate this point, Figure 1 shows the astrometric residuals of the
entire dataset against gravity-only solutions based on two consecutive
apparitions. Except for a handful of outliers, the residuals of the fitted
astrometric observations are consistent with the assumed observational
uncertainties. However, residuals of the observations outside the two fitted
apparitions are clearly too large to be explained by astrometric errors or
ephemeris uncertainties. This behavior is not caused by only a handful of
isolated observations. Instead, every apparition has several tens of
observations with excessively large residuals. Moreover, the failure to
predict astrometric positions manifests for every possible choice of
consecutive apparitions included in the fit.

Figure 1: Astrometric residuals against the best gravity-only fit to the
astrometry from 2003 to 2008 (top panel), from 2008 to 2013 (middle panel), or
from 2013 to 2018 (bottom panel). Dots correspond to Right Ascension (R.A.)
and crosses correspond to Declination (Dec.). Outliers rejected from the fit
are indicated with circles. The shaded area corresponds to observations not
included in the fit. The dashed line and solid line represent the 1-$\sigma$
ephemeris uncertainties of the orbital solution in R.A. and Dec.,
respectively.

###  3.1 Yarkovsky effect

The Yarkovsky effect, a nongravitational perturbation due to anisotropic
thermal re-emission of absorbed solar radiation (Vokrouhlický et al., 2015),
is a reasonable first explanation for the failure to reproduce the data. In
order to model this effect, we include a simple transverse acceleration
$A_{2}(\text{1 au}/r)^{2}$, where $r$ is the heliocentric distance and $A_{2}$
is an estimable parameter (Farnocchia et al., 2013). The fit to the full
observational dataset is now satisfactory with a $\chi^{2}$ of 145.1 and a
weighted RMS of 0.50. The best fit produces an estimate $A_{2}=(212.7\pm
3.6)\times 10^{-14}$ au/d2, i.e., a detection with signal-to-noise ratio of
58. This estimate is consistent when using subsets of the data arc over three
apparitions:

  * •

$A_{2}=(212.5\pm 9.0)\times 10^{-14}$ au/d2 when fitting the first three
apparitions from 2003 to 2013;

  * •

$A_{2}=(207.9\pm 7.9)\times 10^{-14}$ au/d2 when fitting the last three
apparitions from 2008 to 2018.

Figure 2 demonstrates that fits based on three apparitions that include a
transverse acceleration $A_{2}/r^{2}$ successfully predict the fourth
apparition within the uncertainties. However, the magnitude of the transverse
acceleration far exceeds that which could feasibly be produced by the
Yarkovsky effect. Farnocchia et al. (2013) defined $A_{2,exp}$ as a proxy for
the expected value of $|A_{2}|$ for a given asteroid. This approximation is
derived from the $A_{2}$ value measured for (101955) Bennu, scaled according
to the expected size of the target asteroid. Bennu is a good reference because
if its extreme obliquity and high Yarkovsky detection signal-to-noise ratio.
For 2003 RM, the absolute magnitude $H=19.8$ leads to a range in diameter from
300 to 650 m assuming an albedo between 5% and 25%. Therefore,
$A_{2,exp}<8\times 10^{-14}$ au/d2, which is 27 times smaller than the
observed acceleration. Matching the observed acceleration would require either
an unrealistically low bulk density ($\sim$ 0.1 g/cm3) or a size of tens of
meters, which requires a nonphysical albedo $>10$.

Figure 2: Astrometric residuals against the best fit to the astrometry either
from 2003 to 2013 (top panel) or from 2008 to 2018 (bottom panel) using a
transverse acceleration $A_{2}/r^{2}$. Dots correspond to Right Ascension
(R.A.) and crosses correspond to Declination (Dec.). Outliers rejected from
the fit are indicated with circles. The shaded area corresponds to
observations not included in the fit. The dashed line and solid line represent
the 1-$\sigma$ ephemeris uncertainties of the orbital solution in R.A. and
Dec., respectively.

###  3.2 Cometary nongravitational perturbations

Since the Yarkovsky effect is incompatible with the observed acceleration, we
considered the possibility that 2003 RM is a comet and that its motion is
affected by perturbations due to outgassing. By modeling the transverse
acceleration as $A_{2}g(r)$ (with $g(r)$ from Marsden et al., 1973), we obtain
a satisfactory fit to the data. This fit produces $\chi^{2}=138.7$, a weighted
RMS of 0.49, and $A_{2}=(332.4\pm 5.7)\times 10^{-14}$ au/d2. In fact, this
fit is slightly better than the Yarkovsky solution obtained in the previous
subsection. Table 1 shows the corresponding orbit solution.

Table 1: JPL orbit solution 58 for 2003 RM using the Marsden et al. (1973) $g(r)$ function. The heliocentric orbital elements refer to an osculating epoch of 2017 May 5 TDB and are in the IAU76 ecliptic frame (Seidelmann, 1977). The error bars correspond to 1-$\sigma$ formal uncertainties.  Parameter | Value | Uncertainty | Units
---|---|---|---
Eccentricity | $0.6013493571$ | $2.39\times 10^{-8}$ |
Perihelion distance | $1.1642435313$ | $6.98\times 10^{-8}$ | au
Time of perihelion TDB |  $2018$ July $4.6791744$ | $1.08\times 10^{-5}$ | d
Longitude of node | $336.72534377$ | $9.65\times 10^{-6}$ | ∘
Argument of perihelion | $324.5461040$ | $1.20\times 10^{-5}$ | ∘
Inclination | $10.86102945$ | $3.75\times 10^{-6}$ | ∘
$A_{2}$ | $332.42\times 10^{-14}$ | $5.68\times 10^{-14}$ | au/d2

We also investigate whether or not the fit to the data favors a specific
$g(r)\propto 1/r^{m}$ power law. Figure 3 shows the best-fit $\chi^{2}$ as a
function of $m$. There is a very shallow minimum for $m=9$ with a broad
3-$\sigma$ range ($\Delta\chi^{2}=9$) of acceptable values from $m=1.6$ to
$m=55$.

There is no signal for the radial component $A_{1}g(r)$, i.e., using the
Marsden et al. (1973) model the estimate of $A_{1}=(-2300\pm 2500)\times
10^{-14}$ au/d2 is compatible with zero within the uncertainty. The range of
possible values for $A_{1}$ is compatible with $A_{1}$ being an order of
magnitude greater than $A_{2}$, which is a typical ratio for comets
(Farnocchia et al., 2014a).

Figure 3: $\chi^{2}$ of the orbital fit as a function of the power law
exponent for $g(r)=1/r^{m}$. The full observational arc is included in the fit
and $A_{2}$ is the only nongravitational parameter estimated. The dashed line
corresponds to the Marsden et al. (1973) $g(r)$.

###  3.3 Close approach to or collision with another asteroid

Another possibility is that the orbit of 2003 RM is perturbed by some other
small-body perturber. However, increasing the number of perturbers from 16 to
373 (Farnocchia, 2021) neither reveals any significant close approaches nor
provides a satisfactory fit to any three of the apparitions. While it is
possible that some assumed perturber masses are erroneous, we verified that
this could not account for the lack of fit. Specifically, estimating the
perturber masses as free parameters within reasonable a priori ranges
(similarly to Farnocchia et al., 2021) does not allow a fit to the data. As
further proof, the simultaneous estimate of $A_{2}$ and perturber masses still
leads to a 58-$\sigma$ detection of $A_{2}$. These results clearly favors the
additional nongravitational acceleration over corrections to perturber masses.

With an aphelion of 4.7 au, there is a remote possibility that 2003 RM
experienced a close approach or collision with a small body not included as a
perturber in the force model. This close approach or collision would more
likely have occurred when 2003 RM crossed the main belt, which is densest on
the ecliptic plane (e.g., Jedicke et al., 2015, Sec. 6). However, the
ascending and descending nodes of 2003 RM are at 1.2 au and 3.6 au from the
Sun, respectively. The asteroid distribution has low density at these
distances. Still, the 10∘ inclination of 2003 RM could allow for close
encounters or collisions outside of the ecliptic plane.

The $A_{2}$ signal is present when fitting both the first three apparitions
and the last three apparitions. Therefore, any close approach and collision
would need to take place between the second and the third apparition, i.e.,
between 2008 October 29 and 2013 May 16. During this time window, we estimated
an impulsive velocity variation $\Delta v$ as a function of time as discussed
in Farnocchia et al. (2014b, Sec. 5). To aid the fit to the data, we included
the maximum transverse acceleration allowed by the Yarkovsky effect, i.e.,
$A_{2}=8\times 10^{-14}$ au/d2 (see. Sec. 3.1).

Fig. 4 shows the $\chi^{2}$ of the fit as a function of the epoch during which
the $\Delta v$ was applied.

Figure 4: Top panel: $\Delta\chi^{2}$ of the orbital fit for an orbital
solution with an impulsive variation in velocity $\Delta v$ as a function of
impulse time relative to the lowest $\chi^{2}=285.4$, which corresponds to a
$\Delta v$ on 2011 August 19. Bottom panel: magnitude of $\Delta v$ as a
function of impulse time.

There is clear minimum with $\chi^{2}=285.4$ for an impulsive event on 2011
August 19, with a 3-$\sigma$ range from 2011 February 25 to 2012 March 06.

We rule out that possibility that an impulsive $\Delta v$ would explain the
deviation from a gravity-only model for the following reasons:

  * •

During the time frame surrounding the minimum of $\chi^{2}$, 2003 RM travels
from 4.7 au to 4.1 au from the Sun and from 0.5 au to 0.1 au above the
ecliptic plane. Because this region of the Solar System has a low density of
asteroids (Lagerkvist & Lagerros, 1997; Jedicke et al., 2015), it is extremely
unlikely that an impulsive $\Delta v$ event occured there.

  * •

The $\chi^{2}$ of the fit is significantly higher ($\Delta\chi^{2}\sim 147$)
than that obtained with the transverse nongravitational acceleration model.
This is true despite the fact that a larger number of parameters were used:
four (epoch and components of $\Delta v$) instead of one ($A_{2}$).

  * •

A three-apparition orbit solution with $\Delta v$ estimated provides poor
predictions for the fourth apparition. For example, Fig. 5 shows the
astrometric residuals against a solution based on the last three apparition
that estimates a $\Delta v$ on 2011 August 19. The residuals of the 2003
apparitions are much larger than the ephemeris prediction uncertainties.

Figure 5: Astrometric residuals against the best fit to the astrometry from
2008 to 2018 with an impulsive $\Delta v$ event on 2011 August 9. Dots
correspond to Right Ascension (R.A.) and crosses correspond to Declination
(Dec.). Outliers rejected from the fit are indicated with circles. The shaded
area corresponds to observations not included in the fit. The dashed line and
solid line represent the 1-$\sigma$ ephemeris uncertainties of the orbital
solution in R.A. and Dec., respectively.

##  4 Search for cometary activity

No observer has reported a clear detection of cometary activity for 2003 RM to
the Minor Planet Center. As a result 2003 RM is classified as an asteroid. We
carefully reviewed our observations (see Sec. 2) and 2003 RM appears stellar
in all of them, thus providing no observational evidence of a cometary nature
for this object. Below we provide a detailed analysis for two of the
observations with the highest signal-to-noise ratio (SNR): Pan-STARRS
observations on 2013 August 17, 34 days past perihelion, when 2003 RM was at
heliocentric distance of 1.2 au, and VLT observations on 2018 September 19, 76
days past perihelion, when 2003 RM was at a heliocentric distance 1.5 au.

###  4.1 Pan-STARRS

Figure 6: 2003 RM (in the center, North is up, East is left) as it appeared on
2013 August 17 from Pan-STARRS1 moving North at $0.62$ degrees per day towards
a cell gap. Comparison of the FWHM with visible stars in a larger region than
shown here indicates it was not active.

We searched the image archive of the $1.8$ m Pan-STARRS1 telescope (Wainscoat
et al., 2020; Chambers et al., 2016) located on Maui, Hawai’i. We inspected
the “chip” images of all matching exposures for low-level cometary activity.
These images are normally “warped” to a common fixed plane-of-sky projection
with $0.25^{\prime\prime}$ pixels (to allow for deep stacks for non-asteroid
science) for survey operations, but using them allows for inspection of the
native $0.256^{\prime\prime}$ pixel data without resampling, which can be
useful in identifying sub-pixel coma.

We employed the Vereš et al. (2012) algorithm to fit the point-spread-function
(PSF) of both 2003 RM and all visible field stars that appear in the Gaia-DR2
(Gaia Collaboration et al., 2018) catalog. The full width at half maximum
(FWHM) was the primary metric to search for activity. The fit primarily uses
an asymmetric Gaussian function and adopts a trailed Gaussian for fast moving
objects. This method also reports uncertainties in the fit. For trailed PSFs,
the FWHM corresponds to the direction perpendicular to the motion. The “curve
of growth” aperture flux (essentially a plot of aperture flux as a function of
aperture radius) is dependent on the PSF of a given object and, for large-
enough radii, allows for the identification of faint coma when compared to
comparably bright field stars which do not show such coma.

Figure 7: Left: Stack of 82 VLT frames totalling 3280 s exposure time on 2003
RM. The image is 20′′ wide. The individual frames were subtracted for
background objects. Right: Surface brightness profile of 2003 RM (dots) and of
a scaled field star (line).

Unfortunately, there do not exist many Pan-STARRS images for 2003 RM. However,
the images with higher SNR show it to be consistent with a star-like
appearance. On 2013 August 17, a $60$ second z-band image with SNR $\sim 15$
had a FWHM of $1.01^{\prime\prime}\pm 0.03^{\prime\prime}$ measured
perpendicular to the direction of motion. Since the plane-of-sky motion of
2003 RM was aligned to the stellar minor axis (ie: there is a slight asymmetry
in the PSFs), the major axis FWHM of $0.99^{\prime\prime}\pm
0.06^{\prime\prime}$ must be used for comparison. The object was too trailed
for a curve-of-growth comparison, and is illustrated in Figure 6. On 2013
September 14, a $43$-second g-band image with SNR $\sim 8$ had a FWHM of
$1.22^{\prime\prime}\pm 0.05^{\prime\prime}$, with the motion of 2003 RM
aligned with the stellar major axis. The stellar FWHM minor axis for
comparison was $1.24^{\prime\prime}\pm 0.11^{\prime\prime}$. On 2013 October
24, a pair of $45$-second i-band images with SNR $\sim 6$ were stacked to have
a FWHM of $1.01^{\prime\prime}\pm 0.05^{\prime\prime}$ (minor axis) and
$1.09^{\prime\prime}\pm 0.06^{\prime\prime}$ (major axis) compared to stellar
FWHM $1.00^{\prime\prime}\pm 0.06^{\prime\prime}$ and $1.15^{\prime\prime}\pm
0.04^{\prime\prime}$ respectively. The curve of growth in each image was
comparable, although the object was fainter than the field stars. On 2018
October 17, a $45$-second w-band image with SNR $\sim 8$ had a FWHM of
$1.38^{\prime\prime}\pm 1.06^{\prime\prime}$ (minor axis) and
$1.58^{\prime\prime}\pm 0.07^{\prime\prime}$ (major axis) evidently consistent
with the stellar FWHM of $1.33^{\prime\prime}\pm 0.06^{\prime\prime}$ and
$1.49^{\prime\prime}\pm 0.06^{\prime\prime}$ respectively. The curve-of-growth
was consistent with field stars of similar brightness.

Based on these images, there is no evidence of a non-stellar appearance, and
an object like this would never be reported even as a marginal comet
candidate.

###  4.2 Very Large Telescope

On 2018 September 19, 2003 RM was observed with the Unit Telescope 1 of the
ESO Very Large Telescope (VLT) on Paranal, Chile, with the FOcal Reducer and
low dispersion Spectrograph 2 (FORS2) (Appenzeller et al., 1998). These
observations did not use a filter in order to reach the deepest possible
magnitude and surface brightness for an object with solar color. A total of 82
randomly dithered exposures of 40 s were acquired in Service Mode on 2018
September 19. They were bias and flat corrected (with a twilight flat) and
normalized to an exposure time of 1 s. Remaining low spatial frequency
residuals were removed by dividing the frames again by a “super flat field.”
This was obtained using a median of the science frames – after normalization
of the sky level around the expected position of the object and masking the
background objects.

These frames were then co-aligned using a dozen field objects as reference. A
master “star” stack was produced, which was used to calibrate the field
astrometrically using stars from the Gaia DR2 catalogue (Gaia Collaboration et
al., 2016, 2018). The expected pixel position of 2003 RM on each frame was
computed. The master star stack was then subtracted from each of the
individual frames. This subtraction produced images in which the majority of
the contribution from background objects was removed. The original and star-
subtracted frames were then stacked after being re-centered on the expected
position of the object, using an average with outlier rejection. Another stack
was produced, including only the 55 exposures where the expected position of
the object was at least $10^{\prime\prime}$ from a background object.

Both the total stack of star-subtracted frames (displayed in Fig. 7) and the
partial ones (totalling 3280 s and 2200 s exposure time, respectively) were
searched for dust. Visual inspection of the frame reveal no visible extension
of the object.

A photometric profile of 2003 RM was produced by integrating its light in a
series of circular apertures centered on the object with increasing radii. The
instrumental fluxes in these apertures were converted to surface brightness in
the concentric rings. These were then converted to magnitudes using a
photometric zero point of 27.8 (from Hainaut et al., 2021, for filter-less
observations). The profile of a field star slightly brighter than 2003 RM was
obtained in a similar manner from the “star” stack, and scaled to the
brightness of the object for comparison. These profiles are presented in Fig.
7, showing no divergence down to 28 mag/sq. arcsec.

To quantify the lack of dust, it is worth noting that the profile of the
object is perfectly stellar within the noise level. Dust contributing to up to
5-$\sigma$ could exist within the error bars. Integrating the corresponding
flux from 0.5′′ to $2^{\prime\prime}$ corresponds to a total magnitude of
24.0. Assuming dust grains with a radius of $1\mu$m and with a density of 3000
kg$/$m3 and an albedo of 0.2, we derive a coma mass limit below 4 kg. This
limit should be taken as an order of magnitude estimate based on the numerous
assumptions.

##  5 Heliocentric Light Curve Modeling

Evidence of cometary activity in the form of scattered light from the dust and
nucleus can be found from the shape of the heliocentric light curve even in
the absence of imaged dust. This can occur when there is brightening caused by
light scattered from dust contained within the seeing disk, as in the case of
a gravitationally bound dust coma such as the inner coma of (2060) Chiron
(Meech & Belton, 1990) and perhaps (468861) 2013 LU28 (Slemp et al., 2022) and
(418993) 2009 MS9 (Bufanda et al., 2022).

To derive limits on outgassing, we used the ice sublimation model to compare
the heliocentric light curve brightness to a predicted comet dust production
from outgassing. The model computes the amount of gas sublimating from an icy
surface exposed to solar heating, as described in detail in Meech et al.
(2017). The total brightness within a fixed aperture combines radiation
scattered from the nucleus and dust entrained in the sublimating gas flow and
dragged from the nucleus. The model free parameters include the ice type,
nucleus radius, albedo, emissivity, density, dust properties (sizes, density,
phase function), and fractional active area. 2003 RM has not been well-
characterized so we have significant uncertainties for all of these
parameters.

We used magnitudes from 329 observations as reported to the Minor Planet
Center between September 2003 and November 2018 to model the heliocentric
light curve. This data represents observations from 34 different observatories
reported using several different photometric systems, including some sets with
no filter listed. In order convert all the measurements to a Sloan (SDSS)
r-band, we needed to know surface color (e.g. the taxonomic class) for 2003
RM. Most NEOs belong to the S taxonomic class (Ieva et al., 2020), and based
on this assumption we used the mean SDSS colors from Ye et al. (2016) and the
transformations from Jordi et al. (2010) and Lupton (2005)
101010https://classic.sdss.org/dr4/algorithms/sdssUBVRITransform.html#Lupton2005
to convert the G, V, and R filters to the SDSS r-band. Because there was
significant scatter in the data (which was likely never intended to represent
high-precision photometry), we averaged data points taken on the same night.
Data from five of the observing stations were significantly discrepant with
observations taken at the same time and were not used. The resulting
photometry is shown in Fig. 8 and has a an error of $\sim$0.3 mag (about the
size of the data points).

In Sec. 7 we show that 2003 RM may be associated with the Eos family which
maps to the K-taxonomic class, a subset of the S-type members. These asteroids
tend to have the same visible colors as S-types but slightly lower albedos,
around $p_{v}$ $\sim$ 0.12 (Clark et al., 2009). Table 2 lists the starting
parameters for the model, along with the references for the starting values.
Some of these are based on estimates for S-type asteroids and others are
relevant to active comets.

Figure 8: The best-fitting H2O-sublimation model allows for brightening due to
activity consistent with the scatter in the data. Table 2: Summary of
Sublimation Model Parameters

| Initial | Fit
---|---|---
Sublimation Parameter | Value |  Source† |  H2O |  CO2-Limit
Nucleus Radius [km] |  RN | 0.150 | [1] | 120 | 120
Emissivity | $\epsilon$ | 0.95 | [2] | 0.9 | 0.9
Nucleus Phase function [mag/deg] | $\beta_{nuc}$ | 0.04 | [3] | 0.05 | 0.05
Nucleus Density [kg/m3] | $\rho_{n}$ | 1900 | [4] | 1900 | 1900
Nucleus Albedo |  pv | 0.25 | [1] | 0.12 | 0.12
Coma Phase function [mag/deg] | $\beta_{coma}$ | 0.02 | [5] | 0.02 | 0.02
Grain Density, [kg/m3] | $\rho_{g}$ | 3000 | [6] | 3000 | 3000
Grain Radius, [$\mu$m] | a | 1 | [7] | 1 | 1
Fractional Active Area | FAA | 0.04 | [7] | 0.003 | 0.00006
Inferred gas production [molec/s] | Q |  |  | 3.8E24 | 1.7E23

Note. — †[1] matches observed H-value for high albedo S-types; [2] measured
for 67P by Rosetta (Spohn et al., 2015); [3] measured for an S-type Gehrels &
Taylor (1977); [4] average density for small S-type asteroids ((25143)
Itokawa, (101955) Bennu, (162173) Ryugu, (65803) Didymos); [5] measured for
comets (Meech & Jewitt, 1987); [6] see discussion in Meech et al. (1997); [7]
Assumed starting values for comets.

A good fit to the data is shown in Fig. 8, and the fit parameters are shown in
Table 2 for water. Because of all the nucleus uncertainties, we varied RN,
$\beta_{nuc}$, and FAA. The model is shown plotted against true anomaly (TA=0∘
is at perihelion). Because the data span four apparitions, TA increases by
360∘ at each perihelion. The VLT data were taken about 2.5 months post-
perihelion, at TA=1144∘. The total model curve allowing for activity is
consistent with the envelope of the scatter in the data. The implied dust
production rate at perihelion for the fourth apparition from this model is
0.114 kg/s and converting this to a gas production rate assuming a dust:gas
ratio of 1 (Marschall et al., 2020) gives an inferred Q(H2O) $\sim$
3.8$\times$1024 molec/s. If the photometric scatter were much smaller (e.g.
less than 0.01 mag), then the model would be sensitive to activity at the
level of $\sim$ 1023 molec/s. At this gas production level the integrated
brightness from 1 $\mu$m dust contained within a 2′′ radius aperture would be
$r$-mag 24, consistent with the limits from the VLT data. This limit is just
above the amount of gas production that is required for a nucleus of this size
and density based on the nongravitational accelerations (see Fig. 9 below).

In order to assess how much of the scatter in the data might be due to the
object’s rotation, we examined the time series individual images taken over 2
hours from the VLT on 2018 September 19. We used the flattened sky-subtracted
frames and excluded the frames where 2003 RM was too close to field stars
(during the first hour). The remaining data shows a brightness variation of
$\sim$0.05 mag with about 14 cycles over a span of 50 minutes that, if
statistically significant, would suggest a rapid rotation period of 2003 RM.
More data would be necessary to investigate this further.

These models were run on the assumption that a body this small may no longer
contain ices as volatile as CO2 or CO. However, as discussed in Sec. 7, the
dynamical lifetime of NEOs on orbits like that of 2003 RM is relatively short.
Comet 2P/Encke has an even smaller perihelion than 2003 RM ($q$ = 0.33 au)
with a slightly larger aphelion ($Q$ = 4.09 au). Comet Encke could have
migrated in and become decoupled from Jupiter on a similar timescale. Comet
Encke exhibits a curious behavior: Near perihelion it can have very little
visible dust, but significant gas and is often active at aphelion (Meech et
al., 2001; Fink, 2009). However, this comet still has a very strong CO2
production (Reach et al., 2013). For both CO2 and CO, similar limiting
production rates of $\sim$ 1023 molec/s are found based on the heliocentric
light curve.

##  6 Required Outgassing Production

We consider a variety of assumptions regarding the nature of the outgassing.
The production depends on several unconstrained factors, including the albedo
(and corresponding size) of the body, the temperature of the outgassing gas,
the extent to which the outflowing gas is collimated vs. isotropic, and the
gas/dust mass ratio.

The production rate $Q(X)$ of a given species, $X$, can be calculated via the
conservation of linear momentum — essentially just the “rocket” equation —
using the following equation:

$Q(X)=\,\bigg{(}\,\frac{M_{\rm Tot}}{m_{\rm
X}}\,\bigg{)}\,\,\bigg{(}\,\frac{|\ddot{\mathbf{r}}|}{v_{\rm
Gas}\zeta}\,\bigg{)}\,.$ (1)

In Equation 1, the total mass is given by $M_{\rm Tot}=\rho_{\rm Bulk}V$,
where, $\rho_{\rm Bulk}$ and $V$ are the bulk density and the volume of the
nucleus. $\ddot{\mathbf{r}}$ is the instantaneous acceleration at the
heliocentric position. $m_{\rm X}$ is the mass of the outgassing molecule and
$v_{\rm Gas}$ is the gas outflow velocity. $v_{\rm Gas}$ can be related to the
temperature of the outflowing species by equating the kinetic energy to the
thermal energy, which is calculated using:

$v_{\rm Gas}=\,\bigg{(}\,\frac{8k_{B}T_{\mathrm{Gas}}}{\pi
m_{\mathrm{X}}}\,\bigg{)}^{1/2}\,.$ (2)

In Equation 2, $T_{\mathrm{Gas}}$ is the temperature of the outgassing
species. In Equation 1, the factor $\zeta$ parameterizes the geometry of the
outgassing species. $\zeta$ = 1 corresponds to a fully collimated outflow and
$\zeta$ = 0.5 corresponds to an entirely isotropic hemispherical outflow. For
a complete definition of the parametric $\zeta$ function, see Section 4.1 in
Jewitt & Seligman (2022).

With this construction, we calculate the production rate required to produce
the measured nongravitational acceleration under the assumption of a pure H2O
outgassing. We show the resulting production rates for a range of outgassing
temperature and heliocentric distance in Figure 9. In order to calculate this
production, we assume that the outflow is completely collimated and $\zeta=1$,
a spherical nucleus with radius of $R_{\rm Nuc}=210$ m, a bulk density of
$\rho_{\rm Bulk}=1.9\ {\rm g\,cm^{-3}}$ and the best fit nongravitational
acceleration $A_{2}=213.7\times 10^{-14}$ au/d2 for a $(r/1\text{ au})^{-2}$
dependency on heliocentric distance.

Figure 9: The H2O production rates required to produce the measured
nongravitational acceleration of 2003 RM. We calculate this value using
Equations 1 and 2, with $\zeta=1$, radius of $210$ meters, $\rho_{\rm
Bulk}=1.9$ g cm-3 and $A_{2}=213.7\times 10^{-14}$ au/d2 for a $(r/1\text{
au})^{-2}$ dependency for heliocentric distance $r$.

This production rate calculation can easily be generalized to other
assumptions of the outflow and its composition. The following scaled equation:

$\begin{split}Q(X)\simeq 2.4\times
10^{23}{\rm\,molec\,s^{-1}}\,\bigg{(}\,\frac{\rho_{\rm
Bulk}}{1.9{\rm\,g\,cm^{-3}}}\,\bigg{)}\,\\\ \,\bigg{(}\,\frac{R_{\rm
Nuc}}{210{\rm m}}\,\bigg{)}^{3}\,\,\bigg{(}\,\frac{m_{\rm H_{2}O}}{m_{\rm
X}}\,\bigg{)}^{1/2}\,\,\bigg{(}\,\frac{1}{\zeta}\,\bigg{)}\,\,\bigg{(}\,\frac{100\,{\rm
K}}{T_{\rm Gas}}\,\bigg{)}^{1/2}\,\\\ \,\bigg{(}\frac{|A_{2}|}{213.7\times
10^{-14}{\rm\,au\,d^{-2}}}\,\bigg{)}\,\,\bigg{(}\,\frac{1{\rm\,au}}{r}\,\bigg{)}^{2}\,\end{split}$
(3)

may be used to estimate production rates for a variety of assumptions.

##  7 Dynamical history

2003 RM is currently located adjacent to the Eos family in terms of its
semimajor axis and inclination. Therefore, the most likely source region for
the object is in the mid-to-outer part of the main asteroid belt or among
Jupiter-family comets. We used the medium resolution version of the
evolutionary Near-Earth Object (NEO) population model by Granvik et al. (2018)
to quantify our assessment of the source region. The NEO population model
allows us to estimate the relative likelihood that 2003 RM would have entered
the NEO region through one of the seven escape regions described by that
model. The input for the model assessment are 2003 RM’s orbital elements ($a$,
$e$, $i$) and absolute magnitude ($H$). According to this model, 2003 RM has
an $(86\pm 4)$% and $(11\pm 4)$% probability of originating in the mid-to-
outer asteroid belt and Jupiter-family comet region, respectively. An origin
in the inner part of the asteroid belt, including the Hungaria or Phocaea
groups, is less likely, with a probability of less than 4% in total. Thus, it
appears that an asteroidal origin is more likely than a cometary origin for
2003 RM.

We used the model developed by Toliou et al. (2021) to assess 2003 RM’s
orbital history and lower perihelion distance, $q$. Based on the orbital
elements and absolute magnitude, the model predicts that 2003 RM attained
$q<1\,$au and $q<0.5\,$au with probability of $(42\pm 2)$% and $(5\pm 2)$%,
respectively. Therefore, it is likely that 2003 RM has never experienced
heating beyond that which it would have been exposed to at about 0.5 au.

How much solar heating and volatile release are likely to have occurred in
2003 RM’s past? The average lifetime of NEOs originating in the mid-to-outer
asteroid belt is 300–400 kyr. Approximately 80% of these NEOs will eventually
be ejected from the inner Solar System as a result of a close encounter with
Jupiter (Granvik et al., 2018). The median duration of the time that an
object—currently with a 2003 RM-like orbit— spends on orbits with $q<0.5\,$
au, $q<1\,$ au, and $q<1.3\,$ au is about 40, 100, and 190 kyr, respectively
(Toliou et al., 2021). Because only a small fraction of the total orbital
period occurs close to perihelion, these median times should be reduced by
orders of magnitude to estimate the time spent at heliocentric distances
$r<0.5\,$au, $r<1\,$ au, and $r<1.3\,$ au, respectively. If 2003 RM initially
contained volatiles, it is possible that the object retained some fraction of
them while being an NEA based on these timescales. Moreover, it is believed
that main-belt comets that are typically found in the mid-to-outer asteroid
belt initially contained volatiles. Therefore, the excess nongravitational
acceleration measured for 2003 RM could be caused by weak activity. This is
similar to what might have occurred on 1I/‘Oumuamua.

##  8 Conclusions

Based on an observation arc from 2003 to 2018, it is clear that the motion of
near-Earth asteroid 2003 RM is affected by significant nongravitational
perturbations in the transverse direction. Although transverse
nongravitational accelerations on asteroids are typically caused by the
Yarkovsky effect, the magnitude of the required anomalous acceleration is much
greater than can be explained by this phenomenon. We investigated and ruled
out alternative explanations for the observed orbital deviations such as close
approaches or a collision with another asteroid. Therefore, we conclude that
the most likely source of this nongravitational perturbation seems to be some
form of cometary outgassing, orders of magnitude smaller than what is
typically observed in comets, and that 2003 RM could be a cometary object.
(See Extended Data Fig. 1 from Micheli et al. 2018 and Fig. 4 from Farnocchia
et al. 2014a.)

However, direct imaging does not reveal any indication of cometary activity
and even a statistical analysis of the dynamical history of 2003 RM favors an
asteroidal origin. If 2003 RM is actively sublimating, it is curious that the
object does not display a bright cometary tail. We performed a detailed search
through observational data, especially from Pan-STARRs and the VLT, and we
found no evidence for extended coma or brightening events in the secular light
curve. Therefore, if this object is sublimating, the dust coma is very weak or
non-existent.

We estimated the required levels of H2O outgassing that would be required to
produce the nongravitational perturbations. By invoking the conservation of
linear momentum, we estimated that 2003 RM requires production rates of order
$Q(\mathrm{H}_{2}\mathrm{O})\sim 10^{22}-10^{23}$ molec s-1. We also compared
the photometric lightcurve of 2003 RM with a model of (i) a bare nucleus and
(ii) a nucleus with the addition of scattered light from a dust coma contained
within the unresolved seeing disk. We found that the scatter in the data could
be explained by an inferred outgassing rate of
$Q(\mathrm{H}_{2}\mathrm{O})\sim 10^{24}$ molec s-1, a limit just above the
amount of gas production required to produce the nongravitational
perturbations. Therefore, it is possible but not definitive that 2003 RM could
be exhibiting a low level of activity.

With the information currently available, the nature of 2003 RM remains a
puzzle and it is not clear if the object should be considered as an asteroid
or a comet, if not as a new class of small body in the Solar System. 2003 RM
is not alone in this regard. In an accompanying paper (Seligman et al.
submitted), we report the discovery of similar nongravitational accelerations
in six other small objects in the Solar System. Even more strikingly, the
first discovered interstellar object 1I/‘Oumuamua also had a significant
nongravitational acceleration but no coma.

Additional observations of 2003 RM are necessary to understand the origin of
the detected large nongravitational perturbations, which would in turn help
solve the similar puzzle on ‘Oumuamua’s nature. Specifically, high signal-to-
noise ratio space-based near- and mid-infrared observations could be sensitive
to detection H2O, CO2 and CO outgassing activity even in the absence of micron
sized dust particles. Observations with the James Webb Space Telescope would
be particulary helpful for identifying the source of the acceleration of 2003
RM.

##  9 Acknowledgments

Part of this work was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration (80NM0018D0004). DZS acknowledges financial support from
the National Science Foundation Grant No. AST-17152, NASA Grant No.
80NSSC19K0444 and NASA Contract NNX17AL71A from the NASA Goddard Spaceflight
Center. KJM, JVK and JTK acknowledge support through and award from the
National Aeronautics and Space Administration (NASA) under grant
NASA-80NSSC18K0853. This research has made use of data and/or services
provided by the International Astronomical Union’s Minor Planet Center. This
research has made use of observations collected at the European Southern
Observatory under ESO programme 2101.C-5049(A). Pan-STARRS is supported by the
National Aeronautics and Space Administration under Grant No. 80NSSC18K0971
issued through the SSO Near Earth Object Observations Program.

Copyright 2022. California Institute of Technology.

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