11institutetext: LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen,
92195 Meudon, France
22institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, UK
33institutetext: LGL-TPE, UMR 5276, CNRS, Claude Bernard Lyon 1 University,
ENS Lyon, Villeurbanne Cedex, France
44institutetext: School of Physics, National University of Ireland Galway,
University Road, Galway, Ireland
55institutetext: Space Science Institute, 4765 Walnut St, Suite B, Boulder, CO
80301, USA
66institutetext: LERMA, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, Univ. Paris Diderot,
Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
77institutetext: Southwest Research Institute, San Antonio, TX 78238, United
States 88institutetext: The University of Texas at San Antonio, San Antonio TX
78249, United States

# A molecular wind blows out of the Kuiper belt

Quentin Kral E-mail<EMAIL_ADDRESS>J. E. Pringle 22 Aurélie
Guilbert-Lepoutre 33 Luca Matrà 44 Julianne I. Moses 55 Emmanuel Lellouch 11
Mark C. Wyatt 22 Nicolas Biver 11 Dominique Bockelée-Morvan 11 Amy Bonsor 22
Franck Le Petit 66 G. Randall Gladstone 7788

(Received September 15, 1932; accepted March 16, 1937)

###### Abstract

Context. Gas has been detected in many exoplanetary systems ($>$10 Myr),
thought to be released in the destruction of volatile-rich planetesimals
orbiting in exo-Kuiper belts.

Aims. In this letter, we aim to explore whether gas is also expected in the
Kuiper belt (KB) in our Solar System.

Methods. To quantify the gas release in our Solar System, we use models for
gas release that have been applied to extrasolar planetary systems, as well as
a physical model that accounts for gas released due to the progressive
internal warming of large planetesimals.

Results. We find that only bodies larger than about 4 km can still contain CO
ice after 4.6 Gyr of evolution. This finding may provide a clue as to why
Jupiter-family comets, thought to originate in the Kuiper belt, are deficient
in CO compared to Oort-clouds comets. We predict that gas is still produced in
the KB right now at a rate of $2\times 10^{-8}$ M⊕/Myr for CO and orders of
magnitude more when the Sun was younger. Once released, the gas is quickly
pushed out by the Solar wind. Therefore, we predict a gas wind in our Solar
System starting at the KB location and extending far beyond with regards to
the heliosphere with a current total CO mass of $\sim 2\times 10^{-12}$ M⊕
(i.e. 20 times the CO quantity that was lost by the Hale-Bopp comet during its
1997 passage) and CO density in the belt of $3\times 10^{-7}$ cm-3. We also
predict the existence of a slightly more massive atomic gas wind made of
carbon and oxygen (neutral and ionized) with a mass of $\sim 10^{-11}$ M⊕.

Conclusions. We predict that gas is currently present in our Solar System
beyond the Kuiper belt and that although it cannot be detected with current
instrumentation, it could be observed in the future with an in situ mission
using an instrument similar to Alice on New Horizons with larger detectors.
Our model of gas release due to slow heating may also work for exoplanetary
systems and provide the first real physical mechanism for the gas
observations. Lastly, our model shows that the amount of gas in the young
Solar System should have been orders of magnitude greater and that it may have
played an important role for, e.g., planetary atmosphere formation.

###### Key Words.:

Kuiper belt: general – circumstellar matter – Planetary Systems – Solar wind –
Sun: Heliosphere – interplanetary medium

##  1 Introduction

The past decade was very prolific in terms of detecting gas (mostly CO, C and
O) around main sequence stars, therefore changing the paradigm of evolved
planetary systems that were thought to be devoid of gas after 10 Myr. Indeed,
most bright exoplanetesimal belts show the presence of gas, as demonstrated
recently with ALMA (Moór et al., 2017), and it could be that all these belts
have gas at some level (even if undetectable with current instruments). These
belts, similar to our Kuiper belt, are made of large bodies colliding with
each other and creating dust that can then be observed around extrasolar stars
through its emission in the infrared above that of the star, which can be
resolved at high resolution (showing, e.g., gaps and asymmetries that may be
related to the presence of planets).

Recent models show that the best explanation for the CO gas observed co-
located with exo-Kuiper belts is a secondary production (i.e., the gas is not
a remnant of the young planet-forming disks that persist for less than 10
Myr), where CO is released from planetesimals at a rate proportional to their
collisional frequency (Kral et al., 2016, 2017). Observations of carbon and
oxygen atoms are explained as the daughter species of CO photodissociation
within the framework of this model. The most massive gas disks with CO masses
close to the amount observed in younger planet-forming disks have first been
called hybrid disks (e.g., Kóspál et al., 2013, because the gas was thought to
be primordial but the dust secondary). We can now also explain the gas in
these previously considered hybrid disks as entirely secondary, because CO
released from planetesimals becomes shielded by the carbon produced when it
photodissociates (and by CO itself through self-shielding), which can then
accumulate to large amounts (Kral et al., 2019; Marino et al., 2020).

In addition, recent observations showed that comets in our Solar System start
being active as far as the Kuiper belt distance. The long period comet (3 Myr)
C/2017 K2 (PANSTARRS) exhibited activity as far as 9-16 au, and models show
that dust production (presumably driven by sublimation of CO) needs to have
started at the KB (35 au) to explain the photometric data (Jewitt et al.,
2021). Historically, there are also other comets showing distant CO outgassing
such as C/1995 O1 (Hale-Bopp, Biver et al., 2002), or the short-period comets
(Centaur-like) 29P/Schwassmann-Wachmann 1 and 2060 Chiron (Womack et al.,
2017).

Given all this new knowledge in terms of CO outgassing in comets and
exocomets, we want to explore what it means for our own Solar System. For
instance, what gas production rate do we predict for the current Kuiper belt?
Would sublimation still be active enough in releasing CO as far as the KB
distance that it can be detected? Is the dynamics of the released gas around
our G2-type Sun similar to that previously observed (predominantly though not
exclusively) around young main-sequence stars? If gas is released in the KB,
how can we detect it and how does it affect the system as a whole? These are
the questions we are tackling in this paper.

##  2 Results

To answer these questions, we first extrapolate the gas production rate in our
KB from the most recent extrasolar models. To do so, we compute the dust mass
loss rate in the KB due to collisions using a state-of-the-art model of dust
in our Solar System (Vitense et al., 2012; Morbidelli et al., 2021). Indeed,
according to extrasolar models that fit most observations to date (Kral et
al., 2017), the gas production rate is proportional to the mass loss rate of
the belt’s collisional cascade, and we find (see Appendix B) that $\sim
10^{-9}$ M⊕/Myr of CO gas should be released in the current KB. The model’s
idea (which fits extrasolar observations) is that large planetesimals are
composed of $\sim$10% of CO (see Kral et al., 2017) that is released along
with collisions that produce the observed dust (but the detailed physical
mechanism is not constrained), either at the top (large km bodies) or further
down the collisional cascade, but before solid bodies are ground down to dust
and expelled by radiation pressure. We also use a more direct approach relying
on the counting rate of the New Horizons dust counter to determine the dust
production rate (rather than a numerical model) and arrive at the same value
for the gas production rate. We also test a different more physically
motivated model for releasing the CO and assume it comes from the slow heating
provided by the Sun over long timescales, which warms up large bodies at
greater depths as time goes by, and releases the CO in these increasingly
deeper layers. We find that after 4.6 Gyr of evolution, only bodies larger
than about 4 km can still contain CO (smaller bodies would have lost it
already), and all together they release CO at a rate of $\sim 2\times 10^{-8}$
M⊕/Myr. In this model, a single 30 km radius planetesimal would release around
$10^{-14}$ M⊕/Myr, i.e., it is much lower than what can be detected with
missions targeting specific KBOs (e.g., Lisse et al., 2021). Fig. 1 shows the
temporal evolution of the release rate, which goes down with time as only
larger and larger bodies can participate as time goes by (see Appendix A). We
note that this means that sampling the material in the KB now would not lead
to the primordial volatile composition of planetesimals. We also test this
slow stellar-driven heating model on more massive belts (similar to those
observed around extrasolar systems) and show that it provides the right order
of magnitude to explain CO observed around younger exo-systems, which may
provide the first physical explanation for their ubiquitous CO presence.

Figure 1:  CO gas production rate in the Kuiper belt $\dot{M}_{\rm CO}$ as a
function of time ($t=0$ is when the gas release starts, i.e., probably after a
few Myr to 10 Myr and the end of the lines on the right is today) predicted by
our sublimation model (see Appendix A). The solid line is for the Kuiper belt
assuming it starts with a low mass similar to the current KB mass and the
dashed line is for a more massive belt similar to the archetype $\beta$ Pic
belt.

Comets show a diversity in composition, with a factor of 10-100 variability in
the volatile abundance (Bockelée-Morvan & Biver, 2017). However, in general
this diversity does not appear correlated with the dynamical category. An
exception is for CO, whose abundance relative to water appears depleted in
Jupiter family comets (JFC), by a factor $\sim$4 in average compared to Oort
Cloud comets (Dello Russo et al., 2016). As we show in Fig. 2, the CO
depletion in JFC compared to other comets is also visible if expressed as a
specific CO production rate, i.e., the production rate per unit area $Q_{\rm
CO}/(\pi D^{2}/4)$, where $D$ is the equivalent diameter. The top panel of
Fig. 2 shows this specific production rate, multiplied by $r_{h}^{2}$, as a
function of the heliocentric distance $r_{h}$ of the measurements. As
demonstrated for C/1996 B2 Hyakutake and C/1995 O1 Hale-Bopp (96B2 and H.B. in
the figure), the scaling by $r_{h}^{2}$ corrects to first order for the
distance effects and allows comparison of measurements at different distances.
In the bottom panel of Fig. 2, this distance-corrected specific CO production
rate is shown as a function of $D$. In both panels of Fig. 2, JFCs clearly
appear CO-depleted with respect to Oort-cloud comets (OCCs). As JFCs are
thought to originate from the transneptunian region, in particular the
Scattered Disk (e.g., Duncan, Levison, & Dones, 2004; Weissman et al., 2020)
and most of them have diameters $<$ 5 km, our calculation that only Kuiper
belt bodies larger than 4 km can retain CO over the age of the Solar System
may provide a natural explanation to this behaviour. Interestingly, the
observed cumulative size distribution of JFC may show an excess of comets with
radii 3-6 km (Fernández et al., 2013), similar to the above number, and that
could account for the diversity of CO abundances within JFCs, although
statistics are not sufficient to discern a $Q_{\rm CO}$ vs $D$ trend within
the JFC group. While Fig. 2 is reassuringly consistent with our sublimation
calculations for the Kuiper Belt, we note the following two caveats: (i) the
lack of $>$ 5 km JFCs does not allow us to check our prediction that those
objects would be less volatile-depleted (ii) the low CO production rate of
JFCs may also be related to their repeated perihelion passages on their
current orbits. We note finally that, with the notable exception of
29P/Schwassmann– Wachmann 1, Centaurs, which are dynamically associated with
the Scattered Disk and JFC, also appear CO-depleted compared to Oort-cloud
comets (e.g. 10-50 times less CO production per unit surface for Chiron and
Echeclus compared to Hale-Bopp; Wierzchos, Womack, & Sarid, 2017) in spite of
their large $\sim$100 km size. This is a probable consequence of increased
outgassing over their $10^{6}$-$10^{7}$ years lifetime orbits at giant planet
heliocentric distances.

Figure 2:  CO production rate per km2 for Oort cloud comets (OCCs, orange),
Halley family comets (HFCs, blue), Jupiter family comets (JFCs, red) and
Centaurs (purple) of diameter $D$ scaled by the heliocentric distance $r_{h}$
squared - or in other terms $(Q_{\rm CO}/D^{2})r_{h}^{2}$ \- as a function of
$r_{h}$ (top) and $D$ (bottom). Downwards triangles are for upper limits. H.B.
stands for Hale-Bopp, 20F3 for C/2020 F3 Neowise, 96B2 for C/1996 B2 Hyakutake
and other objects have their full names. The data are listed in Table 2.

Once CO is released, we find that its dynamics is different from that modelled
in extrasolar systems so far (see Appendix E), i.e., the gas does not evolve
viscously inwards as expected in massive disks (Kral et al., 2016). It is due
to two reasons. First the gas quantity we find in the KB is very small and not
in the fluid regime in contrast to systems detected up to now. Second, the
majority of gas detections has been around A-type stars (it is mainly an
observational bias because more gas is expected in these systems according to
models, Matrà et al., 2019), where stellar winds are not important (only stars
cooler than about F5 possess significant convective envelopes and then
magnetic fields that can produce strong stellar winds). In contrast, in the
Solar System, the Solar wind (SW) drives the dynamics of the gas. We find that
once released, CO gets pushed outwards by SW protons on timescales of a few
years (at a rate between $\sim 3$ to 10 au/yr depending on the location, see
Appendix F). Some of this CO gets dissociated and ionized (when interacting
with SW protons, photons from the Sun and/or the interstellar medium, see
Appendix F) on its way out. However, the ionization and dissociation
timescales are of order 100 yr so that CO remains the dominant species up to
$\sim 500$ au (see Appendix D), i.e. well beyond the heliopause (which is the
boundary of the heliosphere where the Solar wind is stopped by the interaction
with the local interstellar medium) at $\sim$ 150 au (Opher et al., 2020). The
daughter products of the CO dissociation, namely C and O, are ionized in
$\sim$ 100 yr, leading to an ionized atomic component beyond $\sim$ 500 au.
These ions will then follow the interstellar magnetic lines and get ejected
further in the interstellar medium (see Appendix G). The model predictions for
CO, C, O (neutral and ionized) as a function of distance to the Sun are shown
in Fig. 3 and a summary of the model is given in Appendix C.

Figure 3:  Results of our model for the number density of CO (solid), C and O
(dashed), C+ and O+ (dotted) and CO+ (dash-dotted) as a function of distance
in our Solar System. The exact shape of the radial profile depends on the
heliopause location and geometry, which is not fully modelled in this paper.
The assumptions behind this plot are described in the appendix F. Note that
the neutral and ionized oxygen number densities are roughly superimposed on
those of carbon.

Our model leads to a gas wind with a total CO mass (up to 2000 au) of $\sim
2\times 10^{-12}$ M⊕ (i.e., 20 times the CO quantity that was lost by the
Hale-Bopp comet during its 1997 passage) and an atomic wind of $\sim 10^{-11}$
M⊕ as summarized in Table 1 (and see Appendix H). The CO density in the belt
is predicted to be $3\times 10^{-7}$ cm-3 and the column densities along the
midplane are of order $\sim 10^{8}$ cm-2 for CO, C, and O. We note that our
predictions for O may increase by a factor a few if planetesimals in the KB
routinely include O2 ices in quantity similar to CO (as may be expected from
recent in situ observations of the comet 67 P/Churyumov–Gerasimenko, Bieler et
al., 2015) as they are even more volatile than CO.

Table 1: Results from our gas release model for the total masses, number densities in the belt and column densities along the line of sight to a star. Species | value
---|---
Total Masses in M⊕
CO | $2\times 10^{-12}$
CO+ | $2\times 10^{-14}$
C0 | $10^{-12}$
C+ | $6\times 10^{-12}$
O0 | $10^{-12}$
O+ | $8\times 10^{-12}$
number densities in cm-3 at the center of the belt (45 au)
CO | $3\times 10^{-7}$
CO+ | $3\times 10^{-10}$
C0 | $4\times 10^{-9}$
C+ | $5\times 10^{-12}$
O0 | $4\times 10^{-9}$
O+ | $5\times 10^{-12}$
column densities along the midplane of the belt in cm-2
CO | $2\times 10^{8}$
CO+ | $7\times 10^{5}$
C0 | $2\times 10^{7}$
O0 | $2\times 10^{7}$
C+ | $10^{7}$
O+ | $10^{7}$

In our Solar System, some cometary models predict that planetesimals should
still be outgassing in the KB (see Appendix B) at a low rate (Jewitt et al.,
2008) and it was recently validated through observations (Jewitt et al.,
2021). Upper limits in CO from sub-mm studies targeting specific KBOs show
that ALMA can detect CO outgassing rates of $2\times 10^{-8}$ M⊕/Myr for a
specific comet at KB distances (Jewitt et al., 2008). However, in the case of
the KB, the release is more diffuse as the emission comes from many KBOs and
it would be difficult to observe because of the lack of spatial contrast
compared to extrasolar systems where most emission comes from a few beams. We
find that Planck and ALMA (in a total power array mode) do not have enough
sensitivity to detect the CO rotational lines of the diffuse wind (see
Appendix J). The gas accumulated in the midplane of the KB along the line of
sight to a background star would create some absorption in the UV on the star
spectrum that could be identified as gas in our Solar System. However, we find
that only future instruments may be able to detect this faint absorption. The
most promising technique would be to use in-situ missions similar to New
Horizons to detect emission of resonance line scattering of carbon and/or
oxygen excited by the Sun’s UV light (see Appendix J). We find that a super-
Alice instrument similar to the current Alice UV probe on New Horizons (Stern
et al., 2008) but with a larger effective area would reach the low column
density level predicted for atoms in the KB. Super-Alice could be built with
current technology.

As detailed in Appendix I, we also explored the CO release from Centaurs and
find that their current CO mass loss rate for bodies larger than 4 km is of
the same order of magnitude as that predicted by our model for the KB.
However, Centaurs being closer to the Sun, CO would be blown out by the SW
much faster than in the KB, thus reducing the total CO mass or column that
could be observed. We note that the spatial and velocity distributions of this
gas is very different (much closer in and faster) to that of gas released in
the KB, which could allow future observations to distinguish both components.

The presence of current gas predicted by our model in our Solar System would
not impact the dynamics of bodies (dust or planetesimals) evolving around the
KB. However, we note that in the past, when the Kuiper belt was much younger
and heavier, the release of CO would have been orders of magnitude larger
(above the solid line in Fig. 1) and the gas dynamics would have also been
much different (e.g., in the fluid/hydrodynamic regime), potentially leading
to some interesting effects, such as delivering some CO mass from the KB to
planetary atmospheres as proposed recently for extrasolar systems (Kral et
al., 2020). Indeed, in more massive disks, gas becomes optically thick to the
SW and it does not get pushed outwards. Instead, gas drifts inwards because of
viscous evolution and it can end up accreted onto planets (see Appendix E).
The initial KB may have been much more massive before potential dynamical
instabilities (e.g., Gomes et al., 2005) and could have led to CO outgassing
rate close to the dashed line in Fig. 1, hence providing CO that falls onto
the young planets in greater quantity than other potential sources such as
impacts (see comparison between different CO sources in Kral et al., 2020).
This is a whole new study that emerges naturally from this work and will be
tackled in a different paper.

##  3 Conclusions

We predict the existence of a gas wind in our Solar System starting at the
Kuiper belt and extending farther out. Our model shows that large kilometre-
sized planetesimals can still lose volatiles after billions of years of
evolution due to the slow heating from the Sun, which warms bodies up at
greater depths as time goes by. This finding may provide a clue as to why
Jupiter-family comets, thought to originate in the Kuiper belt, are deficient
in CO compared to Oort-clouds comets. The released CO gas in the Kuiper belt
is constantly produced and then pushed away by the Solar wind, establishing a
quasi steady-state CO disk close to the belt with a calculated current total
CO mass of $\sim 2\times 10^{-12}$ M⊕ (i.e., 20 times the CO quantity that was
lost by the Hale-Bopp comet during its 1997 passage). We predict a CO density
in the belt of $3\times 10^{-7}$ cm-3, as well as, the existence of a slightly
more massive atomic gas wind made of carbon and oxygen (neutral and ionized)
with a mass of $\sim 10^{-11}$ M⊕. This gas cannot be observed with current
instrumentation but could be observed with future in-situ missions (e.g., a UV
instrument similar to Alice/New Horizons but with a larger detector), and may
have played an important role for, e.g., planetary atmosphere formation in the
Solar System youth when the gas release rate was much higher, i.e., when the
Sun was a few tens of Myr old. Lastly, we show that our new model of gas
release due to slow heating of planetesimals by stellar radiation is promising
to explain gas detected in exoplanetary system, which would provide the first
real physical mechanism for the origin of the gas.

###### Acknowledgements.

This paper is dedicated to Inaya. We thank the referee for a prompt and
helpful review. QK thanks Rosine Lallement and Jean-Loup Bertaux for
discussions about the Solar Wind, Heliopause, and Local ISM. QK thanks
François Lévrier and Clément Walter for providing information about Planck. QK
thanks Andrew Shannon for discussions about latest Kuiper belt collisional
models.

## References

  * Axford et al. (1963) Axford, W. I., Dessler, A. J., Gottlieb, B. 1963, ApJ, 137, 1268
  * Balbus & Hawley (1991) Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214
  * Beust et al. (1989) Beust H., Lagrange-Henri A. M., Vidal-Madjar A., Ferlet R., 1989, A&A, 223, 304
  * Bieler et al. (2015) Bieler A., Altwegg, K., Balsiger, H., Bar-Nun,A ., Berthelier, J. -J, Bochsler, P., Briois, C. et al., 2015, Natur, 526, 678
  * Biver et al. (1999) Biver N., Bockelée-Morvan D., Crovisier J., Davies J. K., Matthews H. E., Wink J. E., Rauer H., et al., 1999, AJ, 118, 1850. doi:10.1086/301033
  * Biver et al. (2002) Biver N., Bockelée-Morvan D., Colom P., Crovisier J., Henry F., Lellouch E., Winnberg A., et al., 2002, EM&P, 90, 5
  * Biver et al. (2012) Biver N., Crovisier J., Bockelée-Morvan D., Szutowicz S., Lis D. C., Hartogh P., de Val-Borro M., et al., 2012, A&A, 539, A68. doi:10.1051/0004-6361/201118447
  * Biver et al. (2019) Biver N., Bockelée-Morvan D., Hofstadter M., Lellouch E., Choukroun M., Gulkis S., Crovisier J., et al., 2019, A&A, 630, A19. doi:10.1051/0004-6361/201834960
  * Biver et al. (2021) Biver N., Bockelée-Morvan D., Lis D. C., Despois D., Moreno R., Crovisier J., Colom P., et al., 2021, A&A, 651, A25. doi:10.1051/0004-6361/202140765
  * Bockelée-Morvan et al. (2001) Bockelée-Morvan D., Lellouch E., Biver N., Paubert G., Bauer J., Colom P., Lis D. C., 2001, A&A, 377, 343. doi:10.1051/0004-6361:20011040
  * Bockelée-Morvan et al. (2004a) Bockelée-Morvan D., Biver N., Colom P., Crovisier J., Henry F., Lecacheux A., Davies J. K., et al., 2004, Icar, 167, 113. doi:10.1016/j.icarus.2003.04.001
  * Bockelée-Morvan et al. (2004b) Bockelée-Morvan D., Crovisier J., Mumma M. J., Weaver H. A., 2004, come.book, 391
  * Bockelée-Morvan & Biver (2017) Bockelée-Morvan D., Biver N., 2017, RSPTA, 375, 20160252. doi:10.1098/rsta.2016.0252
  * Brown et al. (1979) Brown, G.N.; Ziegler, W.T., 1979, J. Chem. Eng. Data, 24, 319-30
  * Cordiner et al. (2020) Cordiner M. A., Milam S. N., Biver N., Bockelée-Morvan D., Roth N. X., Bergin E. A., Jehin E., et al., 2020, NatAs, 4, 861
  * Crovisier et al. (1995) Crovisier J., Biver N., Bockelee-Morvan D., Colom P., Jorda L., Lellouch E., Paubert G., et al., 1995, Icar, 115, 213. doi:10.1006/icar.1995.1091
  * Davis et al. (2011) Davis M. W., Gladstone G. R., Greathouse T. K., Slater D. C., Versteeg M. H., Persson K. B., Winters G. S., et al., 2011, SPIE, 8146, 814604. doi:10.1117/12.894274
  * Dello Russo et al. (2016) Dello Russo N., Kawakita H., Vervack R. J., Weaver H. A., 2016, Icar, 278, 301. doi:10.1016/j.icarus.2016.05.039
  * Desch & Jackson (2021) Desch S. J., Jackson A. P., 2021, JGRE, 126, e06807. doi:10.1029/2020JE006807
  * Dialynas et al. (2017) Dialynas K., Krimigis S. M., Mitchell D. G., Decker R. B., Roelof E. C., 2017, NatAs, 1, 0115
  * DiSanti et al. (2017) DiSanti M. A., Bonev B. P., Russo N. D., Vervack R. J., Gibb E. L., Roth N. X., McKay A. J., et al., 2017, AJ, 154, 246. doi:10.3847/1538-3881/aa8639
  * Duncan, Levison, & Dones (2004) Duncan M., Levison H., Dones L., 2004, come.book, 193
  * Feldman et al. (1997) Feldman P. D., Festou M. C., Tozzi P., Weaver H. A., 1997, ApJ, 475, 829. doi:10.1086/303553
  * Fernández et al. (2013) Fernández Y. R., Kelley M. S., Lamy P. L., Toth I., Groussin O., Lisse C. M., A’Hearn M. F., et al., 2013, Icar, 226, 1138. doi:10.1016/j.icarus.2013.07.021
  * Ferrari & Lucas (2016) Ferrari C., Lucas A., 2016, A&A, 588, A133. doi:10.1051/0004-6361/201527625
  * Gladstone et al. (2016) Gladstone G. R., Stern S. A., Ennico K., Olkin C. B., Weaver H. A., Young L. A., Summers M. E., et al., 2016, Sci, 351, aad8866
  * Gladstone et al. (2021) Gladstone G. R., Kammer J. A., Adams D. J., Yung Y. L., Pryor W. R., Strobel D. F., Young L. A., et al., 2021, Icar, 356, 113973
  * Goldsmith & Langer (1999) Goldsmith P. F., Langer W. D., 1999, ApJ, 517, 209
  * Gomes et al. (2005) Gomes R., Levison H. F., Tsiganis K., Morbidelli A., 2005, Natur, 435, 466. doi:10.1038/nature03676
  * Gomes et al. (2008) Gomes R. S., Fernández J. A., Gallardo T., Brunini, A., 2008, ssbn.book, 259
  * Greathouse et al. (2013) Greathouse T. K., Gladstone G. R., Davis M. W., Slater D. C., Versteeg M. H., Persson K. B., Walther B. C., et al., 2013, SPIE, 8859, 88590T. doi:10.1117/12.2024537
  * Groussin et al. (2019) Groussin O., Attree N., Brouet Y., Ciarletti V., Davidsson B., Filacchione G., Fischer H.-H., et al., 2019, SSRv, 215, 29. doi:10.1007/s11214-019-0594-x
  * Gueymard (2018) Gueymard C. A., 2018, SoEn, 169, 434
  * Guilbert-Lepoutre (2012) Guilbert-Lepoutre A., 2012, AJ, 144, 97. doi:10.1088/0004-6256/144/4/97
  * Heays et al. (2017) Heays A. N., Bosman A. D., van Dishoeck E. F., 2017, A&A, 602, A105
  * Horányi et al. (2008) Horányi M., Hoxie V., James D., Poppe A., Bryant C., Grogan B., Lamprecht B., et al., 2008, SSRv, 140, 387
  * Hosteaux et al. (2019) Hosteaux S., Chané E., Poedts S., 2019, A&A, 632, A89
  * Hu et al. (2019) Hu X., Gundlach B., von Borstel I., Blum J., Shi X., 2019, A&A, 630, A5. doi:10.1051/0004-6361/201834631
  * Huebner et al. (1992) Huebner W. F., Keady J. J., Lyon S. P., 1992, Ap&SS, 195, 1
  * Huebner et al. (2006) Huebner W. F., Benkhoff J., Capria M.-T., Coradini A., De Sanctis C., Orosei R., Prialnik D., 2006, hgdc.conf
  * Huebner et al. (2015) Huebner W. F., Mukherjee J., 2015, P&SS, 106, 11
  * Jewitt (2004) Jewitt D. C., 2004, come.book, 659
  * Jewitt et al. (2008) Jewitt D., Garland C. A., Aussel H., 2008, AJ, 135, 400
  * Jewitt et al. (2021) Jewitt D., Kim Y., Mutchler M., Agarwal J., Li J., Weaver H., 2021, arXiv, arXiv:2102.06313
  * Izmodenov et al. (1997) Izmodenov V., Malama Y. G., Lallement R., 1997, A&A, 317, 193
  * Izmodenov et al. (1999) Izmodenov V. V., Lallement R., Geiss J., 1999, A&A, 344, 317
  * Kóspál et al. (2013) Kóspál Á., Moór A., Juhász A., Ábrahám P., Apai D., Csengeri T., Grady C. A., et al., 2013, ApJ, 776, 77. doi:10.1088/0004-637X/776/2/77
  * Kouchi & Yamamoto (1995) Kouchi, A., Yamamoto, T., 1995, Prog, Crystal Growth and Charact. 30, 83
  * Kral et al. (2016) Kral, Q., Wyatt, M., Carswell, R. F., et al. 2016, MNRAS, 461, 845
  * Kral & Latter (2016) Kral Q., Latter H., 2016, MNRAS, 461, 1614
  * Kral et al. (2017) Kral, Q., Matrà, L., Wyatt, M. C., & Kennedy, G. M. 2017, MNRAS, 469, 521
  * Kral et al. (2019) Kral Q., Marino S., Wyatt M. C., Kama M., Matrà L., 2019, MNRAS, 489, 3670
  * Kral et al. (2020) Kral Q., Davoult J., Charnay B., 2020, NatAs, 4, 769
  * Krijt et al. (2018) Krijt S., Schwarz K. R., Bergin E. A., Ciesla F. J., 2018, ApJ, 864, 78
  * Krivov & Wyatt (2021) Krivov A. V., Wyatt M. C., 2021, MNRAS, 500, 718
  * Läuter et al. (2020) Läuter M., Kramer T., Rubin M., Altwegg K., 2020, MNRAS, 498, 3995. doi:10.1093/mnras/staa2643
  * Lellouch et al. (2017) Lellouch E., Moreno R., Müller T., Fornasier S., Santos-Sanz P., Moullet A., Gurwell M., et al., 2017, A&A, 608, A45. doi:10.1051/0004-6361/201731676
  * Lisse et al. (2021) Lisse C. M., Young L. A., Cruikshank D. P., Sandford S. A., Schmitt B., Stern S. A., Weaver H. A., et al., 2021, Icar, 356, 114072
  * López-Patiño et al. (2017) López-Patiño J., Fuentes B. E., Yousif F. B., Martínez H., 2017, PhPro, 90, 391
  * Luu & Jewitt (2002) Luu J. X., Jewitt D. C., 2002, ARA&A, 40, 63
  * Marino et al. (2020) Marino S., Flock M., Henning T., Kral Q., Matrà L., Wyatt M. C., 2020, MNRAS, 492, 4409
  * Masuoka & Samson (1980) Masuoka T., Samson J. A. R., 1980, JCP, 77, 623
  * Matrà et al. (2015) Matrà L., Panić O., Wyatt M. C., Dent W. R. F., 2015, MNRAS, 447, 3936
  * Matrà et al. (2017) Matrà L., Dent W. R. F., Wyatt M. C., Kral Q., Wilner D. J., Panić O., Hughes A. M., et al., 2017, MNRAS, 464, 1415
  * Matrà et al. (2018a) Matrà L., Wilner D. J., Öberg K. I., Andrews S. M., Loomis R. A., Wyatt M. C., Dent W. R. F., 2018, ApJ, 853, 147
  * Matrà et al. (2018b) Matrà L., Marino S., Kennedy G. M., Wyatt M. C., Öberg K. I., Wilner D. J., 2018, ApJ, 859, 72
  * Matrà et al. (2019) Matrà L., Öberg K. I., Wilner D. J., Olofsson J., Bayo A., 2019, AJ, 157, 117
  * McKay et al. (2021) McKay A. J., DiSanti M. A., Cochran A. L., Bonev B. P., Dello Russo N., Vervack R. J., Gibb E., et al., 2021, PSJ, 2, 21. doi:10.3847/PSJ/abd71d
  * Meier (1991) Meier R. R., 1991, SSRv, 58, 1
  * Meyer-Vernet et al. (1998) Meyer-Vernet, N., Issautier, K., 1998, Journal of Geophysical Research, 103, 29705
  * Moór et al. (2017) Moór A., Curé M., Kóspál Á., Ábrahám P., Csengeri T., Eiroa C., Gunawan D., et al., 2017, ApJ, 849, 123
  * Morbidelli et al. (2021) Morbidelli A., Nesvorny D., Bottke W. F., Marchi S., 2021, Icar, 356, 114256
  * Morgado et al. (2021) Morgado B. E., Sicardy B., Braga-Ribas F., Desmars J., Gomes-Júnior A. R., Bérard D., Leiva R., et al., 2021, A&A, 652, A141. doi:10.1051/0004-6361/202141543
  * Nahar (1999) Nahar S. N., 1999, ApJS, 120, 131
  * Nahar & Pradhan (1997) Nahar S. N., Pradhan A. K., 1997, ApJS, 111, 339
  * Nesvorný et al. (2019) Nesvorný D., Vokrouhlický D., Stern A. S., Davidsson B., Bannister M. T., Volk K., Chen Y.-T., et al., 2019, AJ, 158, 132. doi:10.3847/1538-3881/ab3651
  * Ootsubo et al. (2012) Ootsubo T., Kawakita H., Hamada S., Kobayashi H., Yamaguchi M., Usui F., Nakagawa T., et al., 2012, ApJ, 752, 15. doi:10.1088/0004-637X/752/1/15
  * Opher et al. (2020) Opher M., Loeb A., Drake J., Toth G., 2020, NatAs, 4, 675
  * Opitom et al. (2019) Opitom C., Hutsemékers D., Jehin E., Rousselot P., Pozuelos F. J., Manfroid J., Moulane Y., et al., 2019, A&A, 624, A64. doi:10.1051/0004-6361/201834357
  * Planck Collaboration et al. (2014) Planck Collaboration, Ade P. A. R., Aghanim N., Alves M. I. R., Armitage-Caplan C., Arnaud M., Ashdown M., et al., 2014, A&A, 571, A13
  * Poppe et al. (2019) Poppe A. R., Lisse C. M., Piquette M., Zemcov M., Horányi M., James D., Szalay J. R., et al., 2019, ApJL, 881, L12
  * Prialnik et al. (2004) Prialnik D., Benkhoff J., Podolak M., 2004, come.book, 359
  * Rauer et al. (1997) Rauer H., Biver N., Crovisier J., Bockelée-Morvan D., Colom P., Despois D., Ip W.-H., et al., 1997, P&SS, 45, 799,803. doi:10.1016/S0032-0633(97)00016-0
  * Richardson et al. (2019) Richardson J. D., Belcher J. W., Garcia-Galindo P., Burlaga L. F., 2019, NatAs, 3, 1019
  * Roth et al. (2018) Roth N. X., Gibb E. L., Bonev B. P., DiSanti M. A., Dello Russo N., Vervack R. J., McKay A. J., et al., 2018, AJ, 156, 251. doi:10.3847/1538-3881/aae0f7
  * Roth et al. (2020) Roth N. X., Gibb E. L., Bonev B. P., DiSanti M. A., Dello Russo N., McKay A. J., Vervack R. J., et al., 2020, AJ, 159, 42. doi:10.3847/1538-3881/ab536b
  * Rubin et al. (2009) Rubin, M., Hansen, K. C., Gombosi, T. I., Combi, M. R., Altwegg, K., Balsiger, H., 2009, Icarus, 199, 505
  * Rubin et al. (2015) Rubin M., Altwegg, K., Balsiger, H., Bar-Nun,A ., Berthelier, J. -J, Bieler, A., Bochsler, P., et al., 2015, Science, 348, 232
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 500, 33
  * Stern et al. (2008) Stern S. A., Slater D. C., Scherrer J., Stone J., Dirks G., Versteeg M., Davis M., et al., 2008, SSRv, 140, 155
  * Tiscareno & Malhotra (2003) Tiscareno M. S., Malhotra R., 2003, AJ, 126, 3122. doi:10.1086/379554
  * Vitense et al. (2012) Vitense C., Krivov A. V., Kobayashi H., Löhne T., 2012, A&A, 540, A30
  * Wakelam et al. (2012) Wakelam V., Herbst E., Loison J.-C., Smith I. W. M., Chandrasekaran V., Pavone B., Adams N. G., et al., 2012, ApJS, 199, 21
  * Weaver et al. (2011) Weaver H. A., Feldman P. D., A’Hearn M. F., Dello Russo N., Stern S. A., 2011, ApJL, 734, L5. doi:10.1088/2041-8205/734/1/L5
  * Weissman et al. (2020) Weissman P., Morbidelli A., Davidsson B., Blum J., 2020, SSRv, 216, 6. doi:10.1007/s11214-019-0625-7
  * Wierzchos, Womack, & Sarid (2017) Wierzchos K., Womack M., Sarid G., 2017, AJ, 153, 230. doi:10.3847/1538-3881/aa689c
  * Womack et al. (2017) Womack M., Sarid G., Wierzchos K., 2017, PASP, 129, 031001
  * Wyatt (2005) Wyatt M. C., 2005, A&A, 433, 1007
  * Wyatt et al. (2011) Wyatt M. C., Clarke C. J., Booth M., 2011, CeMDA, 111, 1
  * Yamamoto (1985) Yamamoto, T., A&A, 142, 31

##  Appendix A Sublimation calculations

There are three important timescales for gas release through sublimation.
First, the planetesimals need to heat up (via conduction due to the Solar
influx) to above the CO sublimation temperature of $\sim$ 25 K (Huebner et
al., 2006) on the thermal timescale; Second the transition from solid to gas
(sublimation) must happen on the sublimation timescale; and then the gas must
make its way up through the planetesimal pores to finally escape the body on
the gas diffusion timescale.

The thermal timescale to heat a layer of thickness $\Delta p$ is given by
$\tau_{\rm th}=(\Delta p)^{2}/K$, where $K=\kappa/(\rho c_{p})$ is the thermal
diffusivity (in m2/s), which we will assume to be of order $10^{-10}$ for
comet-like objects (Prialnik et al., 2004), with $\rho$ the planetesimal bulk
density in kg/m3, $c_{p}$ its specific heat in J/kg/K and $\kappa$ its thermal
conductivity in J/m/s/K. We note that the thermal diffusivity is smaller for
cometary material compared to solid amorphous water ice because cometary
material is a porous mixture of ices and refractories including organics. The
effects of porosity on the actual effective thermal conductivity (hence
diffusivity) are consequential (e.g., Ferrari & Lucas, 2016; Hu et al., 2019).
This value is consistent with current measurements of the thermal inertia at
the surface of comets (see Groussin et al., 2019, for a review). We calculate
that during the Solar System lifetime, i.e. $t_{S}=4.6$ Gyr, the depth to
which planetesimals can heat up to the equilibrium temperature of $\sim$40 K
(Krijt et al., 2018) is $\sqrt{t_{S}K}\sim 3.8$ km. After 4.6 Gyr, the layers
deeper than 4 km should still retain their primordial temperature of 10-20 K
(Huebner et al., 2006; Krijt et al., 2018) and planetesimals smaller than
about 4 km should have no further gas to release as any primordial CO would
have already been lost.

The sublimation timescale $\tau_{\rm sub}$ is given by $\rho/(SP_{\rm
CO}\sqrt{m_{\rm CO}/(2\pi k_{b}T)})$, where $S=3(1-\Psi)/r_{p}$ is the total
interstitial surface area of the pores (of radius $r_{p}$ of order 1 micron,
Prialnik et al., 2004) of the material per given bulk volume (with $\Psi$ the
porosity taken to be 0.6), $P_{\rm CO}=A_{\rm CO}\exp(-B_{\rm CO}/T)$ is the
CO saturated vapour pressure (with $A_{\rm CO}=0.12631$ in $10^{10}$ Pa, and
$B_{\rm CO}=764.16$ in K, Prialnik et al., 2004), and $m_{\rm CO}$ is the mass
of a CO molecule. For the temperatures and pressures involved, we find that it
takes some $10^{3}$ yr for a solid CO molecule to turn into its gaseous form.

The gas diffusion timescale is given by $\tau_{\rm dif}=3/4(\Delta p)^{2}(2\pi
m_{\rm CO}/(k_{b}T))^{0.5}/(\Psi r_{p})$. For the temperatures involved, we
find that it takes $10^{4}$ yr to diffuse upwards from 4 km deep.

The time to heat up the planetesimals significantly is longer by orders of
magnitude compared to the time to sublimate or diffuse up. Hence, the thermal
timescale will set the gas release rate in planetesimals. Let us model the gas
release rate due to thermal heating over time.

First, we assume that the CO mass contained in $N_{b}$ bodies of size $s$
(taken from a state-of-the-art collisional model of the KB, Morbidelli et al.,
2021) within a layer $\Delta p=\sqrt{Kt}$ deep is $M_{\rm CO}=4/3\pi\rho
f_{\rm CO}N_{b}(s^{3}-(s-\Delta p)^{3})$, where $f_{\rm CO}$ is the CO to
solid mass fraction (around $f_{\rm COinit}=10\%$ in comets).

The derivative of the CO mass that is warmed up by the Sun is ${\rm d}M_{\rm
CO}/{\rm d}t=2\pi\rho N_{b}f_{\rm CO}(s-\sqrt{Kt})^{2}\sqrt{K/t}$. To compute
$f_{\rm CO}$ for different sizes, we calculate for each size bin and at each
timestep $\Delta t$ the potential CO mass $M_{\rm COinit}$ contained in the
$\Delta p$ layer (assuming nothing was lost) as well as the CO mass that was
already lost at time $t$, $M_{\rm lost}=\sum_{t}{\rm d}M_{\rm CO}/{\rm
d}t\,\Delta t$, and we get $f_{\rm CO}=f_{\rm COinit}(1-M_{\rm lost}/M_{\rm
COinit})$, yielding in turn ${\rm d}M_{\rm CO}/{\rm d}t$. This model
reproduces the expectation that after 4.6 Gyr of thermal evolution, it is only
planetesimals bigger than 4 km that can participate in the gas release as
smaller bodies have lost all their CO by that time. The decrease of the gas
release rate with time is mostly due to having less and less bodies that can
participate in releasing CO. The 10-50 km bodies dominate the gas release in
this model. The resulting CO production rate is shown in Fig. 1. A single 30
km radius planetesimal would currently release around $10^{-14}$ M⊕/Myr of CO
in this model, i.e. much lower than what can be detected with missions
targeting specific KBOs (Lisse et al., 2021).

Scattered disk objects are replenished from various sub-populations of the
Kuiper Belt, and possibly the Oort Cloud. Their dynamical lifetime is rather
long ($\sim$1.8 Gyr, Gomes et al., 2008, though smaller than the age of the
Solar System by a factor 2.5) but the thermal effect on CO sublimation during
this period is limited. Objects in the Scattered Disk spend most of their time
at heliocentric distances larger than in the Kuiper Belt. The time spent close
to perihelion is limited, so the layer heated by such passages (10 m at most,
computed from the orbital skin depth, see Prialnik et al. 2004) remains much
smaller than the 4 km where the CO sublimation front is located (after thermal
evolution in the Kuiper Belt). This means that our simple thermal model, which
is only looking at the deepest layer that can release CO (that only depends on
the material as it is a diffusion calculation) will not be affected. We also
note that an SD object will spend most of its time in the KB before going to
the SD and finally be ejected so that the SD phase is not dominant overall.

The equilibrium temperature in the Oort Cloud as computed through the same
energy balance at the surface is extremely low. Other processes may increase
it (e.g., cosmic rays etc, as described for example by Desch & Jackson, 2021)
but the expected equilibrium temperature is roughly 6-10K (Jewitt, 2004),
i.e., well below the sublimation temperature of CO. Therefore, our model does
not predict sublimation of CO for Oort cloud objects.

We note that in our model, the CO released in the KB would be coming from pure
CO ices and not CO trapped in water, CO2, or other less volatile ices (Kouchi
& Yamamoto, 1995), which sublimation temperatures are too high to become gas
at KB distances. Therefore, we only expect the most volatile species to be
able to be released as gas in the KB. The volatiles with sublimation
temperatures lower than 40 K are N2 (22 K), O2 (24 K), CO (25 K), and CH4 (31
K) with the sublimation temperature given in parenthesis (Yamamoto, 1985). The
photodissociation timescales can also affect the observability of these
species. At the KB, we find photodissociation timescales of 25-62 yr for N2,
8-13 yr for O2, 34-86 yr for CO and 3-8 yr for CH4 (where the min/max values
correspond to the Sun at its maximum/minimum activity and the mean over a
Solar cycle of 11 yr should be closer to the longer photodissociation
timescale, Huebner et al., 2015). Now, we use the cometary abundances of
volatiles as a proxy for the abundance of planetesimals in the KB. N2 has been
observed in a few comets from the ground (e.g., for the comet C/2016 R2
(PanSTARRS), Opitom et al., 2019) and in-situ in 67 P (Rubin et al., 2015). At
most the N2-to-CO ratio could be 0.06 (Opitom et al., 2019, but it is one
order of magnitude lower in 67 P) in comets formed at large distances. N2
would thus be even more difficult than CO to observe even though they have
similar photodissociation timescales. O2 has also been observed in the coma of
67 P showing that it may be as abundant as CO on the comet (Bieler et al.,
2015). O2 photodissociates roughly 4 times as fast as CO at the KB and the
daughter neutral oxygen species would accumulate to that created from CO if
planetesimals in the KB routinely include O2 ices. As for methane, the CH4-to-
CO ratio is roughly 0.1 (Bockelée-Morvan & Biver, 2017) and its
photodissociation timescale at the KB is roughly 11 times smaller than for CO.
Therefore, the released CH4 would also be more difficult to observe than CO.

We also note that collisions would increase the predicted rate because they
would expose some fresh CO ices that can be released faster than on a thermal
timescale but the collisional timescale for bodies larger than 4 km are longer
than the age of our Solar System and this contribution will be negligible for
the current KB. Collisions that happened in the early stages of the Trans-
Neptunian disk could have affected the size distribution of small bodies and
release CO on the surface of these bodies but we note that we use a state-of-
the-art size distribution based on observations (Morbidelli et al., 2021),
which already accounts for previous evolution. Therefore the early collisional
evolution is implicitly taken into account in our calculations.

Table 2: Data of JFCs, HFCs, OCCs and Centaurs for which we both have
measurements of $Q_{\rm CO}$ and diameter. $Q_{\rm CO}$ is the CO production
rate in $s^{-1}$ and its error bar is the 1$\sigma$ uncertainty, $D$ is the
equivalent nucleus diameter in km and $r_{h}$ the heliocentric distance of the
object in au. The upper limits are represented by the 3$\sigma$ uncertainties.

Name | $Q_{\rm CO}$ | $D$ | $r_{h}$ | ref
---|---|---|---|---
C/1995O1 | $6.5\pm 0.8\times 10^{27}$ | 69.9 | 14.07 | (1)
C/1995O1 | $2.15\pm 0.17\times 10^{30}$ | 69.9 | 0.917 | (1)
C/1996B2 | $1.2\pm 0.3\times 10^{28}$ | 2.29 | 1.852 | (2)
C/1996B2 | $10.4\pm 2.4\times 10^{28}$ | 2.29 | 0.478 | (2)
C/2020F3 | $1.3\pm 0.5\times 10^{28}$ | 5.0 | 0.71 | (3)
10199 | $<15.0\times 10^{27}$ | 249 | 13.5 | (4,5)
1P | $4.5\pm 2\times 10^{28}$ | 9.3 | 0.92 | (6)
2P | $1.8\pm 0.2\times 10^{26}$ | 7.02 | 0.473 | (7)
8P | $2.4\pm 0.4\times 10^{26}$ | 4.93 | 1.027 | (8)
9P | $8.7\pm 1.6\times 10^{26}$ | 6.07 | 1.506 | (8)
10P | $<7.8\times 10^{27}$ | 9.24 | 1.482 | (9)
19P | $<4.5\times 10^{27}$ | 4.34 | 1.360 | (10)
21P | $4.6\pm 0.8\times 10^{26}$ | 3.6 | 1.18 | (11)
21P | $2.5\pm 0.3\times 10^{26}$ | 3.6 | 1.10 | (11)
22P | $<8.1\times 10^{25}$ | 3.77 | 1.61 | (12)
29P | $3.5\pm 1.5\times 10^{28}$ | 56.3 | 6.0 | (13,14)
41P | $<8.4\times 10^{26}$ | 1.4 | 1.06 | (15)
45P | $1.9\pm 0.3\times 10^{26}$ | 0.92 | 0.555 | (16)
46P | $<3.3\times 10^{25}$ | 1.2 | 1.13 | (17)
67P | $3.4\pm 2.5\times 10^{26}$ | 3.81 | 1.28 | (18,19)
73PC | $7\pm 2\times 10^{25}$ | 1.61 | 0.950 | (8)
81P | $>5.3\times 10^{25}$ | 4.17 | 1.74 | (12)
95P | $1.3\pm 0.6\times 10^{28}$ | 155 | 8.5 | (20,21)
95P | $<8.1\times 10^{27}$ | 155 | 8.48 | (22,21)
95P | $<5.1\times 10^{28}$ | 155 | 8.9 | (4,21)
103P | $2.6\pm 0.3\times 10^{25}$ | 0.68 | 1.064 | (23)
174P | $8.0\pm 3.0\times 10^{26}$ | 64 | 6.13 | (14)

References: (1): Biver et al. (2002); (2): Biver et al. (1999); (3): Biver al.
(in prep) (4): Bockelée-Morvan et al. (2001); (5): Morgado et al. (2021); (6):
Feldman et al. (1997); (7): Roth et al. (2018); (8): Dello Russo et al. (2016,
and references therein); (9): Biver et al. (2012); (10): Bockelée-Morvan et
al. (2004a); (11): Roth et al. (2020); (12): Ootsubo et al. (2012); (13):
Crovisier et al. (1995); (14): Wierzchos, Womack, & Sarid (2017); (15): Biver
et al. (2021); (16): DiSanti et al. (2017); (17): McKay et al. (2021); (18):
Biver et al. (2019); (19): Läuter et al. (2020); (20): Womack et al. (2017);
(21): Lellouch et al. (2017); (22): Rauer et al. (1997); (23): Weaver et al.
(2011);

##  Appendix B Gas production rate calculations

We use two different techniques to estimate the gas production rate in the KB.
First, we assume that the model of gas production that fits detections and
non-detections of gas in extrasolar systems (Kral et al., 2017, 2019) is valid
for the KB (Model 1). Second, we test a more physically motivated model that
works out the sublimation rates of the bodies in the KB to then derive the
total gas production rate in the belt (Model 2).

Model 1 states that the gas production rate is proportional to the mass loss
rate of the planetesimal belt collisional cascade. This is because it is
assumed that gas is produced owing to collisions when solids grind down to
dust somewhere along the cascade. The model does not say which solid size
bodies release gas and what is the physics behind it. Rather the model assumes
that all CO contained on a large body ($\sim$10% of its mass) is released
before it is ground down to dust and ejected because of radiation pressure.
The gas release physics is not yet known and it could be due to high-velocity
collisions at the bottom of the cascade, to photodesorption or to sublimation
(which this paper may favour), which are all more active for more massive
belts releasing more dust indeed. Therefore, the CO gas production rate is
roughly equal to the CO fraction of planetesimals times the dust mass loss
rate (which is the rate at which mass is passed down the cascade from one bin
to the other, which is constant throughout the cascade as the new mass
injected at the top of the cascade is lost at the bottom of it at steady-
state, Wyatt et al., 2011).

We first compute the mass loss rate from a state-of-the-art model of the KB
(Vitense et al., 2012). Using the cross-section density per size decade $A$
(in m2/m3) derived from their simulations (Vitense et al., 2012), we can
compute the total mass of bodies in a disk of area $S_{\rm KB}=2\pi R_{\rm
KB}\Delta R_{\rm KB}$ and scale height $H\sim 0.4R_{\rm KB}$ (Luu & Jewitt,
2002) as $M_{d}(s)=A(4/3\pi s^{3}\rho)S_{\rm KB}(2H)/(\pi s^{2})$ in a given
size bin $s$. For the KB location $R_{\rm KB}$ and its width $\Delta R_{\rm
KB}$, we take 45 and 10 au, respectively (Luu & Jewitt, 2002). Then, we derive
the dust mass loss rate as being $\dot{M}_{d}(s)=M_{d}(s)/t_{\rm surv}(s)$,
where $t_{\rm surv}(s)$ is the lifetime of a solid of size $s$ taken from
collisional simulations (Vitense et al., 2012). Assuming a constant CO
fraction $f_{\rm CO}$ of 0.1 on solid bodies, we then derive the CO gas
production rate as being $\dot{M}_{\rm COtot}=f_{\rm CO}\dot{M}_{d}(s)$, which
is constant at all sizes $s$ (because at steady-state the rate of solids that
are broken up by collisions between large bodies is equal to the dust mass
loss rate due to radiation pressure). Indeed, making the calculation using the
small micron-sized dust grains at the bottom of the cascade or for larger
bodies at collisional equilibrium, we find $\dot{M}_{\rm COtot}=10^{-9}$
M⊕/Myr.

We also derive the mass loss rate based on measurements of the student dust
counter on the New Horizons mission (Horányi et al., 2008). The number of
particles between 0.5 and 5 microns hitting the student dust counter in the KB
at 45 au, moving at $v_{\rm NH}\sim 14$ km/s, is estimated to be
$F_{d}=1.5\times 10^{-4}$ m-2 s-1 (Poppe et al., 2019). The total mass of
grains is then given by $M_{d}[0.5-5{\mu}m]=(F_{d}/v_{\rm NH})\,m^{*}S_{\rm
KB}(2H)$, where $m^{*}$ is the mean mass of a particle in the 0.5-5 micron
size range given a particle size distribution slope in -2.5 as it is in the
PR-drag (rather than collisional) regime (Wyatt et al., 2011; Vitense et al.,
2012). To get the mass loss rate of grains between 0.5 and 5 microns, we
divide by the PR-drag timescale $t_{\rm PR}=400\beta^{-1}(R_{\rm KB})^{2}$ (in
yr, Wyatt, 2005), where we take the ratio between the radiation pressure force
to that of stellar gravity $\beta=0.2$ based on the mean size of a grain of
mass $m^{*}$ (Vitense et al., 2012). However, to get the mass loss rate of the
cascade (and not just that of the grains captured by the student dust
counter), we need to multiply by the number of logarithmic bins up to the size
at which collisions dominate over PR-drag (Wyatt et al., 2011), i.e., up to
$s_{\rm pr}=100$ microns (Vitense et al., 2012). We obtain $\dot{M}_{\rm
COtot}=9\times 10^{-10}$ M⊕/Myr, i.e. a gas production rate very close to that
found with the previous method.

Model 2 uses the sublimation model described in the previous section. We find
that the gas sublimation rate for the KB is dominated by large bodies $>4$ km
after 4.6 Gyr evolution. Below this size, all CO gas was released because the
entire CO inventory has been sublimed already and the gas production rate
drops to zero. To get the final CO production rate, we sum over all the size
bins and find $2\times 10^{-8}$ M⊕/Myr, which is roughly a factor 20 higher
than the previous estimates. The temporal evolution is shown in Fig. 1.

We note that comet sublimation models still predict outgassing at large KB-
like distances and they estimate that a single large Hale-Bopp comet at 40 au
would release about $10^{-10}$ M⊕/Myr of CO (Jewitt et al., 2008). However,
this is valid only if there remains CO ice on the planetesimal surface, but as
we showed previously it would be long gone from the upper layers after 4.6 Gyr
evolution, and our model predicts a rate about 4 orders of magnitude smaller
for a given large planetesimal similar to Hale-Bopp.

We also check whether the sublimation model is consistent with the high
release rates observed around younger more massive stars with ALMA. Let us
take an extreme case, which is that of a belt as massive as that of the
$\beta$ Pic system (i.e., more massive than 1000 M⊕ if we assume the belt to
be composed of bodies up to 100 km, though bodies may be born smaller, Krivov
& Wyatt, 2021). To reach a belt of 1000 M⊕, we scale up the number of bodies
in the KB (of total mass equal to $\sim 0.1$ M⊕) by a factor $10^{4}$ and re-
run our model leading to the dashed line in Fig. 1. This shows that gas
release rates of $\sim 0.1$ M⊕/Myr can be reached with this model when gas is
initially released from the young belt.

The gas release rate in the belt of the $\beta$ Pic system could be up to 0.1
M⊕/Myr (Kral et al., 2016) but we note that the gas release rate necessary to
explain the CO mass observed could be lower, as there could be sufficient
carbon to shield CO from photodissociation in this system (Kral et al., 2019).
The temperature conditions of planetesimals in the KB could be similar to that
of other exo-Kuiper belts, as belts tend to form preferentially at a given
distance from their star ($R\propto L^{0.19}_{\star}$, Matrà et al., 2018b),
and so the resulting belt temperature ($\propto L_{\star}^{0.16}$) is only
weakly dependent on the stellar luminosity (56 K for the $\beta$ Pic
extrapolation) and close to that of the KB. We note that the size distribution
in young systems could be different to that of the Solar System, which could
also increase/decrease the model predictions but this is not taken into
account here. A more thorough study (including collisions for young systems)
of this model (and its free parameters such as the material composition or
temperature of the belt) and whether it can explain all observations (or just
that of the less massive belts for instance) is beyond the scope of this KB-
focused paper.

##  Appendix C Description of the gas evolution model

We now summarize the model for the evolution of the gas released in the KB so
that the reader gets a feel for what mechanisms are at play while reading the
more in-depth sections that follow.

The main ingredients used in the gas evolution model for the KB are summarized
in Table 3 and the main timescales in Table 4. The main ideas go as follows:
1) CO is released from planetesimals in the KB. 2) CO is quickly pushed
outwards with a velocity $\gtrsim$ 3 au/yr due to collisions with high-
velocity ($\sim$ 400 km/s) protons from the solar wind (and a small fraction
of the CO gets ionized and dissociated during these collisions and due to
impinging photons from the Sun and the interstellar medium). 3) Most CO gets
pushed beyond the heliopause (located at $\sim$150 au, Dialynas et al., 2017;
Richardson et al., 2019) before it has time to dissociate or ionize. 4) CO
finally turns into C+O and C gets ionized due to photons from the interstellar
medium, while O gets ionized due to collisions with protons from the local
cloud of the interstellar medium that is colliding with our Solar System. 5)
Then the ionized atoms follow the interstellar magnetic field lines and get
ejected further into the local interstellar medium. More details about each
step of the model are given in the following sections.

Table 3: Processes accounted for in this study for the CO evolution once
released from the Kuiper belt. We show the dominant process at the Kuiper belt
location in the column with a * symbol.

Processes | interactions | quantity of interest | value | *
---|---|---|---|---
CO
Ionization | SW protons | charge exchange cross section (López-Patiño et al., 2017) |  $1.5\times 10^{-15}$ cm2 |
solar photons | photoionization rate (at 1 au) (Huebner et al., 1992) |  $6\times 10^{-7}$ s-1 | x
| SW e- impacts | ionization rate (Rubin et al., 2009) |  $5.49\times 10^{-9}$ cm3/s |
Dissociation | SW protons | dissociation cross section (López-Patiño et al., 2017) |  $1.5\times 10^{-17}$ cm2 |
solar photons | photodissociation rate (at 1 au) (Huebner et al., 2015) |  $5\times 10^{-7}$ s-1 | x
| ISM photons | photodissociation rate (Heays et al., 2017) |  $2.4\times 10^{-10}$ s-1 |
Collisions | SW protons | radial velocity after first collision |  $\sim$ 3 au/yr | x
| collisional frequency (at 45 au) |  $\sim$ 2.7 yr |
| LISM protons | radial velocity after collision | depends on $v_{\rm{CO}}$ |
|  | collisional frequency |  $\sim$ 1.7 yr |
CO+
Dissociation | solar photons | photodissociation rate (at 1 au) (Heays et al., 2017) |  $5\times 10^{-8}$ s-1 |
ISM photons | photodissociation rate (Heays et al., 2017) |  $1\times 10^{-10}$ s-1 | x
Collisions | SW protons | radial velocity after first collision |  $\sim$ 3 au/yr | x
| collisional frequency (at 45 au) |  $\sim$ 24 min |
| LISM protons | radial velocity after collision | depends on $v_{\rm{CO+}}$ |
|  | collisional frequency |  $\sim$ 15 min |
C
Ionization | solar photons | photoionization rate (at 1 au) (Huebner et al., 2015) |  $4\times 10^{-7}$ s-1 |
ISM photons | photoionization rate (Heays et al., 2017) |  $3.4\times 10^{-10}$ s-1 | x
O
Ionization | SW protons | charge exchange cross sectiona (Izmodenov et al., 1997) |  $1.1\times 10^{-15}$ cm2 | x
solar photons | photoionization rate (at 1 au) (Huebner et al., 2015) |  $4\times 10^{-7}$ s-1 |
| ISM photons | photoionization rate | $0$ |

  * •

a We note that some extra ionization comes from electron impact ionization
while crossing the heliosphere but CO mainly photodissociates further out
(Izmodenov et al., 1999).

Table 4: Timescales of the dominant processes at play Processes | Interactions | Timescale
---|---|---
CO
Ionization | solar photons (Huebner et al., 1992) | 107 yr (at 45 au, $\propto 1/r^{2}$)
Dissociation | solar photons (Huebner et al., 2015) | 50 yr (at 45 au, $\propto 1/r^{2}$), 120 yr (at $>$70 au)
Collisions | SW protons | 2.7 yr
Collisions | LISM protons | 1.7 yr
CO+
Dissociation | ISM photons (Heays et al., 2017) | 305 yr (at $>$40 au)
Collisions | SW protons | 24 min
Collisions | LISM protons | 15 min
C
Ionization | ISM photons (Heays et al., 2017) | 94 yr (at $>$40 au)
O
Ionization | SW protons (Izmodenov et al., 1997) | 160 yr (at 45 au, $\propto 1/r^{2}$), 110 yr (at $>$150 au)

##  Appendix D Ionization fraction for the gas in the Kuiper belt

We compute the ionization fraction of the main species we study by equating
the ionization rate to the recombination rate. For the Solar System, we take
the photoionization probabilities at 1 au (Huebner et al., 2015) (in s-1) for
C, O, and CO: $8\times 10^{-7},4\times 10^{-7},6\times 10^{-7}$. The
recombination timescales are based on the modified Arrhenius equation (in
cm3/s) of the form $\alpha(T/300\,{\rm K})^{\beta}\exp(-\gamma/T)$. The
recombination rate for O+ is given by Nahar (1999), $\alpha=3.24\times
10^{-12}$, $\beta=-0.66$, $\gamma=0$, for C+ we use Nahar & Pradhan (1997),
$\alpha=2.36\times 10^{-12}$, $\beta=-0.29$, $\gamma=-17.6$, and for CO+
$\alpha=2.75\times 10^{-7}$, $\beta=-0.55$, $\gamma=0$ are taken from the KIDA
database (Wakelam et al., 2012).

One striking difference with previous work on the subject of gas in planetary
systems (where models were developed mostly for A stars in young systems) is
that in the Solar System, oxygen can be ionized because of the presence of
numerous UV photons at energies greater than the ionization potential of
oxygen (13.6 eV). Solving for the ionization fractions of CO, C and O
analytically, we search for the electron density (expected to be the main
collider here) necessary to get an ionization fraction greater than 0.5 so
that we can later estimate (when we get the electron density from the model)
whether the different species will be ionized or not. For CO, we find
$n_{e}<3\times 10^{-4}(T/30K)^{0.58}$ cm-3. For C (where we also account for
the ionization rate of $3.39\times 10^{-10}$ s-1 from the interstellar medium,
which is greater than the photoionization rate at 45 au, Heays et al., 2017),
we find $n_{e}<91(T/30K)^{0.29}$ cm-3 and for O, we have
$n_{e}<13(T/30K)^{0.66}$ cm-3.

We also estimate the CO ionization rate through SW electron impact. The
density and temperature at the KB of the fast moving electrons ($\sim$ 610
km/s) are $10^{-3}$ cm-3 and $3\times 10^{5}$ K (25.9 eV) (Meyer-Vernet et
al., 1998). The electron impact ionization rate (for an electron energy of 18
eV) is $k_{e}=5.49\times 10^{-9}$ cm3/s, which is multiplied by $25.9/18\sim
1.4$ to account for the highest velocities (Rubin et al., 2009). Equating
$k_{e}n_{e{\rm SW}}n_{\rm CO}$ to the recombination rate of CO+ described
above, we find that an electron density lower than $8\times 10^{-6}$ cm-3 is
necessary to lead to an ionization fraction of CO greater than 0.5 (the slow
moving electrons contribution is of the same order of magnitude albeit
slightly lower, Meyer-Vernet et al., 1998). The photoionization is therefore
much more efficient, and electron impacts from the SW can be neglected for CO
ionization.

In the SW result section, we will use the electron density given by our model
to provide an estimate of the ionization fraction of the different species. In
broad terms (for $T$ close to 20-50 K), C, O, and CO get $>50$% ionized at 45
au for $n_{e}<100$ cm-3, $<10$ cm-3 and $<3\times 10^{-4}$ cm-3, respectively.
So it is most likely that O and C will be close to 100% ionized at 45 au (if
they can be produced and remain at 45 au for long enough, see later), and for
CO it depends on the details that we will investigate in the coming sections
(which are complicated by impacts from SW protons pushing CO away faster than
it can photoionize).

##  Appendix E Spreading timescales

In current and past works about gas in planetary systems, the viscous
evolution of a gas disk is often parameterized using an $\alpha$ model
(Shakura & Sunyaev, 1973), which provides a good description in the fluid
regime when the Knudsen number (mean free path over gas scale height) is lower
than 1. The value of $\alpha$ sets how fast the gas disk spreads as the
viscous timescale $t_{\rm visc}$ is given by $r^{2}/\nu$ with the viscosity
$\nu=\alpha c_{s}^{2}/\Omega$, where $c_{s}$ is the sound speed and $\Omega$
the orbital frequency. A recent theoretical study shows that the
magnetorotational instability (MRI, Balbus & Hawley, 1991) may be able to
develop in the debris disk regime and produce large $\alpha$ values (of the
order of 0.1) owing to the high ionization fraction of the gas in these
systems (Kral & Latter, 2016). Observations seem to favour high $\alpha$
values as well (Kral et al., 2016; Marino et al., 2020), though this may
depend on the emergence of non-ideal MRI effects (such as ambipolar diffusion)
as well as the magnetic field strength (Kral & Latter, 2016). Taking $\alpha$
between $10^{-4}$ and 0.1, we find that the viscous timescale at the KB
location can vary from 1.1 Myr to 1.1 Gyr assuming a gas temperature of 30 K
and a mean molecular weight of 28 (gas dominated by CO).

However, the gas density around the KB may be too low to be in the fluid
regime, and previous considerations used to describe gas in exoplanetary
systems do not apply. When gas density is very low and the Knudsen number
becomes greater than 1, then the non-fluid viscosity can be evaluated as
follows, $\nu_{2}=\lambda_{\rm mfp}\,c_{s}$, with $\lambda_{\rm mfp}$ the mean
free path of a gas particle equal to $(n_{\rm gas}\sigma_{\rm col})^{-1}$,
where $n_{\rm gas}$ is the gas density and $\sigma_{\rm col}$ its collisional
cross-section. Since the gas scale height is $c_{s}/\Omega$, this regime may
happen when $n_{\rm gas}<\Omega/(\sigma_{\rm col}c_{s})$. Let us consider the
most favourable case of collisions between charged particles (with a greater
collisional cross-section) and take $\sigma_{\rm col}=\pi R^{2}_{\rm cc}$,
where $R_{\rm cc}$ is the cross-sectional radius for a proton calculated by
equating kinetic energy to electrostatic energy so that we get $R_{\rm
cc}=e^{2}/(6\pi\epsilon_{0}k_{\rm b}T_{\rm gas})$, with $e$ the elementary
charge and $\epsilon_{0}$ the vacuum permittivity. Thus, we find that the non-
fluid regime appears when $n_{\rm gas}<n_{\rm crit}=3\times 10^{-6}$ cm-3. As
for neutral atoms, Coulomb collisions will happen with charged particles. To
find the cross-section of the, e.g., C+-O collisions, we account for the fact
that the ion induces a dipole on the neutral atom, which gives birth to an
electric repulsion (Beust et al., 1989). The cross-section for neutral-ion
interactions is roughly $10^{4}$ times smaller than for proton-proton
collisions, hence increasing $n_{\rm crit}$ by a factor $10^{4}$.

Assuming the CO gas production rate of $2\times 10^{-8}$ M⊕/Myr derived
earlier, we find that the ionized carbon produced from CO photodissociation
must remain at 45 au for $>$ 25 yr to reach a gas density greater than $n_{\rm
crit}$ and therefore be in the fluid regime. For the case of neutral atoms
(e.g. CO, C or O), they should remain for $>$ 0.25 Myr. We will calculate in
the coming section how long the CO, C or O can survive at a given position
owing to the action of the Solar wind, which we find acts on smaller
timescales than viscous spreading in the Solar System. We find that, due to
the SW, the gas density cannot increase sufficiently to be in the fluid
regime, which is going to drastically change the dynamics of gas compared to
previous studies.

##  Appendix F Effect of Solar wind on gas

Let us consider the effect of the Solar wind on the KB gas ring. The Solar
wind medium density is $\sim$8 cm-3 at the location of the Earth (1 au), which
translates in a density of $n_{\rm SW}=4\times 10^{-3}$ cm-3 at 45 au
(Hosteaux et al., 2019) (assuming a $1/r^{2}$ drop-off, i.e. constant radial
velocity). We can first calculate the time between collisions with the SW for
each molecule, which is given by $1/(\sigma_{\rm X-SW}n_{\rm SW}v_{\rm SW})$,
where $\sigma_{\rm X-SW}$ is the cross-section of interaction between X and
protons from the SW and $v_{\rm SW}$ is the Solar wind velocity of about 400
km/s. We consider that the cross-section of interaction between ionized SW
particles and CO is set by the induced dipole between CO and a proton such as
described above in the spreading timescale section. We find that the induced
dipole cross-section is $\sigma_{\rm CO-SW}\sim 7\times 10^{-14}$ cm2. For CO,
we find that there is a collision with a SW proton every 2.7 yr (i.e., before
it has time to dissociate or ionize). For an ionized species (e.g., CO+ and
proton) with a higher cross section (see previous section), it is only about
24 minutes. Similar calculations in the local cloud of the interstellar medium
(using a velocity of 26 km/s and proton density of 0.1 cm-3) lead to 1.7 yr
and 15 min for neutral and ionized species, respectively (Richardson et al.,
2019).

We can estimate the rate of collisions with the SW protons in the KB for a
given density of atoms or molecules as

$W_{\rm SW}=V_{\rm KB}n_{\rm X}\sigma_{\rm X-SW}n_{\rm SW}v_{\rm SW},$

where $V_{\rm KB}$ is the KB volume and $n_{\rm X}$ the density of X.

###  F.1 Basics of the model based on former literature knowledge

Let us start our calculations as if we were only aware of the models developed
for extrasolar systems (mostly for young A stars) as it will ease the
transition to a Solar System model where the addition of the Solar wind adds
another layer to currently used models.

First, we consider the effect of the SW on the CO molecules for which we first
assume a number density $n_{\rm CO}$ of $5\times 10^{-6}$ cm-3, which is
expected if CO can survive for about 50 years against photodissociation (which
is close to that value at 45 au in our Solar System). We calculate the mean
loss rate of CO due to SW interactions as $\dot{M}_{\rm SW}=W_{\rm SW}\mu
m_{p}$ assuming that after each collision the high impact velocity will give
an outwards kick to the CO molecule. Indeed, working out the momenta for the
CO molecule and the high-velocity proton and using that it is conserved and
that the collision is perfectly inelastic, we find $\@vec{p}_{\rm
tot}=\@vec{p}_{\rm proton}^{\rm\,\,init}+\@vec{p}_{\rm
CO}^{\rm\,\,init}=\@vec{p}_{\rm proton+CO}^{\rm\,\,final}$. Therefore, we can
solve for the final CO velocity vector that we find is 13.8 km/s radially and
4.3 km/s azimuthally. It is indeed unbound already after the first kick. We
note that if the collision is perfectly elastic, the final radial velocity
could be twice as great. In reality, when the CO and the proton stick
together, the collision will indeed lead to a radial velocity of 13.8 km/s
(2.9 au/yr) and otherwise, it will be a value in the range 13.8-27.6 km/s.
Given the charge exchange cross section between SW protons and CO given in
Table 3, we find that exchanges will happen on a $\sim$100 yr timescale, i.e.,
the aftermath of the collision is most likely CO but the detailed modelling of
the collision geometry and properties goes beyond our simple model. We note
that there could be multiple collisions before reaching the heliopause at
$\sim$ 150 au. For instance, if the first collision between CO and a proton
happens at 45 au after 2.7 yr then the CO will travel radially for a bit
longer than 2.7 yr (as the timescale scales as $(r/45)^{2}$) and we calculate
(using a mean collision time over the distance travelled) that the next
collision will happen at $\sim$ 55 au and so it will be outside the main KB
and its radial velocity will have doubled. The next collision will happen
$\sim$ 10 yrs later when the particle is at $\sim$ 110 au, close to the
heliopause. After that there are no more collisions with protons from the SW
as the SW proton density becomes too small given the fast velocity CO moves at
(8.7 au/yr or 41 km/s). Next collisions will be with the protons from the
local interstellar medium, whose density is around 0.1 cm-3 (Izmodenov et al.,
1997), finally equalizing the velocities to around 26 km/s after a few years.
For ions, the collisions with protons (be they from the SW or interstellar
medium) are very fast and it takes a couple of tens of minutes to reach 400
km/s before the heliopause or 26 km/s beyond it. The transitions are smoother
than described above and the variation of the velocity with distance is
represented in Fig. 4.

Figure 4:  Sketch of the velocity of particles as a function of distance to
the Sun for neutrals (solid) and ions (dashed). The exact shape of the radial
profile depends on the heliopause location, which is not fully modelled in
this paper (and assumed symmetric as recently proposed by new studies). The
assumptions behind this plot are described in the appendix section about the
effect of SW.

Thus, the CO mass loss rate due to the SW (when it can accumulate for the
photodissociation timescale of 50 years due to the Sun radiation) would be
$\dot{M}_{\rm SW}\sim 4\times 10^{-7}$ M⊕/Myr (where $m_{p}$ is the proton
mass and we take $\mu=28$), which is higher than the CO gas production rate we
found in this study so that CO will actually be removed before it has time to
photodissociate, which will reduce the total CO density we used so far.

###  F.2 Improvement and further complexities of the model

To find the number density of CO given its production rate and loss through SW
interactions, we equate $\dot{M}_{\rm SW}$ and $\dot{M}_{\rm COtot}$ and find
$n_{\rm CO}=3\times 10^{-7}$ cm-3, which is more than one order of magnitude
smaller than the value previously found without properly accounting for the
dynamical effect of the SW. This is because indeed, there is a collision every
2.7 years at 45 au with high velocity protons so that CO gets removed roughly
20 times faster than under the action of photodissociation alone. CO will move
outwards before being eventually photodissociated by UV photons from the
interstellar radiation field after roughly 120 yr, i.e., at a distance roughly
ten times that of the KB. However, once in the local interstellar medium
(i.e., beyond the heliopause at $\sim$ 150 au, Dialynas et al., 2017), the
velocity of CO molecules will be slowed down by collisions with the local
interstellar medium protons every few years.

From the CO number density calculation, it is also clear that the CO will be
fully ionized by the UV photons from the Sun because we estimate that the
electron density is $<3\times 10^{-4}$ cm-3 assuming that electrons come from
CO ionization (see CO+ densities in Fig. 3). However it is not instantaneous
and it will take approximately 100 yr to photoionize a given molecule of CO at
45 au (Huebner et al., 2015). Also, there will be some charge exchanges
between CO and protons from the SW leading to CO+ that may happen faster than
photoionization. The cross section of exchange for protons with an energy of
$\sim 1$ keV is $\sim 1.5\times 10^{-15}$ cm2 so that exchanges with a CO
molecule will happen every 134 yr, at a slightly lower rate than
photoionization. We note that photoionization leading to C+ happens 13 times
less often and 15 times for O+ (Rubin et al., 2009). As for the collisions
with protons from the SW, they lead to C+ 5 times less often and to O+ 10
times less often.

As soon as CO becomes CO+, it will leave the system very quickly as collisions
with the Solar wind protons happen every 24 min rather than 2.7 yr and the
velocity becomes very quickly equal to that of SW protons (i.e., $\sim$400
km/s). But CO will start heading outwards after 2.7 yr anyway and may become
CO+ on its way out. The final quantity of CO in the KB is therefore indeed set
by the frequency of impacts with the SW of 2.7 yr. But we note that the escape
of CO outside of the KB is not instantaneous as the molecule will travel
radially at roughly 14 km/s (or $\sim 2.9$ au per yr) after an impact with a
SW proton, and it will take a gas particle in the middle of the KB (45 au)
roughly 1.7 yr to reach 50 au, so that CO can slightly accumulate before
leaving the belt. Multiplying the CO number density we obtained with
$\dot{M}_{\rm SW}=\dot{M}_{\rm COtot}$ by (1.7+2.7)/2.7, we obtain the mean CO
density in the belt $n_{\rm CO}$ equal to $5\times 10^{-7}$ cm-3, which will
roughly scale as $1/r^{2}$ until it dissociates or ionizes. This leads to a
surface density at 45 au $\Sigma_{\rm CO}=\mu m_{p}n_{\rm CO}(2H)\sim
10^{-16}$ kg m-2.

###  F.3 Modelling of the outer regions of the KB

We note that CO will photodissociate in $\sim$120 yr beyond 45 au while
photoionization will operate on a timescale of 107 ($r$/45 au)2 yr and proton
collisions leading to CO+ in 134 ($r$/45 au)2 yr. Because CO moves at a rate
of 2.9 au/yr after a collision with a proton, it will be at hundreds of au
after 100 yr and the timescales for photoionization and ionization by protons
from the SW become $>10,000$ yr. Therefore, CO photodissociates before it has
time to be ionized. However, the time for CO to move outside of the KB is on
average 4.4 yr and some small fraction of CO will have time to ionize and
photodissociate. Using an exponential decay law for the time evolution of
ionization and photodissociation, we find that 7.3% of CO will be ionized at
45 au and 3.6% will be photodissociated. If we assume that most electrons in
the KB come from CO ionisation then this gives an electron density $n_{e}$ at
least greater than $7.3\%\times n_{\rm CO}=2\times 10^{-8}$ cm-3. CO+ will
reach the SW proton velocity in a few hours as it collides every 24 minutes
with them. Therefore, we expect the CO+ density in the belt to be $7.3\%\times
n_{\rm CO}\times(13.8/400)=2.5\times 10^{-3}\,n_{\rm CO}=7\times 10^{-10}$
cm-3.

After CO eventually photodissociates, an atomic gas component will appear.
This leads to Fig. 3 where we show the down-wind profile of gas and assume
that the heliosphere is not too asymmetric and close to a ball-shape as
recently proposed (Dialynas et al., 2017; Opher et al., 2020, and we do not
model the specificities of the heliosheath). We also assume that ions move
faster than neutrals as given by Fig. 4 to work out the relative gas
densities. CO produced in the KB would then cross the heliopause (at $\sim$
150 au) after roughly 20 yr. Assuming that CO moves at 26 km/s after the
heliopause (Opher et al., 2020), CO will be mostly photodissociated (after 120
yr in total) at $\sim$ 500 au (on the downwind side and slightly further in on
the upwind side because CO gets pushed backwards once it reaches the
heliopause). The carbon and oxygen atoms will eventually ionize. The
photoionization timescale for C is 94 yr (owing to the interstellar medium
photons). For O it takes $>$13 kyr at $>$400 au from the Sun to become ionized
(as only the Sun’s photons are energetic enough) but O will cross the
heliopause and encounter protons from the local interstellar medium and charge
exchanges can then happen that will operate in $\sim$ 110 yr (Izmodenov et
al., 1997). Therefore, carbon and oxygen will ionize quickly (we assume 100 yr
for both in Fig. 3). The ionized carbon and oxygen will start dominating at
$\sim 500$ au (on the downwind side). They will then follow the interstellar
magnetic field lines (see next section) and get ejected further into the local
interstellar medium.

One of the main conclusions of this model is that CO will move outwards and
almost no gas released from the KB will be able to make it inwards towards
Neptune. However, we note that it could have been different in the past, as
the KB was much more massive and the gas release rate should have been high
enough for the gas to be in the fluid regime. In this situation, the gas
becomes more optically thick to collisions with protons and may have time to
evolve viscously inwards (as described in the viscous evolution section)
rather than being pushed outwards, but consideration of this regime is not the
purpose of the current paper.

##  Appendix G Interaction with the magnetic field

Let us now analyse the dynamics of an ion produced in the KB choosing CO+ for
the example below.

We choose a density $n=10^{-9}$ cm-3 of CO+ as a proxy (as it is close to the
value we find in the previous section). It leads to a mean free path of
$2\times 10^{17}$ cm for proton-proton collisions (i.e. $>$10,000 au),
$3\times 10^{19}$ cm for charged-neutral collisions and $3\times 10^{21}$ cm
for neutral-neutral species collisions (see SW section for the different
cross-sections). We can then compare these values with the gyroradius
$r_{g}=mv_{\rm kep}/(eB)$. With the interplanetary magnetic field $B\sim 0.1$
nT at the KB (it is 6 nT at Earth and scales as $1/r$, Axford et al., 1963),
we obtain $r_{g}=8\times 10^{8}$ cm for a molecule of CO. We can also compare
to the relevant lengthscale of the problem $L=R_{\rm KB}(c_{s}/v_{\rm kep})$,
which we find equal to $L\sim 2\times 10^{13}$ cm.

The gyroradius being smaller than both the mean free path and the relevant
lengthscale, we conclude that ionized species produced in the KB will follow
the interplanetary magnetic field lines and escape the Solar System.

The same reasoning applies to ionized particles beyond the heliopause in the
local interstellar medium, which will then follow the interstellar magnetic
lines.

##  Appendix H Gas mass, density and column calculations

Assuming a CO gas production rate of $2\times 10^{-8}$ M⊕/Myr and that CO
escapes the KB in 4.4 yr (see SW section), we get a total CO mass in the KB
(40-50 au) of $\sim 10^{-13}$ M⊕, which is roughly the total CO mass that was
lost by the Hale-Bopp comet in 1997. This mass translates into the previously
calculated mean number density of $3\times 10^{-7}$ cm-3 in the KB. The total
CO mass (up to 2000 au) obtained with our model is equal to $2\times 10^{-12}$
M⊕ or roughly 20 times the CO mass that was lost by the Hale-Bopp comet during
its 1997 passage.

For the carbon and oxygen wind that forms from the CO photodissociated
molecules, we find that the total atomic masses (up to 2000 au) are $\sim
6\times 10^{-12}$ M⊕ and $\sim 8\times 10^{-12}$ M⊕ for neutral and ionized
species, respectively.

For the column densities $N_{X}$ of species $X$, we integrate along the
midplane outwards so that $N_{X}=\int_{R_{\rm in}}^{R_{\rm out}}n_{X}\,{\rm
d}R$, with $R_{\rm in}=45$ au and $R_{\rm out}=2000$ au, and Table 1
summarizes all these calculations.

##  Appendix I Comparison to other sources of CO in the Solar System

Comets release abundant CO when they approach the Sun (e.g. Bockelée-Morvan et
al., 2004b). For instance, the 60 km diameter Hale-Bopp comet released a CO
mass of $\sim 10^{-13}$ M⊕ during its 1997 passage near the Sun (mostly at its
perihelion). This means that 20 such comets could produce together a CO mass
comparable to that we predicted for the KB. However, the CO comet production
is local, anisotropic and concentrated near the Sun. Furthermore, after each
comet passage, CO is quickly blown out by the strong Solar wind and escapes at
a speed of several au per month. So this CO source cannot in the long run
accumulate and compete with the CO production from the Kuiper belt.

Centaurs are transient bodies with a dynamical lifetime of $\sim
10^{6}-10^{7}$ yr located between Jupiter and Neptune (Tiscareno & Malhotra,
2003). They are expected to originate in the KB (Tiscareno & Malhotra, 2003).
In spite of their large $\sim$100 km size, the observable Centaurs appear CO-
depleted compared to Oort-cloud comets by a factor 10 to 50 (Wierzchos,
Womack, & Sarid, 2017). However, there are more than $10^{5}$ Centaurs larger
than about 4 km (Nesvorný et al., 2019) that may still release CO and
contribute to a global and diffuse CO gas disk mainly between 5-30 au in
addition to the CO released in the KB. An order of magnitude of the CO gas
quantity that could be released by Centaurs in steady state can be obtained as
follows. Using the size distribution given by Nesvorný et al. (2019), and
integrating over all bodies between 4 and 50 km diameter (OSSOS observation
range and in agreement with our model that only bodies larger than 4 km may
still have CO in sub-surface), we find that Centaurs have a total surface area
of $3.4\times 10^{7}$ km2, which is $3\times 10^{3}$ larger than the area of
comet Hale-Bopp.

Assuming that the release rate of Centaurs per unit area is roughly 10 times
lower than Hale-Bopp (and it can be up to 50 times, Wierzchos, Womack, &
Sarid, 2017), using that the CO release rate of Hale-Bopp at 10 au (the mean
distance of Centaur perihelia is close to 13 au but closer in for active
centaurs) is $10^{28}$ mole s-1 or $2\times 10^{-9}$ M⊕/Myr (Biver et al.,
2002), and assuming that roughly 10% of Centaurs are active (Guilbert-
Lepoutre, 2012), we find a mean CO outgassing rate of $6\times 10^{-8}$ M⊕/Myr
for Centaurs (or $\sim$ $10^{-8}$ M⊕/Myr if the release rate of Centaurs is
assumed to be 50 times lower than Hale-Bopp at the same distance, rather than
10 times). This order of magnitude shows that Centaurs could potentially
equally contribute to the CO mass loss rate as do planetesimals in the KB as
derived in this paper.

However, the CO gas released by Centaurs is also blown out by the SW much
faster than in the KB (e.g., more than 20 times because of the increased
proton density at $\sim$10 au), hence decreasing the total mass or column
density of CO as compared to that in the KB. A complete modelling of the CO
gas released by Centaurs and its evolution is complex and beyond the scope of
this paper. However, we also note that the two sources of CO gas could be
differentiated from their different spatial distributions by measuring the
column at different distances from the Sun (e.g., when an in-situ mission
similar to New Horizons with increased sensitivity moves outwards to the KB,
see next section).

##  Appendix J Detecting the gas belt

Conceivably, there are different ways of detecting this gas around the Kuiper
belt. It may not be exhaustive but the obvious possibilities would be: 1)
detection of CO emission at mm-wavelength; 2) detection in the UV (e.g. a
resonant carbon line) in absorption against a background star located in the
ecliptic; 3) in-situ detections with future missions similar to New Horizons.

First, we need to evaluate the population levels for the different lines. Due
to the low quantity of electrons we find, the collider density is probably not
enough to reach local thermal equilibrium (LTE) and the population levels
would be set by the radiation impinging onto the different species. We use a
non-LTE code developed for gas in debris disks (including fluorescent
excitation, Matrà et al., 2015, 2018a). For the radiation field, we include
the CMB and the light from the Sun using a state-of-the-art Solar spectrum
(Gueymard, 2018). As can be seen in Fig. 5, we explore the population levels
for a range of excitations and temperatures given the current uncertainties on
these. For electron densities below $\sim$ 10 cm-3 (which is the most likely
given our model), the population levels are in the radiation regime and they
converge to a given value. What is clear from this plot is that (no matter
what the gas temperature is) roughly half of the CO molecules are in the first
level and 40% in the ground state. We will use these values to make flux
predictions for different lines below.

Figure 5:  Population rotational J-levels of CO for a range of temperatures
and electron densities. The left side shows the radiation dominated regime
(which is the most likely regime for the KB) while the right side has enough
colliders to reach LTE.

Now, let us go through the different possibilities.

###  J.1 Detectability of CO rotational lines in the mm

The Planck mission was used to make Galactic maps of CO using its great
sensitivity (Planck Collaboration et al., 2014). Let us check whether any CO
gas in the KB could affect these detections and/or whether CO could be seen on
Planck maps looking towards the ecliptic. The Planck sensitivity in band 3 (CO
J=1-0 transition at 115 GHz) is roughly 1 K km/s (Planck Collaboration et al.,
2014).

Using the population levels we derived, we compute the column density of CO
that is needed to get a 3$\sigma$ detection with Planck (with eq. 9 of
Goldsmith & Langer, 1999). We find that $2\times 10^{15}$ cm-2 of CO is needed
to get a detection, which is orders of magnitude higher than the CO column
predictions ($\sim 10^{8}$ cm-2) from our model. The ALMA non-interferometric
total power array mode is more sensitive than Planck given its larger
collecting area with 4 $\times$ 12 m antennas. Pushing it to its limit, we
find that observing the KB for 1000 hours (as of now, 3000 hr per year are
devoted to this mode), we would go down to a sensitivity of 0.45 mK km/s,
which would lead to a detection for a CO column density of $10^{12}$ cm-2,
which is still much higher than our CO column predictions. However, using the
auto-correlation mode of ALMA with 50 antennas (Cordiner et al., 2020), the
sensitivity could be 50/4=12.5 times smaller and the detection threshold would
then be $\sim 10^{11}$ cm-2. Future arrays connecting more numerous and larger
antennas in their non-interferometric mode or using autocorrelation (total
power) spectra of interferometric data as recently done for observing comet
tails with ALMA (Cordiner et al., 2020) would allow to go much deeper.
However, we note that we would need to be able to go on and off on the target,
which may be difficult given the extent of the KB.

###  J.2 Detectability of carbon atoms in the UV in absorption against a
background star

We now quantify the absorption signal that would be obtained observing a
bright background star that would happen to lie nearby in the ecliptic plane
(i.e., the line-of-sight would go through the KB). We take a bright Sirius-
like star at a few pc for the example. To estimate the detectability of gas in
the KB, we target a strong C I resonant line at 1656.9284 Angstroms as other
lines will be of similar or lower strength. We take the Einstein coefficients
from the NIST database and compute the optical depth of the line (using eq. 3
of Matrà et al., 2017). We assume a FWHM of the line of about 10 km/s and
using the same non-LTE code as described for CO, we find that the ground state
level for carbon is populated at the 99.9% level. We then find an optical
depth $\tau_{\nu}$ of order $10^{-5}$ for this line.

For an optically thin gas, the flux density of the line is
$I_{\nu}=I_{\nu,{\rm bkg}}\exp(-\tau_{\nu})$, where $I_{\nu,{\rm bkg}}$ is the
background star flux density. The signal-to-noise ratio of the star (SNR) is
$I_{\nu,{\rm bkg}}/\sigma$, where $\sigma$ is the noise level. For a line
detection, we need $I_{\nu,{\rm bkg}}-I_{\nu}>3\sigma$ at the line centre,
which implies an SNR $>3/\tau_{\nu}=3\times 10^{5}$.

To check whether HST/STIS could detect such a faint signal, we use their
Exposure Time Calculator. For a Sirius-like star, we find that the star is too
bright to be observed directly with STIS. After using an ND filter, we find
that we can reach a S/N of order 100 (for 2 hours exposure). However, to get a
detection here, we would need a S/N of order $10^{5}$, which is too much for
HST and could only be tackled by future instruments.

###  J.3 Detectability of gas with an in-situ mission

Our Solar System has the advantage over exoplanetary systems in that we can
send probes to study its complexity. One such probe is the recent New Horizons
mission, which was dedicated to study Pluto (and its satellites) as well as a
Kuiper belt object (named Arrokoth).

To detect CO with these in-situ missions, the way to go is to look for
absorption of species against the Sun or to look for emission of resonance
lines (Gladstone et al., 2016). For absorption of CO, its ground state has a
complex absorption cross-section at wavelengths $<1000$ Angstroms (Masuoka &
Samson, 1980) at around $10^{-17}$ cm2. With an equivalent of the Alice (UV
imaging spectrometer) instrument that is onboard New Horizons, a drop in
brightness of $\sim$ 1% could be detected for a long exposure (Gladstone et
al., 2021). Therefore, CO column densities between the instrument and the Sun
of the order of $10^{15}$ cm-2 may be detected with this technique. This is
clearly not enough to detect the CO level predicted by our model.

For some species (mostly atoms), resonance cross-sections can be several
orders of magnitude larger than for absorption, which can allow detections of
much lower levels of gas in the KB. We now quantify the emission from
resonance line scattering for the OI triplet at 1304 Angstrom. To get an order
of magnitude of the oxygen upper limit, we consider that there is no
background emission and a detection with a signal-to-noise ratio of 3 would
require about $C=10$ counts in the wavelength bin of the emission. For a
brightness $I=gN$ (in photons/cm2/s/($4\pi$ sr), with $g$ the number of
photons scattered per unit time and per atom and $N$ the column density of
neutral oxygen), the number of counts in a time $t$ is equal to (Meier, 1991)

$C=gN(\Omega/(4\pi))A_{\rm eff}t,$

where $\Omega$ is the smallest of the solid angles of the target or detector
pixel, and $A_{\rm eff}$ the effective area of the instrument (the aperture
area times all the yields and reflectivites) at a given wavelength (Stern et
al., 2008). The pixel size of, e.g., the Alice instrument onboard New Horizons
is much smaller than the KB gas disk solid angle so that we use the Alice
pixel size of $0.1\times 0.3$ degrees2 to get $\Omega=9.1\times 10^{-6}$ sr.
At the wavelength of the OI 1304 triplet, the effective area is $A_{\rm
eff}=0.17$ cm2 (Stern et al., 2008). The $g$ factor (see Table IV of Meier,
1991) is rescaled at 45 au to get $g=9.4\times 10^{-10}$ s-1. Integrating the
number of counts for 1000 h, we get the upper limit $N=2.4\times 10^{10}$
cm-2. It is indeed much more promising than for absorption but given our
column density predictions for neutral oxygen ($2\times 10^{7}$ cm-2), this is
not doable for now with e.g. Alice. Only future in-situ instruments with
larger effective apertures and larger pixel sizes could reduce this time: a
super-Alice is needed.

For instance, if we use a 7 deg $\times$ 0.3 deg MCP (micro channel plate) for
the detector instead of the current 7 deg $\times$ 0.1 deg, we gain a factor
3, with a resolution that is still sufficient to distinguish lines of interest
(e.g., OI at 130.4 nm). Summing over all pixels, instead of just on the
$0.1\times 0.3$ degree2 pixel mentioned above (which is possible because the
emission region is larger than the MCP size), the gain factor becomes 70 as
$\Omega$ becomes $6.4\times 10^{-4}$. The effective area $A_{\rm eff}$ can
also be improved by a factor $\sim 3$ using MgF2-coated optics (as on the Juno
mission, which possess an Alice-like spectrograph, Davis et al., 2011;
Greathouse et al., 2013). All together, and without increasing the aperture
size, we gain a factor 200. Using a larger primary would also allow to collect
photons faster and go down in sensitivity. For instance, going to a 6 times
larger primary (24x24cm) would gain another factor 36 (current airglow
aperture is $4\times 4$ cm2, Stern et al., 2008), i.e., in total a $\sim$7000
gain factor, which is enough (with some slack to account for model
uncertainties) to detect the oxygen predicted by our model in the KB with a
reasonable exposure time. This type of super-Alice instrument could be planned
with current technology and may fly in the future. We also note that if
planetesimals in the KB also contain O2 in similar quantity to CO as may be
the case in the comet 67 P, our overall prediction for the OI line would also
increase by a factor of a few.


* Yeomans et al. (2019) Michael Yeomans, Anuj Shah, Sendhil Mullainathan, and Jon Kleinberg. 2019.  Making sense of recommendations.  _Journal of Behavioral Decision Making_ 32, 4 (2019), 403–414.
  * Yin et al. (2019) Ming Yin, Jennifer Wortman Vaughan, and Hanna Wallach. 2019\.  Understanding the effect of accuracy on trust in machine learning models. In _Proceedings of the 2019 chi conference on human factors in computing systems_. 1–12.
  * Yu et al. (2019) Kun Yu, Shlomo Berkovsky, Ronnie Taib, Jianlong Zhou, and Fang Chen. 2019.  Do i trust my machine teammate? an investigation from perception to decision. In _Proceedings of the 24th International Conference on Intelligent User Interfaces_. 460–468.
  * Zhang et al. (2020a) Linkun Zhang, Yuxia Lei, and Zhengyan Wang. 2020a.  Long-Text Sentiment Analysis Based on Semantic Graph. In _2020 IEEE International Conference on Embedded Software and Systems (ICESS)_. IEEE, 1–6.
  * Zhang et al. (2023) Shao Zhang, Jianing Yu, Xuhai Xu, Changchang Yin, Yuxuan Lu, Bingsheng Yao, Melanie Tory, Lace M Padilla, Jeffrey Caterino, Ping Zhang, et al. 2023\.  Rethinking human-ai collaboration in complex medical decision making: A case study in sepsis diagnosis.  _arXiv preprint arXiv:2309.12368_ (2023).
  * Zhang et al. (2020b) Yunfeng Zhang, Q Vera Liao, and Rachel KE Bellamy. 2020b.  Effect of confidence and explanation on accuracy and trust calibration in AI-assisted decision making. In _Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency_. 295–305.
  * Zhang et al. (2022) Zheng Zhang, Ying Xu, Yanhao Wang, Bingsheng Yao, Daniel Ritchie, Tongshuang Wu, Mo Yu, Dakuo Wang, and Toby Jia-Jun Li. 2022.  StoryBuddy: A Human-AI Collaborative Chatbot for Parent-Child Interactive Storytelling with Flexible Parental Involvement.  _CHI’2022_ (2022).
  * Zunic et al. (2020) Anastazia Zunic, Padraig Corcoran, and Irena Spasic. 2020\.  Sentiment analysis in health and well-being: systematic review.  _JMIR medical informatics_ 8, 1 (2020), e16023.