# Systematics and Consequences of Comet Nucleus
Outgassing Torques

David Jewitt1,2 1Department of Earth, Planetary and Space Sciences, UCLA, 595
Charles Young Drive East, Los Angeles, CA 90095-1567
2Department of Physics and Astronomy, University of California at Los Angeles,
430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547
<EMAIL_ADDRESS>
###### Abstract

Anisotropic outgassing from comets exerts a torque sufficient to rapidly
change the angular momentum of the nucleus, potentially leading to rotational
instability. Here, we use empirical measures of spin changes in a sample of
comets to characterize the torques and to compare them with expectations from
a simple model. Both the data and the model show that the characteristic spin-
up timescale, $\tau_{s}$, is a strong function of nucleus radius, $r_{n}$.
Empirically, we find that the timescale for comets (most with perihelion 1 to
2 AU and eccentricity $\sim$0.5) varies as $\tau_{s}\sim 100r_{n}^{2}$, where
$r_{n}$ is expressed in kilometers and $\tau_{s}$ is in years. The fraction of
the nucleus surface that is active varies as $f_{A}\sim 0.1r_{n}^{-2}$. We
find that the median value of the dimensionless moment arm of the torque is
$k_{T}$ = 0.007 (i.e. $\sim$0.7% of the escaping momentum torques the
nucleus), with weak ($<$3$\sigma$) evidence for a size dependence $k_{T}\sim
10^{-3}r_{n}^{2}$. Sub-kilometer nuclei have spin-up timescales comparable to
their orbital periods, confirming that outgassing torques are quickly capable
of driving small nuclei towards rotational disruption. Torque-induced
rotational instability likely accounts for the paucity of sub-kilometer short-
period cometary nuclei, and for the pre-perihelion destruction of sungrazing
comets. Torques from sustained outgassing on small active asteroids can rival
YORP torques, even for very small ($\lesssim$1 g s-1) mass loss rates.
Finally, we highlight the important role played by observational biases in the
measured distributions of $\tau_{s}$, $f_{A}$ and $k_{T}$.

††slugcomment: Revised 2021 March 17

##  1 INTRODUCTION

The dynamical lifetimes of short-period comets are about 0.5 million years,
some $10^{-4}$ of the age of the solar system, while the physical lifetimes
are at least an order of magnitude shorter still (Levison and Duncan 1997).
Several processes potentially limit the physical lifetimes, including complete
devolatilization of the nucleus, formation of a global, refractory mantle that
stifles outgassing, and rotational disruption from outgassing torques
(Samarasinha 1986, Jewitt 1992, 1997). Knowledge of the physical lifetimes is
important both for understanding the populations of the Kuiper belt and Oort
cloud source reservoirs (with shorter lifetimes requiring larger source
populations in order to maintain steady-state) and for understanding the
evolutionary properties of comets when in the terrestrial planet region.

Reaction forces from sublimation exert a torque that can change both the
magnitude and the direction of the spin of a cometary nucleus (Whipple 1961).
It has long been noticed that the characteristic timescale for changing the
spin can be very short (Samarasinha 1986, Jewitt 1992, 1997), and that the
lifetimes of cometary nuclei when active may be determined by spin-up to the
point of rotational disruption. Originally proposed in the near absence of
relevant rotational and physical data on cometary nuclei, we now possess a
better physical characterization of the comets and several reliable
measurements of nucleus spin changes that can be used to better-define the
spin-up process.

Spin-up has been discussed in the refereed literature by Jewitt (1997),
Gutierrez et al. (2003), Samarasinha and Mueller (2013), Steckloff and
Jacobson (2016), Mueller and Samarasinha (2018), Kokotanekova et al. (2018),
Rafikov (2018), and Steckloff and Samarasinha (2018). In this paper we first
recap the simple spin-up model of Jewitt (1997), then describe recent
measurements to establish the empirical nucleus spin-up timescale as a
function of nucleus parameters. We then consider the consequences of this
timescale for the spin evolution of outgassing cometary nuclei. Finally, we
discuss the role of observational bias.

##  2 SCALING RELATIONS

A torque applied to a rotating nucleus changes the vector angular momentum,
resulting in both excited (non-principal axis) rotation and a changing spin
period. Excited rotational states have been reported, perhaps most beautifully
in 1P/Halley (Samarasinha and A’Hearn 1991). However, non-principal axis
rotation is generally much more difficult to detect than changes in the
magnitude of the spin, manifested as time-dependence in the period deduced
from lightcurve photometry (Gutierrez et al. 2003). Therefore, we focus on the
effect on the spin rate of a torque exerted by non-uniform mass loss. We write
the scalar torque as

$T=k_{T}(r_{n})V_{th}r_{n}\overline{\dot{M}}$ (1)

where $r_{n}$ is the radius, $\overline{\dot{M}}$ [kg s-1] is the average rate
of mass loss, $V_{th}$ is the speed with which material is ejected and
$k_{T}(r_{n})$ is the “dimensionless moment arm”. The momentum is dominated by
outflowing gas and, therefore, $\overline{\dot{M}}$ and $V_{th}$ refer to the
gas production rate and speed, respectively, and momentum in the dust is
ignored. Quantity $k_{T}$ is equal to the fraction of the outflowing momentum
that exerts a torque on the nucleus. The limiting values are $k_{T}$ = 0,
corresponding to isotropic ejection with no net torque and $k_{T}$ = 1,
corresponding to collimated ejection in a direction tangent to the surface.
The spin angular momentum is $L=I\omega$, with $I$ equal to the moment of
inertia and angular speed of the rotation $\omega=2\pi/P$, where $P$ is the
instantaneous rotation period. The shapes of cometary nuclei are typically
irregular and $I$ cannot be generally defined. For simplicity, we represent
the nucleus by a homogeneous sphere, for which $I=(2/5)M_{n}r_{n}^{2}$, where
$M_{n}=(4/3)\pi\rho_{n}r_{n}^{3}$ is the nucleus mass and $\rho_{n}$ is the
nucleus density. Equivalently,

$L=(8\pi/15)\rho_{n}r_{n}^{5}\omega.$ (2)

Then, defining the characteristic timescale for spin-up by the torque as
$\tau_{S}=L/T$, we obtain from Equations 1 and 2 (c.f. Jewitt 1997)

$\tau_{s}=\left(\frac{16\pi^{2}}{15}\right)\left(\frac{\rho_{n}r_{n}^{4}}{k_{T}(r_{n})V_{th}P}\right)\left(\frac{1}{\overline{\dot{M}}}\right).$
(3)

As noted above, $\tau_{s}$ and $P$ in Equation (3) can be extracted from
lightcurve observations (e.g. Kokotanekova et al. 2018). The other parameters
in Equation (3) deserve comment, as follows.

Density, $\rho_{n}$: Only the density of the nucleus of 67P/Churyumov-
Gerasimenko has been directly measured. Published values for this, and for a
range of nuclei studied using less direct techniques, are compatible with
$\rho_{n}$ = 500 kg m-3 (Groussin et al. 2019), which we adopt here.

Speed, $V_{th}$: The momentum of the ejected material originates in the
thermal motions of gas produced by sublimated cometary ice. To first order, we
take the speed of the sublimated gas as the mean thermal speed,
$V_{th}=(8kT/(\pi\mu m_{H}))^{1/2}$, where $k=1.38\times 10^{-23}$ J K-1 is
the Boltzmann constant, $T$ is the temperature of the sublimating surface,
$\mu$ is the molecular weight and $m_{H}=1.67\times 10^{-27}$ kg is the mass
of the hydrogen atom. Setting $\mu$ = 18 for water, the dominant cometary
volatile, and $T$ = 330 K for the hemispheric temperature at 1 AU, we obtain
speed $V_{th}\sim$ 677 m s-1. At 2 AU, we find $T$ = 233 K and $V_{th}$ = 522
m s-1. The distance dependence of the speed is weak (because $V_{th}\propto
T^{1/2}$ and $T\propto r_{H}^{-1/2}$), a fact confirmed by high resolution
spectroscopic measurements giving $V_{th}\propto r_{H}^{-1/4}$ over the range
1 $\leq r_{H}\leq$ 8 AU (Biver et al. 2002). Sublimation depresses the
temperature below the local blackbody value to $T$ = 205 K near $r_{H}$ = 1
AU, corresponding to $V_{th}$ = 490 m s-1. Noting the narrow range of
heliocentric distances (1 $\lesssim r_{H}\lesssim$ 2 AU) over which most of
the comets of this study were observed, we neglect any heliocentric variation
and set $V_{th}$ = 500 m s-1, which is within a factor $\sim$2 of speeds
measured in cometary gas in this distance range (Biver et al. 2002).

Mass Loss Rates, $\dot{M}$: In the sublimation hypothesis we expect that
activity should be proportional to the nucleus surface area, and write

$\overline{\dot{M}}=4\pi f_{A}(r_{n})\overline{f_{s}(r_{H})}r_{n}^{2},$ (4)

where $4\pi r_{n}^{2}f_{A}(r_{n})$ is the sublimating area of the nucleus,
assumed spherical, and $\overline{f_{s}(r_{H})}$ is the orbitally-averaged
sublimating mass flux (kg m-2 s-1) calculated from the energy balance equation
as described in the Appendix. Quantity $f_{A}$ is known as the “active
fraction”, equal to the ratio of the sublimating area to the surface area of a
sphere having the radius $r_{n}$.

Combining Equations (3) and (4), we have

$\tau_{s}=\left(\frac{4\pi}{15}\right)\left(\frac{\rho_{n}r_{n}^{2}}{k_{T}(r_{n})f_{A}(r_{n})V_{th}P}\right)\left(\frac{1}{\overline{f_{s}(r_{H})}}\right)$
(5)

showing that we should expect the characteristic spin-up timescale to vary as
$\tau_{s}\propto r_{n}^{2}$, but only if $f_{A}k_{T}P$ in the denominator is
independent of $r_{n}$. Period $P$ is measured for each nucleus in this study.
In the next section we calculate $\tau_{s}$, $k_{T}$ and $f_{A}$ from
published data to compare with this expectation.

##  3 EMPIRICAL RELATIONS

###  3.1 Spin-Up Timescale, $\tau_{s}$

The first reviews of cometary nucleus rotation (Sekanina 1981, Whipple 1982)
were published before useful rotation data were available and, as a result,
are mainly of historical interest. The first reliable measurements of a
cometary nucleus rotational lightcurve were those of 49P/Arend-Rigaux,
obtained in the mid-1980s (A’Hearn et al. 1985, Jewitt and Meech 1985). Before
that time it was widely held that the nucleus could not be directly detected
in ground-based observations; the study of low activity comets like 49P/Arend-
Rigaux revealed this belief to be unfounded. However, it remains true that
rotational lightcurves can be directly determined in relatively few comets
because of photometric contamination by coma. Unlike the asteroids, comets
usually exhibit a diffuse appearance due to outgassed material in the coma,
resulting in the dilution of the nucleus rotational lightcurve to unobservable
levels. In some active objects, however, periodic structures including jets
and spirals in the coma can be used to infer the rotation period of the
nucleus, even though the nucleus itself cannot be photometrically isolated
(e.g. Samarasinha and A’Hearn 1991).

In this work, we use only published measurements of nucleus rotation and
rotation changes, for which Kokotanekova et al. (2018) presented a convenient
summary. These authors list (in their Table 2) the measured change in the
rotation period per cometary orbit, $|\Delta P|$, which is related to the
spin-up timescale, $\tau_{s}$, by

$\tau_{s}=\frac{P}{|\Delta P|}P_{K}$ (6)

where $P$ is the measured instantaneous rotation period of the nucleus and
$P_{K}$ is the Keplerian orbital period ($P_{K}=a^{3/2}$, with $P_{K}$ in
years and orbital semimajor axis $a$ in AU). We add rotational measurements of
46P/Wirtanen using data from Farnham et al. (2021), but ignored two, earlier
measurements of this object by Meech et al. (1997) and Lamy et al. (1998)
because their results were discordant yet nearly simultaneous. Exclusion of
46P/Wirtanen from our sample would not change any of the following results.
The measurements are listed in Table (1).

Equation (6) gives a measure of how long the nucleus would take to change from
stationary to its present rotation period, assuming that the orbitally
averaged torque is constant. In most comets, the period drifts slowly, and the
reported period changes are noticed only when comparing determinations made in
different orbits. In comets 41P/Tuttle-Giacobini-Kresak, 46P/Wirtanen and
103P/Hartley, the rotational period varies so quickly that the rate of change,
$dP/dt$, can be measured within a single orbit (Drahus et al. 2011, Knight et
al. 2015, Bodewits et al. 2018, Moulane et al. 2018, Schleicher et al. 2019,
Farnham et al. 2021). Note that, while $\Delta P$ can be positive or negative,
and a given nucleus can be either spinning up or spinning down, we are
interested only in the magnitude of the change, $|\Delta P|$.

Figure (1) shows $\tau_{s}$ as a function of nucleus radius, $r_{n}$, computed
from Equation (6) and the data of Table (1), with illustrative error bars
showing the effect of $\pm$50% uncertainties in $\tau_{s}$. Evidently,
$\tau_{s}$ varies widely in the range $\tau_{s}\sim$ 3 yr (for the very
rapidly accelerating nucleus of 46P/Wirtanen) to $\tau_{s}\gtrsim 10^{4}$ yr
(for 10P/Tempel 2 and 49P/Arend-Rigaux). As a purely empirical diagram, the
figure shows a convincing, model-independent trend for larger values of
$\tau_{s}$ to be associated with larger cometary nuclei, as expected based on
scaling relations (Equation 5, Jewitt 1997) and noted by Samarasinha & Mueller
(2013) and Kokotanekova et al. (2018). It is obvious from the figure that
$\tau_{s}$ and $r_{n}$ are related. Although numerical evidence of this is not
needed, we computed the Spearman $\rho$ correlation coefficient (Press et al.
1992) between log($\tau_{s}$) and log($r_{n}$), finding $r_{s}$ = 0.88 and a
$p$-value of 0.004, indicating a significant correlation. A least-squares fit
of a power law to the comets having non-zero $|\Delta P|$ (red circles in
Figure 1) gives $\tau_{s}=(102\pm 50)r_{n}^{2.2\pm 0.6}$. However, the
significance of the fit should not be exaggerated (the sample is small, the
uncertainties are poorly characterized, and we have ignored uncertainties in
the radii of the comets plotted in Figure 1). For convenience, we simply adopt

$\tau_{s}\sim 100r_{n}^{2}$ (7)

with $\tau_{s}$ expressed in years and $r_{n}$ in kilometers, in the remainder
of this paper, and point to Figure (1) to show that this gives an acceptable
match to the data. It should be understood that this equation strictly applies
to short-period comets with moderate eccentricities and perihelia near 1 AU
(as indicated in Table 1). Timescales for comets of a given size having
different orbital semimajor axes and perihelia would not be fitted by Equation
(7).

###  3.2 Active Fraction, $f_{A}$

Cometary mass loss is driven by the expansion of sublimated gas, the
production rates of which are estimated from the strengths of resonance
fluorescence bands, using a model of the gas spatial distribution. Typically,
the Haser (1957) model, or one of its variants, is used to infer the
production rate from spectroscopic data. In most comets in the terrestrial
planet region, the gas mass is dominated by sublimated water. Accordingly, we
use $\dot{M}=\mu m_{H}Q_{OH}$, where $\dot{M}$ is the mass production rate (kg
s-1), molecular weight $\mu$ = 18, and $Q_{H_{2}O}$ (s-1) is the production
rate, most usually obtained from measures of the OH 3090Å (e.g. A’Hearn et al.
1995) or Lyman $\alpha$ (Combi et al. 2019) bands. Production rates can also
be inferred from the strength of other gas species and even from the cometary
continuum, but these are less reliable than OH production rates because of
uncertainties in the relative abundances of species and of dust. We do not use
other species or dust measurements of production here. We do note that, in
many comets, the derived instantaneous dust mass production rates are larger
than the gas mass production rates (i.e. the ratio dust/gas $>$ 1).
Physically, though, dust speeds are small compared to the gas speed and the
outflow momentum is necessarily dominated by the gas. For these reasons, we
here use only measurements of the gas production rates, and neglect the
momentum carried by solids.

In order to examine $f_{A}(r_{n})$ we combined active area determinations from
the spectroscopic compilation by A’Hearn et al. (1995) with Hubble Space
Telescope-based nucleus radius measurements from Lamy et al. (2004), to find
24 short-period comets common to both datasets (Table 2). We added 126P/IRAS
from Groussin et al. (2004) to make a sample of 25. The use of two main
sources reduces relative errors in the production rates introduced by
different models and interpretations of the data. Unavoidable systematic
errors remain, however, notably from the unmeasured albedos and phase
functions of the comets (however, infrared data examined by Fernandez et al.
(2013) suggest that albedo is not a strong function of nucleus radius). For
this reason, individual values of $f_{A}$ may differ somewhat from those
reported by others in the literature. As an example, consider 103P/Hartley. We
find (Table 2) $f_{A}$ = 0.60, whereas Groussin et al. (2004) reported
$0.3\lesssim f_{A}\lesssim$ 1 and Lisse et al. (2009) reported $f_{A}$ = 1.1.
Values $f_{A}>1$ are occasionally reported in some so-called “hyper-active
comets”, of which 103P/Hartley is one. In such cases, the sublimation is
presumed to come from grains in the coma, rather than from the nucleus
directly. Such grains cannot torque the nucleus.

The dependence of $f_{A}$ on $r_{n}$ for these comets is plotted in Figure
(2), where a strong inverse relation is evident. The Spearman $\rho$
coefficient computed between log($f_{A}$) and log($r_{n}$) has value $r_{s}$ =
-0.53 and a correspondingly low probability of being due to chance of $p$ =
0.008. A least squares fit to all the data gives $f_{A}=0.15\pm
0.06r_{n}^{-2.05\pm 0.47}$. A fit to the eight objects having measured spin
changes gives $f_{A}=0.22\pm 0.08r_{n}^{-2.61\pm 0.52}$. The absolute
uncertainties may be larger than indicated, and dominated by systematic
effects intrinsic to both the measurements and their interpretation. For
example, the “Haser” model gives a simplistic representation of the gas coma
and production rates, with uncertainties which are both systematic and
difficult to characterize. The effective sublimating area is estimated through
the adoption of a thermophysical sublimation model whose parameters are
themselves numerous and uncertain. In addition, while there is no reason to
remove the three largest nuclei from the plot, the effect of doing so would be
to render $f_{A}$ independent of $r_{n}$, within the uncertainties111All three
of the largest nuclei have $f_{A}<10^{-2}$, whereas this is true of only 2 of
22 (9%) of the smaller nuclei. Assuming this same fraction, the chance of
finding the three largest nuclei with $f_{A}<10^{-2}$ is 0.09${}^{3}\sim
7\times 10^{-4}$, which is significant at the $>$3$\sigma$ level of
confidence. . For all these reasons, and in order to avoid giving the
appearance of undue significance to the relation in Figure (2), we elect
merely to note that the variation resembles the power law

$f_{A}\sim 0.1r_{n}^{-2}$ (8)

with $r_{n}$ in km. Figure (2) shows that Equation (8) is a useful
representation of the data. Equation (8) applies only for $f_{A}\leq$ 1, which
is true for $r_{n}\gtrsim$ 0.3 km. Smaller nuclei should be entirely active
($f_{A}$ = 1) over their surfaces by this relation.

###  3.3 Moment Arm, $k_{T}$

We are interested to determine the dimensionless moment arm for the torque,
$k_{T}$, since this quantity allows $\tau_{s}$ to be estimated through
Equation (3) for any nucleus. Substituting Equation (6) into (3) and solving
for $k_{T}$ we find

$k_{T}=\frac{16\pi^{2}}{15}\left(\frac{\rho_{n}r_{n}^{4}}{V_{th}}\right)\left(\frac{|\Delta
P|}{P^{2}P_{K}}\right)\left(\frac{1}{\overline{\dot{M}}}\right)$ (9)

In Equation (9), $r_{n}$, $|\Delta P|$ and $P$ are measured quantities
obtained from nucleus photometry and/or periodic coma structures that are
modulated by nucleus rotation, while $P_{K}$ is the orbital period. The mean
mass loss rate, $\overline{\dot{M}}$, is obtained from measurements of
resonance fluorescence band strengths, focusing on the OH 3090Å band as a
measure of the production rate of the dominant volatile, H2O. For practical
reasons, measurements of cometary spectra tend to be taken near 1 to 2 AU.
Comets at distances $r_{H}\ll$ 1 AU appear at small elongations and are
difficult to measure. Most comets at $r_{H}\gg$ 1 AU sublimate weakly and
appear faint. For these reasons, most of the spectroscopically well-observed
comets (Table 1) have perihelia $q\sim$ 1 to 2 AU.

The torque on the nucleus is maximized at perihelion where outgassing is
strongest, but the total torque results from the total mass loss integrated
around the orbit. Accordingly, we apply a correction factor, $\mathcal{S}$, to
properly scale the mass loss rate measured at distance $r_{H}$,
$\dot{M}(r_{H})$, to estimate the orbitally averaged mean mass loss rate,
$\overline{\dot{M}}$, that would be measured if we possessed spectroscopic
data around the orbit. We define the scaling factor, $\mathcal{S}$ using

$\mathcal{S}=\frac{\overline{\dot{M}}}{\dot{M}(r_{H})}$ (10)

where $\overline{\dot{M}}$ is the mass loss rate averaged over one orbit
period, $P_{K}$. The calculation of $\mathcal{S}$ is described in the
Appendix. An obvious objection to the calculation of $\mathcal{S}$ is that the
orbital variation of $\dot{M}$ might not be well represented by the
sublimation model described in the Appendix. For example, seasonal variations
on comets with non-zero obliquity create important pre- vs. post-perihelion
asymmetries, but cannot be incorporated in the model. We acknowledge this
weakness and look forward to future gas production rate measurements taken
more densely at a range of locations around the orbit.

For three of the comets in Table (1) (namely 14P/Wolf, 143P/Kowal-Mrkos,
162P/Siding Spring) we possess upper limits to the period change, $|\Delta
P|$, but a literature search revealed no measurements of the mass loss rates.
Therefore, the moment arm for these three comets cannot be obtained from
Equation (9) and we exclude them from further consideration. Conversely, while
only an upper limit to the period change was set in 49P/Arend-Rigaux, the mass
loss rate has been quantified, and so we retain this plus seven other, better-
measured comets in our sample to determine $k_{T}$ (Table 1).

Values of $\mathcal{S}$ are listed for each nucleus in Table (1). The orbits
of the well-characterized comets are clustered near $q\sim$ 1 to 1.5 AU and
$e\sim$ 0.6, for which typical values are $\mathcal{S}\sim$ 0.1. This means
that the sublimation rate averaged around the orbit of most comets is of order
10% of the rate measured at perihelion. 2P/Encke has a smaller $q$ and larger
$e$, resulting in $\mathcal{S}\sim$ 0.034.

We use Equations (9) and (10) to calculate $k_{T}$ for each nucleus. The
resulting values are listed in the penultimate column of Table (1). Values of
the moment arm range from 2$\times 10^{-4}$ to 4$\times 10^{-2}$; the median
value, $k_{T}$ = 0.007, is an order of magnitude smaller than deduced (before
observations) from an early toy model ($k_{T}$ = 0.05, Jewitt 1997). Our value
for 9P/Tempel ($k_{T}$ = 0.006) compares with the range 0.005$\leq k_{T}\leq$
0.04 found by Belton et al. (2011). Our value for 103P/Hartley ($k_{T}$ =
4$\times 10^{-4}$) is consistent with $k_{T}=4\times 10^{-4}$ from Drahus et
al. (2011).

The data provide some evidence that $k_{T}$ and $r_{n}$ are correlated (Figure
3). The Spearman $\rho$ correlation coefficient between log($k_{T}$) and
log($r_{n}$) is $r_{s}$ = 0.81, with a probability that this or a larger value
could be obtained by chance of $p$ = 0.01. The observed correlation is thus
not statistically significant at the 3$\sigma$ ($p$ = 0.005) level. A power
law fit to the data gives $k_{T}=(1.4\pm 0.8)\times 10^{-3}r_{n}^{1.6\pm
0.5}$. The equation

$k_{T}\sim 10^{-3}r_{n}^{2}$ (11)

adequately represents the data over the range 0.5 $\lesssim r_{n}\lesssim$ 7
km (Figure 3). The maximum possible value, $k_{T}=1$, is reached at
$r_{n}\sim$ 30 km, which is larger than any well-measured cometary nucleus.

###  3.4 Bias Effects

We identify two sources of bias likely to affect determinations of
$f_{A}(r_{n})$. First, most comets are discovered by magnitude-limited
surveys, leading to a “discovery bias” acting against low activity (small
$f_{A}$) comets of a given size. Small nuclei with small $f_{A}$ will be
preferentially undercounted relative to high activity (large $f_{A}$) comets
of equal size because they are fainter. The discovery bias is particularly
acute for small nuclei, potentially pushing such objects with small active
fractions beneath the survey detection threshold.

Second, the determination of $f_{A}$ relies on spectroscopic measurements of
resonance fluorescence bands in gas. These bands are weak in low activity
comets of a given size, which therefore constitute more difficult and less
appealing spectroscopic targets than bright comets (i.e. those with large
active areas). Small nuclei with small $f_{A}$, even if they survive the
discovery bias, are thus likely to be under-reported in spectroscopic surveys
(e.g. A’Hearn et al. 1995, Combi et al. 2018) of cometary activity,
constituting a “spectroscopy bias”.

To examine this effect, we compiled the cumulative distribution of water
production rates from the data listed by A’Hearn et al. (1995), the primary
source of our activity data in Table (2). The distribution shows a change in
slope for $Q_{OH}\lesssim 10^{27}$ s-1, corresponding to about $\dot{M}$ = 30
kg s-1. The local water ice sublimation rate at 1.5 AU is $f_{s}=8\times
10^{-5}$ kg m-2 s-1, corresponding to a sublimating area $\dot{M}/f_{s}\sim$
0.4 km2, equal to the projected area of a circle of radius 0.35 km. This is
consistent with the observation that small nuclei tend to be the most active,
because sub-kilometer nuclei could not produce enough water to be
spectroscopically detected in the A’Hearn survey if $f_{A}\ll$ 1.

Quantitatively, Equations (4) and (8) show that the mass loss rate,
$\dot{M}\propto f_{A}r_{n}^{2}$, is approximately independent of $r_{n}$,
consistent with sublimation from a fixed active area (not fraction).
Substitution into these equations gives $\dot{M}\sim 100$ kg s-1,
corresponding to $Q_{OH}\sim 3\times 10^{27}$ s-1 for a comet sublimating from
the dayside hemisphere at representative distance $r_{H}$ = 1.5 AU (c.f. Table
1). This is close to the limit of the spectroscopic data summarized in A’Hearn
et al. (1995), only 10% of which have $Q_{OH}<2\times 10^{27}$ s-1
($\dot{M}\sim$ 60 kg s-1). Numerous, substantially less productive comets
surely exist but are not spectroscopically attractive targets and are
therefore under-reported.

A different bias likely plays a role in the distribution of the moment arm,
$k_{T}$. A small nucleus with a large $k_{T}$ would, by Equation (3), have a
small spin-up time, leading to rotational instability and the removal of the
nucleus from the observable population. This “survival bias” results in an
observational sample that is naturally depleted of small nuclei having large
values of the dimensionless moment arm, because these nuclei are less likely
to survive (c.f. Drahus et al. 2011). The upper left portion of Figure (3) is
presumably depleted of objects for this reason. Conversely, large nuclei, even
if $k_{T}$ = 1, would take a long time to spin-up under the action of
outgassing torques and so are less susceptible to the survival bias.

The existence of these bias effects does not eliminate the possibility that
there are real size dependencies in $f_{A}$ and $k_{T}$. For example, larger
nuclei may be better able to retain refractory surface mantles than smaller
nuclei because of their larger surface gravity, resulting in more complete
blockage of the gas flow and depressing $f_{A}$ (Rickman et al. 1990). Large
nuclei are also more efficient in the recapture of slowly ejected material
that could build a rubble mantle (Jewitt 2002). Samarasinha and Mueller (2013)
suggested that torques from multiple sources on highly active (large $f_{A}$)
nuclei should more nearly cancel out than on weakly active (small $f_{A}$)
nuclei. This would lead to small nuclei having small $k_{T}$, as is suggested
by Figure (3). Unfortunately, we do not yet possess information sufficient to
distinguish such effects from those due to the detection, spectroscopic and
survival biases.

##  4 CONSEQUENCES

###  4.1 Paucity of Small Nuclei

Equations (3) and (7) show that small nuclei are particularly susceptible to
outgassing torques and therefore to potential rotational breakup (Jewitt 1992,
1997, Samarasinha 2007), consistent with the observed paucity of small nuclei
(Fernandez et al. 2013, Bauer et al. 2017). Indeed, the spins of nuclei
smaller than a critical radius, $r_{c}\sim$ 0.1 to 0.3 km, can be
substantially modified within a few orbits. The observed fragmentation of the
small nucleus of 332P/Ikeya-Murakami (radius $r_{n}\leq$ 0.28 km) is a
particular example of a sub-kilometer nucleus likely suffering rotational
instability. Perhaps not coincidentally its rotation period, $P$ = 2 hour, is
very short (Jewitt et al. 2016). Rapid period changes observed in the small
nuclei of comets 41P/Tuttle-Giacobini-Kresak (radius 0.7 km, Bodewits et al.
2018, Schleicher et al. 2019), 46P/Wirtanen (0.6 km, Farnham et al. 2021) and
103P/Hartley (0.6 km, Drahus et al. 2011) also indicate strong torques and
incipient rotational instability.

In addition to potential destruction by spin-up, a spherical nucleus of mass
$M=4\pi\rho_{n}r_{n}^{3}/3$ also experiences the loss of volatiles. The true
timescale for devolatilization, $\tau_{dv}$, is an intractable function of the
time-varying active fraction, $f_{A}$, and the dynamical evolution of the
comet, with some evidence that these two are interconnected (Rickman et al.
1990). A crude estimate may be obtained from $\tau_{dv}\sim
M/\overline{\dot{M}}$, with $\overline{\dot{M}}$ given by Equation (4), giving

$\tau_{dv}\sim\frac{\rho_{n}r_{n}}{3f_{A}\overline{f_{s}}}.$ (12)

We compare $\tau_{dv}$ with $\tau_{s}$ as a function of nucleus radius in
Figure (4), which updates Figure 2 from Jewitt (1997) to incorporate the new
findings about the radius dependence of $f_{A}$ (Equation 8) and $k_{T}$
(Equation 11). We computed the orbitally-averaged $\overline{f_{s}}$ for
hemispheric sublimation from comets having perihelion $q$ = 1.5 AU and
eccentricity $e$ = 0.5, representative of those in Table (1), finding
$\overline{f_{s}}=2\times 10^{-5}$ kg m-2 s-1. We set $P$ = 15 hours, this
being the median period from Table (1) and we assume $f_{A}=0.1r_{n}^{-2}$ for
$r_{n}\geq$ 0.3 km (Equation 8) and $f_{A}$ = 1 otherwise. The resulting
sublimation lifetime from Equation (12) is shown in Figure (4) as a solid red
line. The spin-up time is shown in blue for two assumptions about the radius
dependence of $k_{T}$. First, the dashed blue line shows $\tau_{s}(a)$, the
timescale computed assuming that Equation (11) holds for all $r_{n}$, even
though we possess no constraining data for $r_{n}<$ 0.3 km. Second, the dash-
dot blue line shows $\tau_{s}(b)$ computed assuming that $k_{T}$ “saturates”
to its value at $r_{n}$ = 0.3 km, namely $k_{T}=10^{-4}$, for $r_{n}<$ 0.3 km,
and otherwise follows Equation (11). These two assumptions reflect our lack of
knowledge of the size dependence of the moment arm, but usefully demonstrate a
range of possible behaviors. Finally, the black curves in Figure (4) show the
combined lifetimes, $\tau=(\tau_{s}^{-1}+\tau_{dv}^{-1})^{-1}$, with the lower
(yellow circles, $\tau_{s}(a)$) and upper (green diamonds, $\tau_{s}(b)$
branches reflecting the two models for $k_{T}(r_{n})$ at $r_{n}<$ 0.3 km. We
emphasize that Figure (4) is simplistic (real nuclei are not spherical, the
bulk density is assumed, we have neglected seasonal effects, the model of
equilibrium water ice sublimation is no doubt too simple) and is also specific
to orbits with $q$ = 1.5 AU and $e$ = 0.5. Timescales can be scaled from the
figure to other orbits in inverse proportion to $\overline{f_{s}}$.

Figure (4) shows that the spin-up timescales are shorter than the
devolatilization timescale for all comets with $r_{n}\gtrsim$ 0.1 km,
regardless of which model for $k_{T}(r_{n})$ is used. This size range
encompasses all cometary nuclei measured to-date, and shows the importance of
spin-up. Very few sub-kilometer nuclei are known, relative to power law
extrapolations from larger sizes (e.g. Meech et al. 2004), consistent with
their rapid destruction. For example, measurements of short-period comets in
the 1 to 5 km radius range reveal a differential power law size distribution,
$n(r_{n})dr_{n}\propto r_{n}^{-3.3\pm 0.3}dr_{n}$ (Bauer et al. 2017) while
Fernandez et al. (2013) found $n(r_{n})dr_{n}\propto r_{n}^{-2.9\pm
0.2}dr_{n}$. If these power laws extrapolated to smaller radii, we should
expect the number of nuclei with $r_{n}>$ 0.1 km radius to be $\sim$100 times
the number with $r_{n}>$ 1 km. Even given the observational bias against the
detection of smaller objects, this seems unlikely to be true. Crater counts in
the Kuiper belt source region reveal an impactor population with differential
index $q=-1.7\pm 0.3$ in the radius range $0.1\lesssim r_{n}\lesssim$1 km
(Singer et al. 2019). Setting aside the question of how the source population
could be flatter than the nucleus size distribution, $q$ = -1.7 would still
give a population of $r_{n}>$ 0.1 km comets some 10${}^{0.7}\sim$5 times
larger than that of $r_{n}>$ 1 km comets, inconsistent with the data. But,
regardless of the size distribution of Kuiper belt objects, the strong size
dependence of the lifetimes shown in Figure (4) explains the paucity of small
nuclei.

Figure (4) also shows the median dynamical lifetime of short-period comets
(Levison and Duncan 1997), marked by a long-dashed horizontal line at
$\tau_{dyn}=4\times 10^{5}$ yr. A dotted black horizontal line shows,
$\tau_{L}=1.2\times 10^{4}$ yr, the physical lifetime inferred by the same
authors as necessary to match the inclination distribution of the comets. We
see that, while devolatilization of the larger nuclei is very slow (and may be
impossible due to the formation of impermeable refractory surface mantles not
accounted for here) spin-up times are $\tau_{s}\lesssim\tau_{L}$ for all
comets with $r_{n}\lesssim$ 10 km. Almost all studied comets are smaller than
10 km in radius. For example, of the 25 comets in Table (2), only one
(28P/Neujmin) is larger than 10 km in radius. Therefore, the spins of all
measured comets are liable to have evolved from their source-region values in
response to outgassing torques.

###  4.2 Long Nucleus Rotation Periods

Figure 5 compares the rotation period distribution of cometary nuclei from
Table (1) with that of small asteroids from Waszczak et al. (2015). For the
latter, we selected asteroids with absolute magnitudes 13 $\leq H\leq 18$ in
order to sample objects similar in size to those of the comets. The median
period of nuclei from Table (1) is $P_{n}$ = 15.0 hours (12 objects). The
median period of the 3883 small asteroids is $P_{a}$ = 6.35 hours. The medians
and the cumulative distributions of the periods are obviously inconsistent
(Figure 5), a conclusion buttressed by the K-S test, which gives the
probability that the two distributions could be drawn by chance from the same
parent as $<10^{-4}$. We also compared the asteroid distribution with the list
of comet rotation periods compiled by Kokotanekova et al. (2017) with the same
result; the K-S probability that the two distributions could be drawn from the
same parent is $<10^{-4}$.

The simplest explanation is that the median period difference reflects the
role of density in setting the critical period for rotational instability. In
the absence of cohesive forces, the critical period, $P_{C}$, at which
equatorial centripetal acceleration equals local gravity, varies with density
as $P_{C}\propto\rho^{-1/2}$, and also depends on the body shape. Periods in
the ratio $P_{n}/P_{a}\sim$ 2.4:1 would indicate densities in the ratio
$\rho_{a}/\rho_{n}\sim$ 5.8:1. The nominal nucleus density is $\rho_{n}$ = 500
kg m-3 (Groussin et al. 2019) while the average densities of C-type and S-type
asteroids are reportedly $\sim$1500 kg m-3 and $\sim$3000 kg m-3 (Hanus et al.
2017), indicating ratios $\rho_{a}/\rho_{n}\sim$ 3 and 6 respectively. Within
the uncertainties on $P_{n}/P_{a}$, these expected and observed ratios are
probably compatible.

However, other effects may also contribute to $P_{n}/P_{a}$. Torques drive
nucleus rotations equally towards shorter and longer values, but drive the
median period towards longer values. This is because nuclei torqued to periods
shorter than $P_{C}$ should be destroyed, leaving a survivor distribution
biased towards longer-lived, longer period objects. This effect is a function
of nucleus size, with small nuclei more affected than large nuclei given the
size dependence of $\tau_{s}$. While not detectable in the existing meagre
observational sample, it should be sought in the future as more abundant and
accurate data become available.

Lastly, observational biases inherent in the methods of period determination
play a potentially crucial role in Figure (5). For example, rotational
modulation of the photometry in active comets is limited by aperture averaging
to periods longer than the aperture crossing time (Jewitt 1991), imposing a
bias against short periods that is not present in photometry of asteroids and
other point sources. A similar bias affects rotation periods determined from
rotation-modulated coma structures (spiral arms and arcs, c.f. Samarasinha and
A’Hearn 1991) because short-period nuclei will produce tightly-wrapped spirals
that are more difficult to resolve than open spirals from longer period
nuclei. Disentangling these and other bias effects will be difficult. Ideally,
we need a comet rotation sample based only on well-sampled bare-nucleus
photometry in order to make an accurate comparison with the asteroids.

###  4.3 Destruction of Sungrazing Comets

Thousands of small sungrazing (small perihelion) comets are known (Battams and
Knight 2017). Most are members of the so-called Kreutz group, with perihelia
in the 0.01 to 0.02 AU range, and are thought to be products of the recent
disruption of a larger precursor body (Sekanina and Chodas 2002, 2007).
Despite not impacting the photosphere (the radius of the Sun is $R_{\odot}$ =
0.005 AU), few Kreutz sungrazers survive perihelion, and the same is true for
members of the Kracht, Marsden and Meyer groups, which have similar or
slightly larger perihelia. Instead, observations indicate that the sungrazers
are destroyed (or, more precisely, rendered invisible) before they reach peak
solar insolation at perihelion. For example, photometric measurements of three
Kreutz comets show peak brightness near $r_{H}\sim$ 12 $R_{\odot}$ ($\sim$0.06
AU), with subsequent fading on the way to perihelion (Knight 2010). Could
rotational disruption be responsible?

We can answer this question most directly for C/2005 S1, which is one of the
best-observed Kreutz sungrazing comets. This object lost sodium (presumably by
desorption from minerals) at rate $\dot{M_{Na}}$ = 2 kg s-1 when at $r_{H}$ =
12 $R_{\odot}$ (0.06 AU) (Knight 2010). Sodium is merely the most readily
observed optical species; others are surely present but undetected, and may
carry more mass. Therefore we conservatively interpret $\dot{M_{Na}}$ as
setting a lower limit to the rate of loss of mass from C/2005 S1. We assume
$\rho_{n}$ = 500 kg m-3, $k_{T}$ = 0.007, $V_{th}$ = 103 m s-1, $P$ = 5 hour
and note that the estimated radii of most sungrazers fall in the range
$1\lesssim r_{n}\lesssim$ 50 m (Knight 2010). The radius of C/2005 S1 is
estimated to be $\sim$10 m but, to be conservative and so to over-estimate
$\tau_{s}$, we set $r_{n}$ = 50 m. Then, Equation (3) gives the
extraordinarily short characteristic time $\tau_{s}<1.3\times 10^{5}$ s (about
1.5 day). This timescale is only $\sim$1% of the $\sim$month-long free-fall
time from 1 AU to the Sun, providing ample opportunity for mass loss torques
to spin-up and rotationally disrupt the nucleus, if it has a weak, comet-like
structure. Once the nucleus breaks up, the resulting components themselves are
subject to fragmentation on even shorter timescales, resulting in the
catastrophic destruction of the object (c.f. Sekanina and Chodos 2002). The
peak brightness of C/2005 S1 occurred at $r_{H}\sim$ 14 R⊙ (Figure 9 of Knight
2010) suggesting that this marks the point of fragmentation. Since we assumed
that the radius is at top end of the range given by Knight (2010), we can
infer that rotational breakup is an important destructive process for all
smaller Kreutz comets.

We cannot conclude that rotational break-up is the only destructive process,
and many others of potential importance have been elucidated by Brown et al.
(2015). For example, sungrazers that enter the Sun’s Roche sphere (radius
$\sim 2R_{\odot}$ or $\sim$0.01 AU) could, if strengthless, be sheared apart
by solar tides. Sublimation of water ice, if present, is also very strong.
Using the approach from Section 4.1, a 50 meter radius water ice body at
$r_{H}$ = 0.06 AU would sublimate away on the timescale $\tau_{sub}\sim
1.8\times 10^{5}$ s (a few days ). Thus, if ice is present, devolatilization
through sublimation can compete with spin-up at this size.

However, rotational disruption does not require the presence of ice in
sungrazers, only of mass loss. In fact, we are not aware of direct evidence
for ice in C/2005 S1 and there is little evidence for it in any other
sungrazers. For example, the emission spectrum of C/(1965 S1) Ikeya-Seki at
$r_{H}\sim$ 0.3 AU was dominated by metal lines (Na, Ca, Cr, Co, Mn, Fe, Ni,
Cu, V) probably released by thermal desorption or sublimation of rocks
(Slaughter 1969), possible because of the high temperatures found near the
Sun. This raises the possibility that some sungrazing comets are not comets at
all, but asteroids (rocks) scattered in to orbits with small perihelia and
disintegrating in the heat of the Sun.

###  4.4 Main Belt Comets

The rotations of small asteroids may be influenced by radiation (“YORP”)
torques, with a timescale for spin-up approximately given by

$\tau_{Y}\sim\psi\left(\frac{r_{n}}{1~{}\textrm{km}}\right)^{2}\left(\frac{r_{H}}{1~{}\textrm{AU}}\right)^{2}$
(13)

where $\psi=1.3\times 10^{13}$ s, $r_{n}$ is in kilometers and $r_{H}$ in AU
(Jewitt et al. 2017). By this equation, a 1 km asteroid in a circular orbit at
3 AU has $\tau_{Y}\sim$ 4 Myr. The relation is very approximate because the
YORP effect is sensitive to (mostly unknown) specific details of each
asteroid, including the shape, rotation vector and detailed thermophysical
properties (Statler 2009); Equation (13) is just a guide to the order of
magnitude of the YORP timescale.

Most asteroids have a refractory composition and sublimate negligibly under
the Sun’s radiation field. However, a sub-population known as the “active
asteroids” lose mass, generating comae and dust tails that are obvious in
optical data (Jewitt 2012). The causes of activity in these objects are many
and varied, ranging from impact, to rotational instability, thermal fracture,
desiccation stresses and the sublimation of near-surface ice (Hsieh and Jewitt
2006). Active asteroids driven by ice sublimation are referred to as “main
belt comets”.

If present, outgassing torques on the nuclei of main-belt comets will exceed
the YORP torque when $\tau_{s}<\tau_{Y}$. Combining Equations (3) and (13)
gives (c.f Jewitt et al. 2017)

$\dot{M_{C}}>\frac{16\times 10^{12}\pi^{2}\rho_{n}}{15\psi
V_{th}Pk_{T}}\left(\frac{r_{n}}{1~{}\textrm{km}}\right)^{2}\left(\frac{1~{}\textrm{AU}}{r_{H}}\right)^{2}$
(14)

for the critical mass loss rate above which the resulting torque exceeds that
from YORP. Then, substituting $\rho_{n}$ = 1500 kg m-3 (to take account of the
larger density of asteroids), representative asteroid period $P$ = 5 hours,
$k_{T}$ = 0.007, we find that sublimation torques are dominant over YORP when

$\dot{M_{C}}\gtrsim
0.02\left(\frac{r_{n}}{1~{}\textrm{km}}\right)^{2}\left(\frac{1~{}\textrm{AU}}{r_{H}}\right)^{2}$
(15)

with $\dot{M_{C}}$ in kg s-1. For example, on a $r_{n}$ = 1 km body at $r_{H}$
= 2.5 AU, sustained mass loss rates as small as $\overline{M_{C}}=3\times
10^{-3}$ kg s-1 could generate a torque larger than the YORP torque. Such tiny
mass loss rates fall below the current spectroscopically detectable limits
($\dot{M}\sim$ 1 kg s-1, Jewitt 2012) and therefore the existence of
sublimation spin-up of asteroids cannot be directly tested. Working against
the influence of outgassing torques on icy asteroids is the observation that
strong outgassing is highly time-variable, with main-belt comets spending a
large fraction of the total time in an inactive or weakly active state (Hsieh
and Jewitt 2006)

As a specific example, we consider the disrupted outer-belt active asteroid
P/2013 R3, whose precursor body broke into numerous $\sim$100 m scale pieces
(Jewitt et al. 2017). Sustained comet-like sublimation as small as (Equation
15) $\dot{M}\gtrsim 3\times 10^{-5}$ kg s-1 could, in principle, have driven
this precursor to breakup on a timescale short compared to the YORP timescale.
Mass loss at such a low level would be completely unobservable using existing
techniques. Low albedo ice exposed at the subsolar point at $r_{H}$ = 3 AU
sublimates in equilibrium with sunlight at the rate 2.8$\times 10^{-5}$ kg m-2
s-1, meaning that a strategically located ice patch of only $\sim$ 1 m2 could
generate a YORP-beating torque. Temporarily larger rates of sublimation could
have the same effect. So, while we possess no evidence that P/2013 R3 was
rotationally disrupted by sublimation torques, neither can we reject this
possibility. Hybrid schemes are also possible. For instance, an initial
breakup of a body triggered by impact or the YORP torque could expose
previously buried water ice to the Sun, leading to sublimation and the rapid
spin-up and disintegration of the fragments by outgassing torques. Such hybrid
schemes might be necessary to prevent the otherwise very rapid spin-up of ice-
containing asteroids in the main-belt.

##  5 DISCUSSION

The observations establish beyond reasonable doubt both the importance of the
outgassing torque in comets and the major role played by observational
selection effects. To further emphasize these points, we refer to Figure (6),
which shows the moment arm, $k_{T}$, plotted against the active fraction,
$f_{A}$, with the sizes of the plot symbols shown in proportion to the radii
of the nuclei. Figure (6) illustrates three points. First, the discovery bias
against the detection of small cometary nuclei is evident from the top-heavy
distribution of nucleus sizes. Sub-kilometer nuclei are under-counted in
optical surveys relative to their intrinsic proportion in the comet size
distribution. Second is the additional bias against small comets with small
active fractions, $f_{A}$, because for a given nucleus radius the coma
production rate (and hence the coma brightness and detectability) scales in
proportion to $f_{A}$. Small, weakly active nuclei are pushed beneath the
survey sensitivity limits leaving only small nuclei with large $f_{A}$, like
41P/Tuttle-Giacobini-Kresak, 46P/Wirtanen and 103P/Hartley (all with
$f_{A}>0.3$) in the figure. Third, is the survival bias against small nuclei
having large $k_{T}$; such objects have short spin-up times leading them to be
depleted in number by rapid breakup. The notable outlier to this trend is
41P/Tuttle-Giacobini-Kresak, which is a small nucleus with a large moment arm
and an empirical spin-up time that is exceedingly short (Figures 1 and 3).
Howell et al. (2018) suggested that 41P might be in an excited rotational
state which, if true, would invalidate its inclusion and improve the
correlation with the remaining comets in Figures (3) and (6). At the other end
of the scale, the massive nucleus of 10P/Tempel 2 can sustain a large $k_{T}$
while still having a very long spin-up time. The absence of large nuclei with
small $k_{T}$ (lower left in Figure 6) cannot be attributed to observational
or survival bias.

The dashed blue line in Figure (6) shows the relation $f_{A}k_{T}=10^{-4}$,
which evidently describes the observations (with the exception of 41P/Tuttle-
Giacobini-Kresak) rather well. Re-arranging Equation (9) and substituting
Equation (4) for $\overline{\dot{M}}$ we obtain

$k_{T}f_{A}=\frac{4\pi\rho_{n}}{15V_{th}P_{K}}\left(\frac{|\Delta
P|r_{n}^{2}}{P^{2}\overline{f_{s}}}\right)$ (16)

where the orbitally-averaged sublimation flux, $\overline{f_{s}}$, is
calculated as described in the Appendix.

Given that $\rho_{n}$, $V_{th}$ and $P_{K}$ are approximately the same for all
comets in the study, the inference to be drawn from Equation (16) and Figure
(6) is that $|\Delta P|r_{n}^{2}/(P^{2}\overline{f_{s}})$ is constant. This
quantity, appearing in parentheses in Equation (16), corresponds to the “X
parameter” discussed by Samarasinha and Mueller (2013), Mueller and
Samarasinha (2018) and Steckloff and Samarasinha (2018). Steckloff and
Samarasinha (2018) concluded that the near constancy of $X$ (but not for
41P/Tuttle-Giacobini-Kresak, as noted by Bodewits et al. 2018) implied that
“the net sublimative torque experienced by a comet nucleus depends
predominantly on its size and heliocentric distance, independent of nucleus
age, shape, local topography, and active fraction.” Instead, Equations (1) and
(4) show that the torque must depend on $f_{A}$, but that size-dependent
trends in $f_{A}$ are largely canceled by those in the moment arm $k_{T}$,
such that $k_{T}f_{A}\sim$ constant (c.f. Figures (2) and (3)). The
approximate constancy of the $X$ parameter is seen as a product of these
opposing size-dependent trends.

Finally, a limitation of this and all investigations of nucleus rotation is
the implicit assumption that the outgassing properties of each nucleus,
including $f_{A}$ and $k_{T}$, remain fixed in time. In fact, the comets are
dynamic and evolving bodies whose properties change both stochastically and in
response to dynamical and thermal evolution. As the surface and angular
pattern of the mass loss evolve, the magnitude and possibly the direction of
the sublimation torque might change. Exactly this circumstance was reported in
46P/Wirtanen (Farnham et al. 2021), when the period change in the $\sim$50
days before perihelion was largely cancelled by the change after it. Just as
is the case with the YORP torque, whose magnitude and direction change in
response to even minimal disturbances of the surface (Statler 2009, Cotto-
Figueroa et al. 2015), secular evolution of the sublimation torque vector can
slow the rate of change of the nucleus angular momentum relative to the
relations presented here.

##  6 SUMMARY

Anisotropic outgassing exerts a torque which can change the spin of the
cometary nucleus. We parametrise the outgassing torque in terms of the radius
of a spherical nucleus, $r_{n}$, the fraction of the surface which is active,
$f_{A}$, the dimensionless moment arm, $k_{T}$, the period, $P$, and the
characteristic spin-up time, $\tau_{s}$. Based on a simple model we expect
that $\tau_{s}\propto r_{n}^{2}/(f_{A}k_{T}P)$ (Equation 5). Using published
rotational measurements of short-period comet nuclei with $0.5\lesssim
r_{n}\lesssim 7$ km and with perihelia $q\sim$ 1 to 2 AU, we find that

  1. 1.

The empirical spin-up times follow $\tau_{s}\sim 100r_{n}^{2}$, with
$\tau_{s}$ in years and $r_{n}$ in kilometers.

  2. 2.

The fractional active areas vary as $f_{A}\sim 0.1r_{n}^{-2}$.

  3. 3.

The median dimensionless moment arm is $k_{T}$ = 0.007 with weak evidence for
a size dependence $k_{T}\sim 10^{-3}r_{n}^{2}$.

Consequences of the short timescales include

  1. 1.

Sub-kilometer nuclei are rapidly destroyed, explaining their paucity relative
to power-law extrapolations from larger sizes. This result is independent of
the size distribution in the Kuiper belt source population.

  2. 2.

The spin-up times of sungrazing comets (most of which are small,
$r_{n}\lesssim$ 50 m) are shorter even than the free-fall time to the Sun,
consistent with their observed failure to survive passage through perihelion.

  3. 3.

Weak mass loss torques on small main-belt asteroids, even at immeasurably
small mass loss rates $\lesssim$ 1 g s-1, surpass the YORP torque and, if
sustained, can control the spin-state.

  4. 4.

The angular momenta of short period comets $\lesssim$10 km in radius are, on
average, not primordial.

Finally, we highlight a) flux-limited biases in optical and spectroscopic
surveys against the discovery and measurement of nuclei with small $f_{A}$ and
b) a survival bias against small nuclei with large moment arms, $k_{T}$,
because these objects are quickly spun-up to rotational instability and
removed from the observable population. These importance of these biases
should be assessed in future work.

I thank Jane Luu, Pedro Lacerda and the anonymous referee for helpful comments
on this work. Appendix In order to evaluate Equation (10), we consider energy
balance for a sublimating surface, neglecting conduction, in the form

$\frac{L_{\odot}(1-A)}{4\pi r_{H}^{2}}=\chi\left[\varepsilon\sigma
T^{4}+f_{s}(r_{H})L(T)\right].$ (A1)

Here, $A$ and $\varepsilon$ are the Bond albedo and emissivity of the
sublimating surface, $L_{\odot}$ is the solar luminosity, $r_{H}$ is
heliocentric distance expressed in meters, $\sigma$ is the Stefan-Boltzmann
constant and $L(T)$ is the temperature-dependent latent heat of sublimation.
We assume $A$ = 0.04, $\varepsilon$ = 1 while noting that solutions to
Equation (A1) are insensitive to both quantities. Parameter $\chi$ is a
dimensionless number that expresses the distribution of absorbed energy over
the nucleus, varying between $\chi$ = 1 for a flat surface oriented
perpendicular to the Sun-comet line and $\chi$ = 4 for an isothermal sphere.
We adopt $\chi$ = 2 as the intermediate case, corresponding to hemispheric
warming of a spherical nucleus. We solved Equation (A1) using thermodynamic
parameters for water ice tabulated by Brown and Ziegler (1980) and Washburn
(1926). The equilibrium temperature, $T$, was calculated as a function of
$r_{H}$, which was in turn computed as a function of time by solving Kepler’s
equations

$r_{H}(t)=a(1-e\cos(E(t))$ (A2)

$E(t)-e\sin(E(t))=2\pi(t-T_{0})/P_{K}.$ (A3)

Here, $E(t)$ is the eccentric anomaly and $T_{0}$ is the time of perihelion.
The specific sublimation rate $f_{s}$ was then used to evaluate
$\overline{\dot{M}}$ from various combinations of $a$ and $e$ using Equations
(10) and (4). The average sublimation rate is

$\overline{{f_{s}}}=\frac{1}{P_{K}}\int_{0}^{P_{K}}f_{s}(r_{H})dt$ (A4)

where the integral is taken around the orbit and, since $\dot{M}\propto
f_{s}$, Equation (10) becomes

$\mathcal{S}(r_{H})=\frac{\int_{0}^{P_{K}}f_{s}(r_{H})dt}{P_{K}f_{s}(r_{H})}.$
(A5)

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Table 1: Sublimation Spin-Up  Name | $a\tablenotemark{a}$ | $e\tablenotemark{b}$ | $q\tablenotemark{c}$ | $r_{n}\tablenotemark{d}$ | $P_{K}\tablenotemark{e}$ | $P\tablenotemark{f}$ | $|\Delta P|\tablenotemark{g}$ | $\dot{M}/r_{H}\tablenotemark{h}$ |  $\mathcal{S}$iiScale factor, from Equation (10) | $\overline{\dot{M}}\tablenotemark{j}$ | $\tau_{s}\tablenotemark{k}$ | $10^{3}k_{T}\tablenotemark{l}$ | ReferencemmReferences: B19 = Biver et al. 2019, C20 = Combi et al. 2020, E17 = Eisner et al. 2017, G12 = Gicquel et al. 2012, J03 = Jewitt et al. 2003, K18 = Kokotanekova et al. 2018, L04 = Lamy et al. 2004, M12 = Macquet 2012, R18 = Roth et al. 2018, W17 = Wilson et al. 2017
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| [AU] |  | [AU] | [km] | [yr] | [hr] | [min] | [kg s${}^{-1}]$ |  | [kg s${}^{-1}]$ | [yr] |  |
2P/Encke | 2.215 | 0.848 | 0.337 | 2.4 | 3.30 | 11.0 | 4 | 1110/0.46 | 0.034 | 38 | 540 | 14 | L04, K18, R18
9P/Tempel | 3.146 | 0.510 | 1.542 | 3.0 | 5.58 | 40.9 | 13.5 | 140/1.50 | 0.210 | 30 | 1014 | 6 | L04, K18, G12
10P/Tempel | 3.067 | 0.536 | 1.423 | 5.3 | 5.37 | 8.9 | 0.27 | 600/1.40 | 0.170 | 102 | 10,600 | 8 | L04, K18, W17
14P/Wolf | 4.247 | 0.357 | 2.729 | 3.0 | 8.80 | 9.0 |  $<$4.2 | — | — | — |  $>$1130 | — | F13, K18
19P/Borrelly | 3.611 | 0.624 | 1.358 | 2.2 | 6.86 | 29.0 | 20 | 1800/1.35 | 0.13 | 596 | 600 | 0.6 | L04, K18, M12
41P/TGK | 3.085 | 0.661 | 1.046 | 0.7 | 5.42 | 34.8 | 1560 | 100/1.05 | 0.078 | 8 | 4 | 36 | L04, K18, C20
46P/Wirtanen | 3.093 | 0.659 | 1.055 | 0.6 | 5.44 | 9.15 | 12 | 390/1.06 | 0.078 | 30 | 250 | 0.2 | L04, F21, C20
49P/Arend-Rigaux | 3.525 | 0.619 | 1.343 | 4.2 | 6.62 | 13.0 |  $<$0.23 | 48/1.38 | 0.098 | 5 |  $>$22,000 |  $<$0.2 | L04, K18, E17
67P/C-G | 3.465 | 0.641 | 1.244 | 2.0 | 6.45 | 12.0 | 21 | 300/1.24 | 0.150 | 45 | 220 | 13 | L04, K18, B19
103P/Hartley | 3.470 | 0.695 | 1.058 | 0.6 | 6.46 | 18.2 | 120 | 450/1.06 | 0.052 | 23 | 60 | 0.4 | A11, D13, C20
143P/Kowal-Mrkos | 4.298 | 0.409 | 2.542 | 4.8 | 8.90 | 17.0 |  $<$6.6 | — | — | — |  $>$58.8 | — | J03, K18
162P/Siding-Spring | 3.050 | 0.596 | 1.232 | 7.0 | 5.30 | 33.0 |  $<$25 | — | — | — |  $>$550 | — | F13, K18

Table 2: Active Fraction Measurements  Name | $r_{n}\tablenotemark{a}$ | $A\tablenotemark{b}$ | $f_{A}\tablenotemark{c}$
---|---|---|---
2P/Encke | 2.4 | 0.7 | 0.010
4P/Faye | 1.8 | 2.7 | 0.066
6P/d’Arrest | 1.6 | 1.7 | 0.052
9P/Tempel | 3.1 | 5.2 | 0.043
10P/Tempel | 5.3 | 0.7 | 0.002
19P/Borrelly | 2.2 | 6.6 | 0.109
21P/Giacobini-Zinner | 1.0 | 7.4 | 0.590
22P/Kopff | 1.7 | 12.3 | 0.339
26P/Grigg-Skjellerup | 1.3 | 0.1 | 0.005
28P/Neujmin | 10.7 | 0.5 | 0.0004
31P/Schwassmann-Wachmann | 3.1 | 7.9 | 0.066
41P/Tuttle-Giacobini-Kresak | 0.7 | 6.0 | 0.970
43P/Wolf-Harrington | 1.8 | 2.2 | 0.054
45P/Honda Mrkos Pajdusakova | 0.8 | 0.2 | 0.020
46P/Wirtanen | 0.6 | 1.9 | 0.431
47P/Ashbrook-Jackson | 2.8 | 4.4 | 0.044
49P/Arend-Rigaux | 4.2 | 0.5 | 0.002
59P/Kearns-Kwee | 0.8 | 1.6 | 0.197
67P/Churyumov-Gerasimenko | 2.0 | 1.3 | 0.026
68PKlemola | 2.2 | 0.5 | 0.008
74P/Smirnova-Chernykh | 2.2 | 36.3 | 0.597
78P/Gehrels | 1.4 | 0.3 | 0.011
81P/Wild | 2.0 | 4.1 | 0.081
103P/Hartley | 0.8 | 4.8 | 0.595
126P/IRAS | 1.6 | 3.4 | 0.110

Figure 1: Empirical spin-up timescale, $\tau_{S}$, vs. nucleus radius,
$r_{n}$, from Equation (6) and Table (1). Filled red circles show comets in
which period changes have been detected. Filled yellow diamonds show comets in
which only observational limits to period changes have been set. Sample error
bars show a $\pm$50% uncertainty in $\tau_{s}$. The solid line shows Equation
(7). Logarithmic slopes of 1, 2 and 3 are illustrated.

Figure 2: Nucleus active fraction, $f_{A}$, as a function of nucleus radius,
$r_{n}$. The straight lines indicate $f_{A}\propto r_{n}^{-x}$ with x =
1.5,2.0,2.5 as marked.

Figure 3: Dimensionless moment arm, $k_{T}$, vs nucleus radius, $r_{n}$. The
median $k_{T}$ = 0.007 is marked by a dashed horizontal line and a fit is
added to guide the eye. Data from Table (1).

Figure 4: Model lifetimes as a function of nucleus radius with respect to
spin-up ($\tau_{s}$, blue lines, from Equation 5) and devolatilization
($\tau_{dv}$, solid red line, from Equation 12). As discussed in the text, the
dash-dot blue line, $\tau_{s}(a)$, assumes $k_{T}=10^{-4}$ for $r_{n}\leq$ 0.3
km and Equation (11) otherwise. The dashed blue line, $\tau_{s}(b)$, assumes
that Equation (11) applies at all radii. The combined lifetimes are shown as a
thick black line with lower (yellow circles) and upper (green diamonds)
branches labeled $\tau(a)$ and $\tau(b)$, corresponding to the two models for
$k_{T}(r_{n})$. Horizontal dashed and dotted black lines show the dynamical
lifetimes of Jupiter family comets and their estimated active lifetimes,
respectively, from a model by Levison and Duncan (1997).

Figure 5: Cumulative distributions of the rotation periods listed for (solid
red line) comet nuclei in Table (1) and for (dashed black line) small
asteroids from the sample of Waszczak et al. (2015). The eye and the K-S test
confirm that these distributions are not consistent.

Figure 6: Dimensionless moment arm, $k_{T}$, vs active fraction, $f_{A}$. The
diameters of the symbols are proportional to the nucleus diameters. Arrows
show the direction of increasing discovery bias (which acts against small,
weakly active comets because of their faintness) and increasing survival bias
(which acts against small comets with large $k_{T}$ because of their
vulnerability to rotational breakup, c.f. Equation 3). The red line and circle
marked B11 and D11 show, for comparison, values for 9P/Tempel by Belton et al.
(2011) and 103P/Hartley by Drahus et al. (2011), respectively. The dashed blue
line indicates $f_{A}k_{T}=10^{-4}$ and is not a fit to the data.