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The first reviews of cometary nucleus rotation (Sekanina1981; Whipple1982) were published before useful rotation data were available, and, as a result, are mainly of historical interest. The first reliable measurements of the rotational lightcurve of a cometary nucleus were those of 49P/Arend–Rigaux, obtained in the mid-1980s (A’Hearn et al.1985; Jewitt & Meech1985). 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 such as 49P/Arend–Rigaux revealed this belief to be unfounded. However, it remains true that rotational lightcurves can be directly determined in relatively few comets, due to photometric contamination by coma. Unlike 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 & A’Hearn1991).

In this work, we use only published measurements of nucleus rotation and rotation changes, for which Kokotanekova et al. (2018) has presented a convenient summary. These authors list (in their Table2) the measured change in the rotation periodper cometary orbit, ∣ΔP∣, which is related to the spin-up timescale,τ_s, by

whereP is the measured instantaneous rotation period of the nucleus, andP_K is the Keplerian orbital period (P_K =a^3/2, withP_K given in years, and orbital semimajor axis,a, in au). We add rotational measurements of 46P/Wirtanen, using data from Farnham et al. (2021), but ignore 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 Table1.

Table 1. Sublimation Spin-up

Name	aa	eb	qc	rnd	PKe	Pf	∣ΔP∣g	h	i	j	τsk	103kTl	Referencem
	(au)		(au)	(km)	(yr)	(hr)	(minutes)	(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

Notes.

^aOrbital semimajor axis. ^bOrbital eccentricity. ^cPerihelion distance. ^dNucleus radius (Lamy et al. (2004). ^eOrbital period. ^fRotation period (Kokotanekova et al. (2018), except 46P from Farnham et al. (2021). ^gRotation change per orbit (Kokotanekova et al. (2018), except 46P from Farnham et al. (2021). ^hReported mass-loss rate and the distance at which it was measured (A’Hearn et al.1995). ⁱScale factor, from Equation (10). ^jOrbit average mass-loss rate,. ^kSpin-up timescale, from Equation (6). ^lDimensionless moment arm ×10³, from Equation (9). ^mReferences: 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 = Maquet (2012), R18 = Roth et al. (2018), W17 = Wilson et al. (2017).

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Equation (6) gives a measure of how long the nucleus would take to change from stationary to its current 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 ΔP 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, ∣ΔP∣.

Figure1 showsτ_s as a function of nucleus radius,r_n, computed based on Equation (6) and the data from Table1, with illustrative error bars showing the effect of ±50% uncertainties inτ_s. Evidently,τ_s varies widely in the rangeτ_s ∼ 3 yr (for the very rapidly accelerating nucleus of 46P/Wirtanen) toτ_s ≳ 10⁴ 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 ofτ_s to be associated with larger cometary nuclei, which is as expected based on scaling relations (Equation (5), Jewitt1997), and has been noted by Samarasinha & Mueller (2013), and Kokotanekova et al. (2018). It is obvious from the figure thatτ_s andr_n are related. Although numerical evidence of this is not needed, we computed the Spearmanρcorrelation coefficient (Press et al.1992) between log(τ_s) and log(r_n), findingr_s = 0.88, and ap-value of 0.004, indicating a significant correlation. A least-squares fit of a power law to those comets having non-zero ∣ΔP∣ (red circles in Figure1) gives. 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 Figure1). For convenience, we simply adopt

withτ_s expressed in years, andr_n in kilometers, in the remainder of this paper, and point to Figure1 to show that this provides 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 Table1). Timescales for comets of a given size, having different orbital semimajor axes and perihelia, would not be fitted by Equation (7).

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Cometary mass loss is driven by the expansion of sublimated gas, the production rates of which are estimated based on 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, where is the mass production rate (kg s⁻¹), molecular weightμ = 18, and (s⁻¹) is the production rate, most usually obtained from measures of the OH 3090 Å (e.g., A’Hearn et al.1995), or Lyα (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, owing to 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, however, dust speeds are small compared to the gas speed, and the outflow momentum is necessarily dominated by the gas. For these reasons, we use only measurements of the gas production rates here, and neglect the momentum carried by solids.

In order to examinef_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 data sets (Table2). 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 Fernández et al. (2013) suggest that albedo is not a strong function of nucleus radius). For this reason, individual values off_A may differ somewhat from those reported by others in the literature. As an example, consider 103P/Hartley. We find (Table2)f_A = 0.60, whereas Groussin et al. (2004) reported 0.3 ≲f_A ≲ 1, and Lisse et al. (2009) reportedf_A = 1.1. Valuesf_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.

Table 2. Active Fraction Measurements

Name	rna	Ab	fAc
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//HondaMrkosPajdusakova	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

Notes.

^aNucleus radius, (km), from Lamy et al. (2004) except 126P/IRAS from Groussin et al. (2004). ^bActive area, km², from A’Hearn et al. (1995) except 126P/IRAS from Groussin et al. (2004), and 41P/TGK from Bodewits et al. (2018). ^cActive fraction,.

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The dependence off_A onr_n for these comets is plotted in Figure2, where a strong inverse relation is evident. The Spearmanρ coefficient, computed between log(f_A) and log(r_n), has the valuer_s = −0.53, and a correspondingly low probability of being due to chance ofp = 0.008. A least-squares fit to all the data gives. A fit to the eight objects having measured spin changes gives. 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 via 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 renderf_A independent ofr_n, within the uncertainties. ³ For all these reasons, and in order to avoid giving the appearance of undue significance to the relation in Figure2, we elect merely to note that the variation resembles the power law

withr_n given in km. Figure2 shows that Equation (8) is a useful representation of the data. Equation (8) applies only forf_A ≤ 1, which is true forr_n ≳ 0.3 km. Smaller nuclei should be entirely active (f_A = 1) over their surfaces, based on this relation.

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We are interested to determine the dimensionless moment arm for the torque,k_T, as this quantity allowsτ_s to be estimated via Equation (3) for any nucleus. Substituting Equation (6) into (3), and solving fork_T, we find that

In Equation (9),r_n, ∣ΔP∣, andP are measured quantities obtained from nucleus photometry and/or periodic coma structures that are modulated by nucleus rotation, whileP_K is the orbital period. The mean mass-loss rate,, is obtained from measurements of resonance fluorescence-band strengths, focusing on the OH 3090 Å 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 1–2 au. Comets at distancesr_H ≪ 1 au appear at small elongations, and are difficult to measure. Most comets atr_H ≫ 1 au sublimate weakly, and appear faint. For these reasons, most of the spectroscopically well-observed comets (Table1) have periheliaq ∼ 1–2 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,, to properly scale the mass-loss rate measured at distancer_H,, to estimate the orbitally averaged mean mass-loss rate,, that would be measured if we possessed spectroscopic data around the orbit. We define the scaling factor,, using

where is the mass-loss rate, averaged over one orbit period,P_K. The calculation of is described in theAppendix. An obvious objection to the calculation of is that the orbital variation of might not be well-represented by the sublimation model described in theAppendix. For example, seasonal variations for comets with non-zero obliquity create important pre- versus 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 Table1 (i.e., 14P/Wolf, 143P/Kowal–Mrkos, and 162P/Siding Spring) we possess upper limits to the period change, ∣ΔP∣, but a literature search revealed no measurements of the mass-loss rates. Therefore, the moment arm for these three comets cannot be obtained using 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, so as to determinek_T (Table1).

Values of are listed for each nucleus in Table1. The orbits of the well-characterized comets are clustered nearq ∼ 1 to 1.5 au, ande ∼ 0.6, for which typical values are 0.1. This means that the sublimation rate, averaged around the orbit of most comets, is on the order of 10% of the rate measured at perihelion; 2P/Encke has a smallerq and largere, resulting in 0.034.

We use Equations (9) and (10) to calculatek_T for each nucleus. The resulting values are listed in the penultimate column of Table1. Values of the moment arm range from 2 × 10⁻⁴ to 4 × 10⁻²; the median value,k_T = 0.007, is an order of magnitude smaller than that deduced (before observations) from an early toy model (k_T = 0.05, Jewitt1997). Our value for 9P/Tempel (k_T = 0.006) compares with the range 0.005 ≤k_T ≤ 0.04 found by Belton et al. (2011). Our value for 103P/Hartley (k_T = 4×10⁻⁴) is consistent withk_T = 4 × 10⁻⁴ as given by Drahus et al. (2011).

The data provide some evidence thatk_T andr_n are correlated (Figure3). The Spearmanρ correlation coefficient between log(k_T) and log(r_n) isr_s = 0.81, with a probability that this, or a larger value, could be obtained by chance ofp = 0.01. The observed correlation is therefore not statistically significant at the 3σ (p = 0.005) level. A power-law fit to the data gives. The equation

adequately represents the data over the range 0.5 ≲r_n ≲ 7 km (Figure3). The maximum possible value,k_T = 1, is reached atr_n ∼ 30 km, which is larger than any well-measured cometary nucleus.

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We identify two sources of bias likely to affect determinations off_A(r_n). First, most comets are discovered by magnitude-limited surveys, leading to a “discovery bias” acting against low-activity (smallf_A) comets of a given size. Small nuclei with smallf_A will be preferentially undercounted, relative to high activity (largef_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 off_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 smallf_A, even if they survive the discovery bias, are therefore likely to be under-reported in spectroscopic surveys (e.g., A’Hearn et al.1995; Combi et al.2019) 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), as the primary source of our activity data in Table2. The distribution shows a change in slope forQ_OH ≲ 10²⁷ s⁻¹, corresponding to about = 30 kg s⁻¹. The local water-ice sublimation rate at 1.5 au isf_s = 8 × 10⁻⁵ kg m⁻² s⁻¹, corresponding to a sublimating area 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, as sub-kilometer nuclei could not produce enough water to be spectroscopically detected in the A’Hearn survey iff_A ≪ 1.

Quantitatively, Equations (4) and (8) show that the mass-loss rate,, is approximately independent ofr_n, consistent with sublimation from a fixed active area (not fraction). Substitution into these equations gives s⁻¹, corresponding toQ_OH ∼ 3 × 10²⁷ s⁻¹ for a comet sublimating from the dayside hemisphere at representative distancer_H = 1.5 au (see Table1). This is close to the limit of the spectroscopic data summarized in A’Hearn et al. (1995), only 10% of which haveQ_OH < 2 × 10²⁷ s⁻¹ ( 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 probably plays a role in the distribution of the moment arm,k_T. A small nucleus with a largek_T would, based on 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 (see Drahus et al.2011). The upper-left portion of Figure3 is presumably depleted of objects for this reason. Conversely, large nuclei, even ifk_T = 1, would take a long time to spin-up under the action of outgassing torques, and as such, are less susceptible to the survival bias.

The existence of these bias effects does not eliminate the possibility of genuine size dependencies inf_A andk_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 the depression off_A (Rickman et al.1990). Large nuclei are also more efficient in the recapture of slowly ejected material that might build a rubble mantle (Jewitt2002). Samarasinha & Mueller (2013) suggested that torques from multiple sources on highly active (largef_A) nuclei should more nearly cancel out than on weakly active (smallf_A) nuclei. This would lead to small nuclei having smallk_T, as suggested by Figure3. Unfortunately, we do not yet possess information sufficient to distinguish such effects from those due to the detection, spectroscopic, and survival biases.

Equations (3) and (7) show that small nuclei are particularly susceptible to outgassing torques, and therefore to potential rotational breakup (Jewitt1992,1997; Samarasinha2007), consistent with the observed paucity of small nuclei (Fernández et al.2013; Bauer et al.2017). Indeed, the spins of nuclei smaller than a critical radius,r_c ∼ 0.1–0.3 km, can be substantially modified within a few orbits. The observed fragmentation of the small nucleus of 332P/Ikeya–Murakami (radiusr_n ≤ 0.28 km) is a particular example of a sub-kilometer nucleus, likely to be suffering rotational instability. Perhaps not coincidentally, its rotation period,P = 2 hr, 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 also experiences the loss of volatiles. The true timescale for devolatilization,τ_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, with given by Equation (4), giving

We compareτ_dv withτ_s as a function of nucleus radius in Figure4, which updates Figure 2 from Jewitt (1997) to incorporate the new findings with respect to the radius-dependence off_A (Equation (8)) andk_T (Equation (11)). We computed the orbitally averaged for hemispheric sublimation from comets having perihelionq = 1.5 au, and eccentricitye = 0.5, representative of those in Table1, finding kg m⁻² s⁻¹. We setP = 15 hr, this being the median period from Table1, and we assume forr_n ≥ 0.3 km (Equation (8)), andf_A = 1 otherwise. The resulting sublimation lifetime from Equation (12) is shown in Figure4 as a solid red line. The spin-up time is shown in blue for two assumptions regarding the radius-dependence ofk_T. First, the dashed blue line showsτ_s(a), the timescale computed assuming that Equation (11) holds for allr_n, even though we possess no constraining data forr_n < 0.3 km. Second, the dashed–dotted blue line showsτ_s(b), computed assuming thatk_T “saturates” to its value atr_n = 0.3 km, i.e.,k_T = 10⁻⁴, forr_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 Figure4 show the combined lifetimes,, with the lower (yellow circles,τ_s(a)) and upper (green diamonds,τ_s(b) branches reflecting the two models fork_T(r_n) atr_n < 0.3 km. We emphasize that Figure4 is simplistic (real nuclei are not spherical, the bulk density is assumed, we have neglected seasonal effects, and the model of equilibrium water-ice sublimation is no doubt too simple), and is also specific to orbits withq = 1.5 au ande = 0.5. Timescales can be scaled from the figure to other orbits in inverse proportion to.

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Figure4 shows that the spin-up timescales are shorter than the devolatilization timescale for all comets withr_n ≳ 0.1 km, regardless of which model fork_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 1–5 km radius range reveal a differential power-law size distribution, (Bauer et al.2017), while Fernández et al. (2013) found. If these power laws extrapolated to smaller radii, we should expect the number of nuclei withr_n > 0.1 km radius to be ∼100 times the number withr_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 indexq = − 1.7 ± 0.3 in the radius range 0.1 ≲r_n ≲ 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 ofr_n > 0.1 km comets some 10^0.7 ∼ 5 times larger than that ofr_n > 1 km comets, which is inconsistent with the data. However, regardless of the size distribution of Kuiper Belt objects, the strong size-dependence of the lifetimes shown in Figure4 explains the paucity of small nuclei.

Figure4 also shows the median dynamical lifetime of short-period comets (Levison & Duncan1997), marked by a long-dashed horizontal line atτ_dyn = 4 × 10⁵ yr. A dotted black horizontal line shows,τ_L = 1.2 × 10⁴ yr, the physical lifetime inferred by the same authors as necessary to match the inclination distribution of the comets. We note 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 areτ_s ≲τ_L for all comets withr_n ≲ 10 km. Almost all studied comets are smaller than 10 km in radius. For example, of the 25 comets in Table2, 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.

Figure5 compares the rotation-period distribution of cometary nuclei from Table1 with that of small asteroids from Waszczak et al. (2015). For the latter, we selected asteroids with absolute magnitudes 13 ≤H ≤ 18 in order to sample objects similar in size to those of the comets. The median period of nuclei from Table1 isP_n = 15.0 hr (12 objects). The median period of the 3883 small asteroids isP_a=6.35 hr. The medians, and the cumulative distributions of the periods, are clearly inconsistent (Figure5), 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.

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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 asP_C ∝ρ^−1/2, and also depends on the body shape. Periods in the ratioP_n/P_a ∼ 2.4:1 would indicate densities in the ratioρ_a/ρ_n ∼ 5.8:1. The nominal nucleus density isρ_n = 500 kg m⁻³ (Groussin et al.2019), while the average densities of C-type and S-type asteroids are reportedly ∼1500 kg m⁻³ and ∼3000 kg m⁻³ (Hanus et al.2017), indicating ratiosρ_a/ρ_n ∼ 3 and 6, respectively. Within the uncertainties forP_n/P_a, these expected and observed ratios are probably compatible.

However, other effects may also contribute toP_n/P_a. Torques drive nucleus rotations equally toward shorter and longer values, but drive the median period toward longer values. This is because nuclei torqued to periods shorter thanP_C should be destroyed, leaving a survivor distribution biased toward 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 ofτ_s. While not detectable in the existing meager 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 Figure5. For example, rotational modulation of the photometry in active comets is limited by aperture averaging to periods longer than the aperture crossing time (Jewitt1991), imposing a bias against short periods that is not present in the photometry of asteroids and other point sources. A similar bias affects rotation periods determined from rotation-modulated coma structures (spiral arms and arcs, see Samarasinha & A’Hearn1991), 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.

Thousands of small sungrazing (small perihelion) comets are known (Battams & Knight2017). Most are members of the so-called Kreutz group, with perihelia in the 0.01–0.02 au range, and are thought to be products of the recent disruption of a larger precursor body (Sekanina & Chodas2002,2007). Despite not impacting the photosphere (the radius of the Sun isR_⊙ = 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 nearr_H ∼ 12R_⊙ (∼0.06 au), with subsequent fading on the way to perihelion (Knight et al.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 via desorption from minerals) at a rate of = 2 kg s⁻¹ when atr_H = 12R_⊙ (0.06 au) (Knight et al.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 as setting a lower limit to the rate of loss of mass from C/2005 S1. We assumeρ_n = 500 kg m⁻³,k_T = 0.007,V_th = 10³ m s⁻¹, andP = 5 hr, and note that the estimated radii of most sungrazers fall in the range 1 ≲r_n ≲ 50 m (Knight et al.2010). The radius of C/2005 S1 is estimated to be ∼10 m but, to be conservative, and so to overestimateτ_s, we setr_n = 50 m. As such, Equation (3) gives the extraordinarily short characteristic time ofτ_s < 1.3 × 10⁵ s (about 1.5 day). This timescale is only ∼1% of the ∼month-long freefall 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 (see Sekanina & Chodas2002). The peak brightness of C/2005 S1 occurred atr_H ∼ 14R_⊙ (Figure 9 of Knight et al.2010) suggesting that this marks the point of fragmentation. Since we assumed that the radius is at the top end of the range given by Knight et al. (2010), we can infer that rotational breakup is an important destructive process for all smaller Kreutz comets.

We cannot conclude that rotational breakup is the only destructive process, and many others of potential importance have been elucidated by Brown et al. (2015). For example, sungrazers entering the Sun’s Roche sphere (radius ∼ 2R_⊙ or ∼0.01 au) could, if strengthless, be sheared apart by solar tides. The sublimation of water ice, if present, is also very strong. Using the approach given in Section4.1, a 50 m radius water-ice body atr_H = 0.06 au would sublimate away on the timescaleτ_sub ∼ 1.8 × 10⁵ s (a few days ). Therefore, 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 atr_H ∼ 0.3 au was dominated by metal lines (Na, Ca, Cr, Co, Mn, Fe, Ni, Cu, and V), probably released by thermal desorption or the sublimation of rocks (Slaughter1969), made possible due to the high temperatures found near the Sun. This raises the possibility that some sungrazing comets are not comets at all, but asteroids (rocks), scattered into orbits with small perihelia, and disintegrating in the heat of the Sun.

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

whereψ = 1.3 × 10¹³ s,r_n is given in kilometers, andr_H in au (Jewitt et al.2017). Based on this equation, a 1 km asteroid in a circular orbit at 3 au hasτ_Y ∼ 4 Myr. The relation is very approximate, because the YORP effect is sensitive to (mostly unknown) specific details of each asteroid, including its shape, rotation vector, and detailed thermophysical properties (Statler2009); Equation (13) is therefore just a guide as 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 (Jewitt2012). 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 & Jewitt2006). 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 whenτ_s <τ_Y. Combining Equations (3) and (13) gives (see Jewitt et al.2017)

for the critical mass-loss rate, above which the resulting torque exceeds that from YORP. Substitutingρ_n = 1500 kg m⁻³ (to take account of the larger density of asteroids), representative asteroid periodP = 5 hr,k_T = 0.007, and we find that sublimation torques are dominant over YORP when

with given in kg s⁻¹. For example, on ar_n = 1 km body atr_H = 2.5 au, sustained mass-loss rates as small as kg s⁻¹ could generate a torque larger than the YORP torque. Such tiny mass-loss rates fall below the current spectroscopically detectable limits ( 1 kg s⁻¹, Jewitt2012), 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 their total time in an inactive, or weakly active state (Hsieh & Jewitt2006)

As a specific example, we consider the disrupted outer-belt active asteroid P/2013 R3, whose precursor body broke into numerous ∼100 m scale pieces (Jewitt et al.2017). Sustained comet–like sublimation as small as (Equation (15)) kg s⁻¹ could, in principle, have driven this precursor to break up on a shorter timescale, as 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 atr_H = 3 au sublimates in equilibrium with sunlight at the rate 2.8 × 10⁻⁵ kg m⁻² s⁻¹, meaning that a strategically located ice patch of only ∼1 m² could generate a YORP-beating torque. Temporarily larger rates of sublimation could have the same effect. As such, 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 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.