Cometary Activity Begins at Kuiper Belt Distances: Evidence from C/2017 K2
David Jewitt, Yoonyoung Kim, Max Mutchler, Jessica Agarwal, Jing Li, Harold Weaver
CCOMETARY ACTIVITY BEGINS AT KUIPER BELTDISTANCES: EVIDENCE FROM C/2017 K2
David Jewitt , Yoonyoung Kim , Max Mutchler , Jessica Agarwal , Jing Li ,Harold Weaver Department of Earth, Planetary and Space Sciences, UCLA, 595 Charles Young DriveEast, Los Angeles, CA 90095-1567 Department of Physics and Astronomy, University of California at Los Angeles,430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547 Institut for Geophysik und Extraterrestrische Physik, Technische UniversitatBraunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germany Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 The Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Road,Laurel, Maryland 20723 [email protected]
Received ; accepted
Astronomical Journal, in press a r X i v : . [ a s t r o - ph . E P ] F e b ABSTRACT
We study the development of activity in the incoming long-period cometC/2017 K2 over the heliocentric distance range 9 (cid:46) r H (cid:46)
16 AU. The cometcontinues to be characterized by a coma of sub-millimeter and larger particlesejected at low velocity. In a fixed co-moving volume around the nucleus we findthat the scattering cross-section of the coma, C , is related to the heliocentricdistance by a power law, C ∝ r − sH , with heliocentric index s = 1 . ± .
05. Thisdependence is significantly weaker than the r − H variation of the insolation as aresult of two effects. These are, first, the heliocentric dependence of the dustvelocity and, second, a lag effect due to very slow-moving particles ejected longbefore the observations were taken. A Monte Carlo model of the photometryshows that dust production beginning at r H ∼
35 AU is needed to match themeasured heliocentric index, with only a slight dependence on the particle sizedistribution. Mass loss rates in dust at 10 AU are of order 10 kg s − , whileloss rates in gas may be much smaller, depending on the unknown dust to gasratio. Consequently, the ratio of the non-gravitational acceleration to the localsolar gravity, α (cid:48) , may, depending on the nucleus size, attain values ∼ − (cid:46) α (cid:48) (cid:46) − , comparable to values found in short-period comets at much smallerdistances. Non-gravitational acceleration in C/2017 K2 and similarly distantcomets, while presently unmeasured, may limit the accuracy with which we caninfer the properties of the Oort cloud from the orbits of long-period comets. Subject headings: comets: general—comets: individual (C/2017 K2)—Oort Cloud
1. INTRODUCTION
The dominant cometary volatile, water, can sublimate appreciably at blackbodytemperatures prevailing out to about 5 AU, corresponding to the orbit of Jupiter (Whipple1950). Comets that are active at much larger distances must be powered by the sublimationof a more volatile material (for example, carbon monoxide or carbon dioxide) or by anotherphysical process (for example, crystallization of amorphous ice, or non-thermal processes).The prime example of an inbound, distantly active comet is the long period C/2017 K2(Pan STARRS) (hereafter “K2”), which was discovered at heliocentric distance r H = 15.9AU and found to be active in pre-discovery data out to r H = 23.7 AU (Jewitt et al. 2017b,Meech et al. 2017, Hui et al. 2018). Comet K2 is especially important because its longorbital period, ∼ a = -5034 AU, perihelion distance q = 1.810 AU, eccentricity e = 1.00036 andinclination i = 87.5 ◦ ) with perihelion expected on UT 2022 December 20. However, theslight excess of the eccentricity above unity is not an indicator of an interstellar origin;numerical integrations reveal a pre-entry, barycentric semi-major axis a ∼ e = 0.9998 and, as noted above, a period ∼ q = 10 AU, according to theseauthors. Comet K2 is thus not dynamically new in the Oort sense, but is a return visitor tothe planetary region. Nevertheless, K2 is entering the planetary region of the solar systemfrom Oort cloud distances (aphelion ∼ (cid:46)
10 K), retaining 4 –negligible heat from the previous perihelion millions of years ago.For these reasons, K2 is a model object in which to follow the development of activityupon approach to the Sun. Indeed, the range of pre-perihelion distances over which K2will be observable, from the pre-discovery observation at 24 AU to perihelion on UT 2022December 20 at 1.8 AU, is unprecedented. Here, we present observations obtained as partof a continuing program to monitor K2 using the Hubble Space Telescope (HST). The datacover the heliocentric distance range from 15.9 AU to 8.9 AU, placing comet K2 in anessentially un-observed cometary realm.
2. OBSERVATIONS
We used data from the UVIS channel of the WFC3 imaging camera on the 2.4 mHubble Space Telescope (HST), taken under observational programs GO 15409, 15423 and15973. Since the ephemeris of K2 is well-known and the size of the coma not large, weread out only a 2058 × (cid:48)(cid:48) × (cid:48)(cid:48) field of view at image scale0.04 (cid:48)(cid:48) pixel − . The point-spread function from WFC3 was measured at 0.085 (cid:48)(cid:48) FWHM (fullwidth at half maximum), corresponding to ∼
600 km at 10 AU. The wide spectral responseof the F350LP filter was chosen to provide maximum sensitivity to low surface brightnessin the coma. The peak system throughput is 29% and the filter takes in most of the opticalspectrum at wavelengths λ >
The absence of a distinct tail (Figure 1), as we previously noted (Jewitt et al 2017,2019), reflects the minimal effect of solar radiation pressure on large ( a ≥ µ m) particles.Some deviations from circular symmetry of the isophotes are present, however, hinting atstructure in the pattern of emission from the nucleus. To enhance this azimuthal comastructure, we divided each image by the annular median, using the excellent software madeavailable by Samarasinha and Larson (2014). The annuli were taken to be 1 pixel (0.04 (cid:48)(cid:48) )wide and divided into 1 ◦ sectors, each centered on the nucleus. Figure (2) shows the results,further smoothed by convolution with a Gaussian function of 3 pixels (0.12 (cid:48)(cid:48) ) half-width athalf-maximum to reduce the noise. Other than for excursions due to imperfectly removedfield objects, the panels show excess fan-like coma projected to the south west of thenucleus, with a central axis of the fan rotating clockwise from position angle 250 ± ◦ in2017 to 170 ± ◦ in 2020. The generally smooth time evolution of the fan orientationsuggests that there is no direct link with the projected anti-solar direction, which changesappreciably over the period of observations (Figure 2). This is likely a result of the extremeforeshortening resulting from the small phase angles at which K2 is observed. Regardlessof the orientation of K2’s rotation axis, the nucleus should not have experienced strongseasonal changes between 2017 and 2020. 6 –The observation on UT 2020 June 18 was taken only 0.08 ◦ from the projected orbitalplane of comet K2, proving that ejection is asymmetric, with more dust ejected to the southin the plane of the sky. We used photometry to measure the scattering cross-section of the coma. The angularsizes of the photometry apertures were scaled to take account of the varying geocentricdistance, ∆, so as to maintain fixed linear radii of 5, 10, 20, 40, 80 and 160 × km whenprojected to the distance of the comet. The use of fixed linear (as opposed to angular)apertures guarantees that we obtain consistent measurements of a fixed volume around thenucleus. This removes the need to make an additional (and uncertain) geometry-dependentphotometric corrections in order to meaningfully compare measurements taken at differenttimes. The smallest aperture (5000 km at the comet) was picked so as to remain largerin angular extent than the 0.4 (cid:48)(cid:48) radius of the PSF of the telescope. The largest aperture(160,000 km at the comet) was picked to match the field of view of WFC3. The backgroundsky brightness and its uncertainty were estimated from the median and dispersion of datanumbers in a concentric annulus as large as permitted by the field of view of WFC3,typically corresponding to inner and outer radii of 400 and 500 pixels (16 (cid:48)(cid:48) and 20 (cid:48)(cid:48) ),respectively. The photometry was calibrated assuming that a G2V source with V = 0 wouldgive a count rate 4.72 × s − in the same filter .The apparent V magnitudes were converted to the scattering cross-section of dust using C = 1 . × p V − . H (1) http://etc.stsci.edu/etc/input/wfc3uvis/imaging/ 7 –where p V is the geometric albedo and H is the absolute magnitude computed from H = V − ( r H ∆) + 2 . Φ( α ) . (2)Quantity Φ( α ) ≤ α to that which would be observed at the same r H and ∆ and α = 0 ◦ . Neither Φ( α ) nor p V is observationally constrained in comet K2 and so we are forced to make assumptionsin order to make progress. We assumed 2 . Φ( α ) = − Bα , with B = 0.02 magnitudesdegree − , consistent with observations of other active comets (c.f. Meech et al. 1986). Thelargest phase angle in our data is α = 6 ◦ (Table 1), so that the effects of even a factor-of-twoerror in Φ( α ) are unimportant. However, we emphasize that we have no specific constraintson the phase function of K2 and the uncertainties could, in principle, be larger. We assume p V = 0.1 in Equation (1) in order to be consistent with results for other cometary dustcomae (Zubko et al. 2017). Results for other values of p V can be simply scaled fromEquation 1. The apparent and absolute magnitudes as well as the derived cross-sections arelisted in Table (2).
3. DISCUSSION3.1. Heliocentric Index, Velocity Law and the Lag EffectHeliocentric Index:
Figure (3) shows the cross-section in the 160 × km fixed-radius aperture as a function of heliocentric distance. We fitted a weighted power-law tothe data by least-squares using C = C r − sH (3) 8 –where C and s are constants and s is the “heliocentric index”. Quantity C is thecross-section in the 160 × km aperture scaled to r H = 1 AU using index s . We find C = (907 ± × km and heliocentric index s = 1 . ± .
05 (solid red line in Figure3). For comparison, in equilibrium sublimation with sunlight, the production rate of anexposed supervolatile should vary as r − H (i.e. s = 2), which is steeper than the measuredvalue by about 17 σ . We attribute this difference to two effects; 1) the heliocentric distancedependence of the dust velocity and 2) a “lag effect”, both of which we describe below. Notethat we ignore the cross-section of the nucleus, C n , in computing the heliocentric index.The limiting nucleus radius r n < C n <
250 km , which is <
1% of even the smallest cross-section (3.8 × km ) measured within the 160 × kmphotometry aperture (Table 2). Dust Velocity:
The gas drag force acting on a spherical particle of density ρ s andradius a is F = C D πa ρ g ( r, r H )( V g − V ) , where C D is a dimensionless drag coefficient oforder unity, ρ g ( r, r H ) is the density of the gas at distance r from the nucleus and r H fromthe Sun, V g the bulk velocity of the gas and V the speed of the particle. Quantity V g − V is the speed of the gas relative to the particle, decreasing as the particle accelerates awayfrom the nucleus. We write ρ g ( r, r H ) = ρ (cid:16) r n r (cid:17) (cid:18) r H (cid:19) (4)with r n being the radius of the nucleus, r H the heliocentric distance in AU, and ρ the gasdensity at the nucleus surface ( r = r n ) normalized to heliocentric distance r H = 10 AU.The r − H dependence in Equation (4) is appropriate for supervolatile ices, in which almostall energy absorbed from the Sun is used to break molecular bonds, leaving little to sustainthermal radiation (e.g. Jewitt et al. 2017). It is not appropriate for water ice, which showsa steeper distance dependence, except at heliocentric distances r H (cid:46) ρ g ( r, r H ) and neglecting the gravitational attraction to the nucleus forsimplicity, the equation of motion is43 πρ s a V dVdr = C D πa ρ (cid:16) r n r (cid:17) (cid:18) r H (cid:19) ( V g − V ) . (5)The terminal speed of the particle, V ∞ , is given by integrating Equation (5) as (cid:90) V ∞ V dV ( V g − V ) = 3 C D a ρ ρ s (cid:18) r H (cid:19) r n (cid:90) ∞ r n drr . (6)Prompted by the low measured speeds of grains in K2, we assume V (cid:28) V g and assume V g is independent of r H , as found empirically by Biver et al. (2002) in the distance range 7 to14 AU, to solve Equation (6) V ∞ = (cid:18) C D ρ ρ s r n a (cid:19) / (cid:18) r H (cid:19) V g . (7)The inverse dependence of V ∞ on r H , for particles of a given size, contributes to the smallvalue of the heliocentric index because the residence time in a given fixed-radius aperturedecreases as the comet approaches the Sun, even as the production rate increases.To calculate V ∞ from Equation (7), we substitute ρ = f s /V g , where f s (kg m − s − ) isthe equilibrium mass sublimation flux evaluated, by definition, at heliocentric distance r H = 10 AU. The hemispheric sublimation rate is given, to first order, by equating the powerabsorbed from the Sun to the power consumed in sublimation, f s = L (cid:12) πr H H , (8) 10 –where L (cid:12) = 4 × W is the luminosity of the Sun, r H is the heliocentric distanceexpressed in meters, and H = 2 × J kg − is the latent heat of vaporization of CO(Huebner et al. 2006). Substituting, we find f s ∼ × − kg m − s − at 10 AU. A moredetailed (numerical) solution of the energy balance equation, including a term for radiationcooling of the ice, gives a slightly smaller f s = 2 × − kg m − s − . (We note that thisnumber is, itself, a probable over-estimate of f s given that sublimation likely proceeds frombeneath a thin, porous mantle, not from the exposed surface). The drag coefficient, C D ,is unknown. For simplicity, we set 3 C d / V g = 130 m s − , ρ s = 500 kg m − and r n ≤ a = 1 mm particle, V ∞ ≤ . − at r H = 14.8 AU (the averagedistance of K2 used to estimate the speed in Jewitt et al. 2019). This is acceptably closeto the measured speed, V ∞ ∼ − , given the simplicity of the model and the fact thatmany of the quantities in Equation (7) are unmeasured.Equation (7) can be written V ∞ = 2 . (cid:16) r n (cid:17) / (cid:18) a (cid:19) / (cid:18)
10 AU r H (cid:19) (9)with a in millimeters, r n in km and r H in AU. This relation is plotted in Figure (4) for a 1mm particle and nucleus radii r n = 3 km (long-dashed blue curve), r n = 6 km (solid blackcurve) and r n = 9 km (short-dashed red curve). The particle speed reported by Jewitt etal. (2019), marked for comparison in the figure, is in reasonable agreement with Equation(9). Lag Effect:
At a given epoch, C is contributed by particles having a wide range ofsizes, ejection speeds and aperture residence times, some of which are very long. Undergas drag acceleration, small, fast-moving particles quickly leave the photometry aperture,but large slowly-moving particles linger longer. In distant comet K2, the residence timescan be extreme. For example, 1 mm particles have characteristic ejection speeds V = 11 –4 m s − (Jewitt et al. 2019) at which speed they take ∼ × s (1.3 years) to travelacross the radius of the 160,000 km aperture. Larger, slower particles would taken evenlonger. Particles released into the coma at any instant thus contribute to a “background”of lingering particles accumulated over the previous year or, depending on their size andejection speed, even longer.To explore this lag effect with a more physical model, we constructed a set of MonteCarlo simulations. The simulations follow the motions of particles ejected following thevelocity law in Equation (9) and starting at heliocentric distance r (cid:63) . We assumed thatthe heliocentric dependence of the gas production rate varies as dM/dt ∝ r − H , needed tomaintain equilibrium with the insolation, and we included particles with dimensionlessradiation pressure parameters in the range β = 10 − to 10 − . These correspond to particleradii in the range 1 mm to 100 mm, respectively.The distribution of particle radii is taken to be a power-law, such that the numberof particles having radii in the range a to a + da is n ( a ) da = Γ a − q da , with Γ and q beingconstant. In Table (3) we list a brief and incomplete summary of values of q reported inthe recent literature. Most are based on matching the isophotal dust distribution in opticalimages but for three comets, 1P/Halley, 67P/Churyumov-Gerasimenko and 103P/Hartley,we also list direct measurements obtained from spacecraft in, or passing through, the coma.There is a tendency for the size distribution to become steeper as the particle size increases.The unweighted mean of the measurements in the Table is q = 3.7 ± q ∼ ± q is themiddle value and the error is half the range. The resulting weighted mean is q = 3.45 ± q = 3.5 (Jewitt et al. 2017, Hui et 12 –al. 2018). Accordingly, we proceed to interpret the heliocentric index using models with q = 3.5.Results are shown in Figure (5), which shows the modeled heliocentric index, s , as afunction of r (cid:63) . Error bars on the model points were calculated from least-squares fits at fiveheliocentric distances spanning the 9 AU to 16 AU heliocentric distance range, to simulatethe way in which we computed s from the HST data. Also marked in the figure are themeasured value, s = 1 . ± .
05, and a shaded region extending ± σ from this value. Weshow models for three values of the size distribution index, q .For the nominal q = 3 . r (cid:63) =36 AU. Adopting q = 3.0, a value smaller than typical of comets (Table 3) the minimumturn-on distance is r (cid:63) ≥
38 AU (c.f. yellow circles in Figure 5; reached by K2 in 2004).Index values q = 4 . r (cid:63) >
34 AU. Model values of s within ± σ of the measured value are obtained for 32 ≤ r (cid:63) ≤
42 AU for 3.0 ≤ q ≤ r (cid:63) to the adopted size distribution is modest unless q ispathologically large or small. The latter possibility is counter-indicated by our own MonteCarlo models of the optical data in which q = 3.5 provides a convincing match to themorphology (Jewitt et al. 2017, Hui et al. 2018). In short, the shallow heliocentric indexrequires that activity in K2 begin at Kuiper belt distances.This lag effect on the photometry is particularly prominent in K2 because of theunusually large distances at which the comet has been observed, because of the largeaverage size (and low speed) of the particles, and because of the head-on viewing geometry,resulting in very long aperture residence times. Published measurements of other cometsare difficult to compare with the present study because, for example, no other studies havesampled in-bound comets over heliocentric distances as large as those considered here.Furthermore, published photometry generally uses fixed angular (not linear) apertures, 13 –requiring an uncertain correction for the changing volume of coma that is measured as r H varies. Sekanina’s (1973) early inference that long-period comets Baade 1955 VI andHaro-Chivara 1956 ejected icy sub-millimeter particles when at 5 to 15 AU, while lackingphotometric support and being less extreme than the case of comet K2, is perhaps the mostsimilar to the picture developed here. The radial distribution of dust within the coma was assessed using the annularphotometry from Table (2). Figure (6) shows the cumulative dust cross-section as afunction of the linear aperture radius, for each of the dates of observation in the table. Thecumulative profiles are similar in shape on each date, consistent with the steady appearanceof the comet, but show progressive increases in brightness at all radii between 2017 and2019. In steady-state, the surface brightness of a coma varies inversely with radius, r ,because of the equation of continuity (Jewitt and Meech 1987). When integrated withrespect to radius, as in Figure (6), the encircled brightness of a steady-state coma shouldvary in proportion to r . This accurately describes the coma of K2 for all measured profilesup to radii ∼ r , the linear radius at which theencircled cross-section is 80% of the peak cross-section determined using the 160,000km radius aperture. Measurements of r are plotted as a function of the heliocentricdistance in Figure (7). The average value is r = (8 . ± . × m (error on the 14 –mean of eight measurements). Travel times for coma particles, estimated using τ = r /V are ∼ V ∼ − speeds of 1 mm particles (Jewitt et al. 2019).Figure (7) shows evidence for a weak gradient. A least-squares fitted power law gives r = (142 ± × r − . ± . H in the range 9 (cid:46) r H (cid:46)
16 AU. The near constancy of r argues for the action of radiation pressure and against the “fading grains” hypothesis, as weargue below.A dust particle launched sunward at speed V will be stopped by radiation pressureat a turn-around distance, (cid:96) , given by (cid:96) = V / (2 βg (cid:12) ). Here, g (cid:12) is the gravitationalacceleration towards the Sun and β is the dimensionless radiation pressure efficiency factor,such that βg (cid:12) is the acceleration of the particle. Quantity β is inversely related to particlesize (Bohren and Huffman 1983) and conveniently approximated by β ∼ − /a , with a expressed in meters. A 1 mm particle has β ∼ − . The solar gravity may be written g (cid:12) = g (cid:12) (1) /r H , where g (cid:12) (1) = 0.006 m s − is the acceleration at 1 AU and r H is in AU.Then, setting V = V ∞ and with the use of Equation (7), we find (cid:96) = 100 (cid:18) C D ρ ρ s r n a (cid:19) V g βg (cid:12) (1) (10)which is independent of heliocentric distance.Substituting into Equation (10) for the parameters as above, we obtain (cid:96) ∼ m, ingood agreement with the measured r = 8 × m. The observation that (cid:96) ∼ r andthe fact that both quantities are approximately independent of r H strongly favor radiationpressure shaping of the coma as the cause of the flattened profiles in Figure (6).The “fading grains” hypothesis is less consistent with the data. In its favor is the factthat the residence times for particles in the coma are very long, ∼ r H decreases. This is because the insolation variesas r − H while the residence time in the coma varies, by Equation (7), as r − H . Therefore,weathering and disaggregation should be more pronounced at smaller r H , leading toshrinkage of the coma upon approach to the Sun. Since this is not observed (Figure 6), wediscount the fading grains hypothesis. The mass of an opaque spherical particle, M , is proportional to its geometriccross-section, C , according to M = 4 ρ s aC/
3, where ρ s is the particle density and a is theparticle radius. An equivalent relation holds for an optically thin collection of spheres, with a replaced by the mean particle radius a , and C being the sum of the cross-sections of allthe particles within the projected photometry aperture. Since C is measurable from thephotometry using Equations (1) and (2), we can estimate the coma mass in K2. At anyinstant, the coma mass within an aperture is M ( r H ) = (4 / ρ s aC ( r H ) . (11)Dust within an aperture of radius r must, in steady state, be replaced on the crossingtimescale τ = r/V . Differentiating Equation (11), setting dC ( r H ) /dt = C ( r H ) /τ andneglecting the numerical multiplier, we estimate the steady-state dust loss rate from dMdt = ρ s a C ( r H ) V ( r H ) r (12)Substituting for V from Equation (9), we have 16 – dMdt = 2 . × − ρ s r (cid:16) r n (cid:17) / (cid:18) a (cid:19) / (cid:18) r H (cid:19) C ( r H ) (13)The rate given by Equation (13) is only an order-of-magnitude estimate of the mass lossrate in dust because of the many unmeasured parameters. For example, the particle density, ρ s , is unmeasured, we possess only an upper limit to the nucleus radius, r n ≤ a , is uncertain to within a factor of at least two, and because thederived cross-sections, C ( r H ), rely on the assumption of the coma albedo, which itself couldbe in error by a factor of two to three. Nevertheless, the equation gives a useful measure ofthe relative mass production rates in dust and their variation with heliocentric distance.We plot Equation (13) in Figure (8) assuming r n = 5 km, a = 1 mm, ρ s = 500 kg m − .We find dM/dt ∼ /r H ) . with dM/dt in kg s − , over the 9 ≤ r H ≤
16 AU range.At the upper end of this distance range, the derived mass loss rate is about a factor of twolarger than obtained by Jewitt et al. (2019) using less complete data and slightly differentassumptions about the particle properties. We regard the difference as insignificant. Massloss rates in K2 at 10 AU are comparable to those found in many comets at 1 AU, indicatingthe large mean particle size and the high level of activity in K2.
Dust mass loss rates implied by Equation (13) can be supplied by freely sublimatingexposed CO of area A ∼ ( dM/dt ) / ( f s f dg ), where f dg is the dust to gas ratio. For example,dust production at rate dM/dt = 400 kg s − at 15 AU, where f s = 1 × − kg m − s − ,requires a CO patch of area A ∼ f − dg km . This sets a lower limit to the radius of aspherical nucleus r n ≥ ( A/π ) / ∼ f − / dg km. Quantity f dg is unmeasured in comet K2,but in short-period comets, values f dg ≥ ≤ f dg ≤
30) have been reported in comet 2P/Encke (Reach et al. 2000), where the large,mass-dominant particles are reminiscent of those in K2. If f dg = 20, for instance, the COcould be supplied from a sublimating area A ∼ and a nucleus of radius r n ≥ f s will be over-estimated in this calculation and a larger sublimating area wouldbe needed.Gas drag forces resulting from the free sublimation of exposed supervolatiles wouldbe sufficient to eject micron-sized grains against the gravity of the nucleus, but incapableof overcoming grain-grain cohesive forces (Gundlach et al. 2015, Jewitt et al. 2019) atthese distances. Paradoxically, gas drag forces can overcome the (weaker) cohesive forcesbinding 1 mm sized particles but cannot eject them against the gravitational attractionto the nucleus because they are too heavy. Therefore, taken at face value, no particles ofany size can be ejected by gas drag, creating the so-called “cohesion bottleneck” whichoperates beyond a critical distance that is controlled by the latent heat of sublimation ofthe responsible volatile and by cohesion. One solution to the bottleneck problem might bepressure build-up inside a porous medium having significant tensile strength, for examplebeneath a mantle or within postulated centimeter sized pebbles as proposed by Fulle etal. (2020).Activity driven by the (exothermic) crystallization of amorphous water ice, leading tothe release of trapped molecules (Prialnik et al. 2004), is possible at distances r H (cid:46)
10 AUbut not at the low equilibrium temperatures found at larger distances (Guilbert-Lepoutre2012). Comet K2 so far shows no evidence for excess activity that might be attributed tocrystallization down to r H = 9 AU (Figure 3), presumably indicating that amorphous ice isnot present in close thermal contact with the surface. However, the interpretation of thisobservation is ambiguous. Amorphous ice could be absent in the nucleus, or it could simply 18 –have migrated to greater depths during a previous approach to the Sun. In the latter case,crystallization might play a future role, detectable by a surge in activity, as K2 approachesperihelion and the surface thermal wave diffuses into the interior. A variety of non-thermalprocesses might also operate at large distances, including thermal fracture and electrostaticsupercharging (Jewitt et al. 2019) but these are probably minor contributors to the activity. Mass loss at rate dM/dt corresponds to global erosion of a spherical nucleus of radius r n at the rate dr n /dt = − ( dM/dt ) / (4 πr n ρ ). At r H = 15 AU, for example, dM/dt ∼
400 kgs − (Figure 8) and a 5 km radius nucleus would shrink at the rate dr n /dt ∼ -8 cm yr − .At r H = 10 AU, dM/dt ∼ − and dr n /dt ∼ −
20 cm yr − . By extrapolation ofEquation (13), a meter or more of surface will be lost by the time K2 reaches r H = 5 AU,where water ice sublimation is expected to begin. As on other comets, instead of beingglobal, mass loss from K2 is likely to be confined to a fraction of the surface of the nucleus,with material lost locally from much smaller areas and greater depths than indicated bythese global average values. We conclude that topography on the nuclei of K2 and otherlong-period comets, even those entering the planetary region for the first time, can besubstantially altered long before reaching the water ice sublimation zone.Cometary mass loss in the outer solar system may also account for one observationalpuzzle concerning the comets. Specifically, numerous observations show that the opticalcolors of cometary nuclei and cometary dust are independent of the dynamical classificationof the comet. Short-period and long-period comets are indistinguishable by their opticalcolors (Jewitt 2015). For example, K2 has B-V = 0.74 ± ± ± ± ± ± The recoil from anisotropic mass loss produces the non-gravitational acceleration ofcometary nuclei. To estimate the magnitude of the non-gravitational acceleration on thenucleus of comet K2, we assume that the ultimate source of the outflow momentum is theexpansion of gas, produced by sublimation at the nucleus surface, into the surroundingvacuum. Dust is dragged from the nucleus by the gas, which escapes at speed, V g . Whilethe momentum is always dominated by the gas, the mass loss rate in more slowly movingdust can rival or exceed the mass loss rate in gas, giving rise to the ratio of dust to gasmass production rates f dg >
1. Then, the gas mass loss rate is f − dg ( dM/dt ) and the forceexerted by the gas is k R ( dM/dt ) V g /f dg . The momentum transfer coefficient, k R , is equal tothe fraction of the outflow momentum transferred to the acceleration of the nucleus, with k R = 1 for collimated ejection and k R = 0 for isotropic emission.The resulting magnitude of the non-gravitational acceleration is α ng = k R V g f dg M n (cid:18) dMdt (cid:19) (14) 20 –where M n = (4 π/ ρr n is the mass of the nucleus, assumed to be spherical and of density ρ , radius r n .Dynamically, what matters is the ratio of α ng to the local acceleration due to the gravityof the Sun, g (cid:12) . We define α (cid:48) = α ng /g (cid:12) and write g (cid:12) = GM (cid:12) /r H , where G = 6 . × − Nkg − m is the gravitational constant, M (cid:12) = 2 × kg is the mass of the Sun and r H isexpressed in meters. Then, we substitute for M n to find α (cid:48) = 3 k R V g πGM (cid:12) ρf dg (cid:18) r n (cid:19) (cid:16) r H (cid:17) (cid:18) dMdt (cid:19) (15)where r n is in kilometers, r H is in AU, dM/dt is in kg s − and α (cid:48) is dimensionless.The outflowing gas travels at approximately the thermal speed, given by V g =(8 kT / ( πµm H )) / , where k = 1 . × − J K − is Boltzmann’s constant, µ is the molecularweight and m H = 1 . × − kg is the mass of hydrogen. Ices like CO ( µ = 28) areso volatile that their sublimation depresses the surface temperature to T ∼
20 to 25 K,approximately independent of heliocentric distance, at which temperature the thermal speedis V g ∼
130 m s − . The best-measured value of the momentum transfer coefficient is k R = 0.5, for 67P/Churyumov-Gerasimenko (Appendix to Jewitt et al. 2020). Measurementsof other comets show that f dg is spread over a wide range, with most of the mass carriedby large particles like those in the coma of K2. Unfortunately, neither k R nor f dg has beenmeasured in comet K2. For the sake of definiteness, we adopt k R = 0.5, f dg = 1, ρ = 500kg m − and substitute dM/dt from Equation (13) into Equation (15). Neglecting a smallresidual heliocentric dependence, we obtain α (cid:48) ∼ × − (cid:18) r n (cid:19) (16)with r n expressed in kilometers. 21 –The solid black and red lines in Figure (9) show α (cid:48) as a function of r n , for assumedvalues of the dust to gas ratio f dg = 1 and 20, respectively. We have plotted α (cid:48) in the radiusrange 0 < r n ≤
10 km, bearing in mind the empirical upper limit to the radius of K2 ( r n ≤ α (cid:48) for well-characterized cometary nuclei, using nucleus radiiand solutions for the non-gravitational parameters A , A and A taken from NASA’s JPLHorizons site . The magnitude of α (cid:48) varies around the orbit, but reaches a maximum valuenear perihelion, where outgassing is strongest. Therefore, we show α (cid:48) computed with r H equal to the perihelion distance of each comet. Specifically, we used α (cid:48) = g ( q ) q GM (cid:12) ( A + A + A ) / (17)where g ( q ) is the function defined by Marsden et al. (1973) to represent the sublimationrate of water ice, evaluated at the perihelion distance, q . We note that, in Equation (17)the total acceleration is dominated by the radial component, A , because the bulk of themass loss is directed sunward from the heated dayside of the nucleus. The resulting pointsplotted in Figure (9) should be regarded as upper limits to α (cid:48) in each case because, unlikesupervolatile CO, the sublimation rate of less volatile water ice falls faster with heliocentricdistance than r − H .Figure (9) shows that, depending on the nucleus radius, the dynamical effect ofoutgassing in K2, even when beyond Saturn, should be comparable to that in comets ofthe inner solar system, for which 10 − (cid:46) α (cid:48) (cid:46) − . However, observations of short-periodcomets are possible over multiple orbits, making visible the cumulative effects of even verysmall α (cid:48) . The detection of non-gravitational acceleration in K2 (and other long-period https : //ssd.jpl.nasa.gov/horizons.cgi
22 –comets for which only single-orbit observations are possible) will be more difficult. Forexample, consider a nucleus with f dg = 1 and r n = 9 km, the maximum possible radiusestimated for K2 (Jewitt et al. 2019). Figure (9) indicates α (cid:48) ∼ × − . The resultingdisplacement of the nucleus caused by non-gravitational acceleration acting continuouslyover the three years from 2017 to 2020 would be an immeasurably small ∼
60 km. Fora smaller nucleus, r n = 3 km, α (cid:48) increases to ∼ − and the resulting displacementwould increase to ∼ (cid:46) (cid:48)(cid:48) in the plane of thesky, comparable to or slightly smaller than the best astrometry reported for comets, andtherefore still very difficult to detect. Moreover, in single-orbit observations, a non-zero α (cid:48) can be easily misinterpreted as caused by an orbital eccentricity slightly different fromthe outgassing-free value. For these reasons, non-gravitational accelerations of long-periodcomets remain undetected in the middle and outer solar system, at least in comets withperihelia q (cid:38)
4. SUMMARY
We present Hubble Space Telescope measurements of inbound, long-period cometC/2017 K2 over the range of heliocentric distances from r H = 15.9 AU to 8.9 AU. We findthat1. The particle properties established in previous observations (a coma of large,slowly-moving grains distributed in a nearly spherical coma) are unchanged acrossthis heliocentric distance range.2. The dust scattering cross-section, measured within a fixed, nucleus-centered volume,varies with heliocentric distance as C e ∝ r − . ± . H . This weak distance dependencereflects an inverse relation between dust ejection speed and distance and also impliesthat a significant fraction of the dust cross-section is carried by ultra-slow, nearlyco-moving particles released at much larger distances.3. The release of ultra-slow particles began at Kuiper belt distances ( r H ∼
35 AU),presumably driven by the sublimation of carbon monoxide (or other supervolatile ice).4. The normalized non-gravitational acceleration, α (cid:48) , even when in the giant planetregion of the solar system, may rival α (cid:48) measured in comets in the terrestrial planetregion. Distant outgassing may set the ultimate limit to the accuracy with which theorbits of long-period comets (and so the structure of the Oort cloud) can be deduced.We thank the anonymous referee for comments. Based on observations made underGO 14939, 15409, 15423 and 15973 with the NASA/ESA Hubble Space Telescope, obtainedat the Space Telescope Science Institute, operated by the Association of Universitiesfor Research in Astronomy, Inc., under NASA contract NAS 5-26555. Y.K. and J.A. 25 –acknowledge funding by the Volkswagen Foundation. J.A.’s contribution was made inthe framework of a project funded by the European Union’s Horizon 2020 research andinnovation programme under grant agreement No 757390 CAstRA. Facilities:
HST. 26 –
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UT Date & Time Tel a DOY a r H b ∆ c α d θ −(cid:12) e θ − V f δ ⊕ g h HST 179 15.869 15.811 3.7 166.3 357.2 +0.592017 Nov 28 17:08 - 17:52 HST 332 14.979 15.133 3.7 17.1 358.3 +1.382017 Dec 18 22:30 - 23:13 HST 352 14.859 15.010 3.7 358.4 357.3 +0.082018 Mar 17 09:28 - 10:34 HST 441 14.331 14.328 2.0 275.6 353.3 -3.972018 Jun 15 15:40 - 16:19 HST 531 13.784 13.668 4.2 181.3 356.3 -0.282019 Oct 03 04:40 - 05:17 HST 1007 10.736 10.871 5.3 70.9 2.1 +5.152019 Dec 17 01:33 - 02:10 HST 1082 10.224 10.536 5.2 1.3 358.0 +0.312020 Mar 14 20:28 - 21:05 HST 1169 9.603 9.634 5.9 278.8 352.5 -5.942020 Jun 18 18:51 - 19:28 HST 1265 8.913 8.626 6.4 178.1 357.3 -0.08 a Day of Year, DOY = 1 on UT 2017 January 01 b Heliocentric distance, in AU c Geocentric distance, in AU d Phase angle, in degrees e Position angle of projected anti-solar direction, in degrees f Position angle of negative heliocentric velocity vector, in degrees g Angle from orbital plane, in degrees h Observations from GO 14939, described in Jewitt et al. (2017)
30 –Table 2. Fixed-Aperture Photometry aUT Date θ (cid:96)/ = 5 10 20 40 80 1602017 Jun 28 0.44 21.59/9.45/2.5 20.80/8.66/5.2 20.04/7.90/10.4 19.34/7.20/19.8 18.83/6.69/32 18.63/6.49/382017 Nov 28 0.46 21.31/9.38/2.6 20.54/8.61/5.4 19.78/7.85/10.9 19.09/7.16/20.4 18.64/6.71/31 —2017 Dec 18 0.46 21.31/9.42/2.6 20.52/8.63/5.3 19.74/7.85/10.9 19.03/7.14/20.8 18.53/6.64/33 18.27/6.38/422018 Mar 17 0.48 21.03/9.31/2.8 20.26/8.54/5.8 19.49/7.77/11.7 18.80/7.08/22.1 18.32/6.60/34 18.13/6.41/412018 Jun 15 0.50 20.81/9.27/2.9 20.05/8.51/5.9 19.29/7.75/11.9 18.57/7.03/23.1 18.02/6.48/38 17.78/6.24/482019 Oct 03 0.63 19.61/9.06/3.6 18.86/8.31/7.1 18.12/7.57/14.1 17.39/6.84/27.6 16.78/6.23/48 16.55/6.00/602019 Dec 17 0.65 19.38/9.01/3.7 18.63/8.26/7.4 17.90/7.53/14.6 17.18/6.81/28.3 16.57/6.20/50 16.31/5.94/632020 Mar 14 0.72 19.02/8.95/3.9 18.27/8.20/7.8 17.53/7.46/15.5 16.83/6.76/29.6 16.23/6.16/51 15.95/5.88/662020 Jun 18 0.80 18.56/8.88/4.2 17.80/8.11/8.5 17.06/7.38/16.8 16.35/6.67/32.4 15.72/6.04/58 15.43/5.75/76 a For each date and aperture radius, (cid:96) (measured in units of 10 km at the comet), the Table lists the apparent magnitude, V,the absolute magnitude, H, and the scattering crossection, C e (in units of 10 km ), in the order V/H/ C e . C e is computed fromH using Equation (1). b θ is the angle subtended by the radius of the smallest (5000 km) aperture, in arcsecond.
31 –Table 3. Size Distribution Indices aComet Method b Radii ( µ m) Index, q Reference1P/Halley In-Situ >
20 3.5 ± > > >
60 3.3 Fulle et al. (1993)67P/Churyumov-Gerasimenko (coma) In-Situ > +0 . − . Marschall et al. (2020)67P/Churyumov-Gerasimenko (trail) Optical >
100 4.1 Agarwal et al. (2010)81P/Wild Optical > ± > > ± > ± a Reported differential power-law size distribution index, q . b In-Situ: measured by a spacecraft in the coma. Optical: remote determination by fitting tail isophotes
32 –Fig. 1.— Images of C/2017 K2 on four epochs shown in the top row with common scalingfrom -0.05 to 2.0 data numbers per second, showing the steady brightening of the comet.The bottom row shows the same four images but with contours spaced by a factor of twoin surface brightness starting from 0.01. Each panel has North to the top, East to the Left,and is 32 (cid:48)(cid:48) tall. The heliocentric distances of the comet are indicated, as are the projecteddirections of the anti-solar vector ( −(cid:12) ) and the negative heliocentric velocity vector ( − V ). 33 –Fig. 2.— (Top row:) Same as Figure 1 to give the scale (vertical yellow bar is 160,000km in length), orientation and direction arrows for comparison with (Bottom row:) Imagesspatially filtered by dividing each image by the annular median. The location of the nucleusis marked in each panel by a yellow circle. 34 – C r o ss - S ec ti on , C e [ k m ] Heliocentric Distance, r H [AU] s = 1s = 2 s = 1.14±0.05 Fig. 3.— Total scattering cross-section of K2 (yellow-filled circles) as a function of theheliocentric distance. The cross-section was computed as described in the text from Equation(1). Equation (3) is shown as a solid red line. Black lines ilustrate heliocentric indices s =1 and 2, as labeled. 35 – n = 9 kmr n = 3kmr n = 6 kmMeasurement P a r ti c l e V e l o c it y V ( mm , AU ) ( m s - ) Heliocentric Distance, r H [AU] Fig. 4.— Velocity of 1 mm radius particles as a function of heliocentric distance. The curveswere computed from Equation (7) using r n = 3 km (long-dashed blue line), r n = 6 km (solidblack line) and r n = 9 km (short-dashed red line), respectively. The yellow-filled circle showsthe speed measured using a Monte Carlo simulation of data on UT 2018 June 15. 36 – H e li o ce n t r i c I nd e x , s Activity Initiation Distance, r * [AU] -- -- -- -- s = 1.14+/-0.05 -- Fig. 5.— Heliocentric index vs. activity initiation distance. Monte Carlo simulations showthe model index for size distribution indices q = 3.0 (yellow circles), 3.5 (red circles) and q =4.0 (green circles), respectively. The measured heliocentric index and its ± σ uncertaintiesare indicated in the yellow-shaded horizontal box. Dates in the middle part of the figureshow the heliocentric distances reached by C/2017 K2 on approach to perihelion. 37 – C r o ss - s ec ti on , C [ k m ] Physical Radius [km x 1000]12
Fig. 6.— Cumulative dust cross-section as a function of aperture radius for each date ofobservation. Power law slopes in C ∝ r x , with x = 1 and 2 are marked for comparison withthe data. 38 – E n c i r c l e d R a d i u s , r [ k m x ] Heliocentric Distance, r H [AU] r H-0.23+/-0.05 r = (80±2) x 10 km Fig. 7.— Coma radius r as a function of heliocentric distance, r H . 39 – D u s t M a ss L o ss R a t e , d M / d t [ kg s - ] Heliocentric Distance, r H [AU]dM/dt = 1050(10/r H ) Fig. 8.— Dust mass loss rate computed from Equation (13) as a function of heliocentricdistance. 40 – -8 -7 -6 -5 -4 α ', f dg = 1 α ', f dg = 20Comets N o r m a li ze d A cce l e r a ti on , α ' Nucleus Radius, r n [km]
21P 1P26P19P81P 9P67P103P 10P2P
Fig. 9.— Normalized non-gravitational acceleration of C/2017 K2 is plotted as a function ofnucleus radius, r n , from Equation (16). Black and red curves show dust to gas ratios f dgdg