A Distribution of Large Particles in the Coma of Comet 103P/Hartley 2
Michael S. Kelley, Don J. Lindler, Dennis Bodewits, Michael F. A'Hearn, Carey M. Lisse, Ludmilla Kolokolova, Jochen Kissel, Brendan Hermalyn
AA distribution of large particles in the coma of Comet103P/Hartley 2
Michael S. Kelley a, ∗ , Don J. Lindler b , Dennis Bodewits a , Michael F. A’Hearn a , Carey M.Lisse c , Ludmilla Kolokolova a , Jochen Kissel d,1 , Brendan Hermalyn e a Department of Astronomy, University of Maryland, College Park, MD 20742-2421, USA b Sigma Space Corporation, 4600 Forbes Boulevard, Lanham, MD 20706, USA c Johns Hopkins University–Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723,USA d Max-Planck-Institut f¨ur Sonnensystemforschung, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany e NASA Astrobiology Institute, Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive,Honolulu, HI 96822, USA
Abstract
The coma of Comet 103P/Hartley 2 has a significant population of large particles observedas point sources in images taken by the
Deep Impact spacecraft. We measure their spatialand flux distributions, and attempt to constrain their composition. The flux distribution ofthese particles implies a very steep size distribution with power-law slopes ranging from − . − .
7. The radii of the particles extend up to 20 cm, and perhaps up to 2 m, but theirexact sizes depend on their unknown light scattering properties. We consider two cases:bright icy material, and dark dusty material. The icy case better describes the particlesif water sublimation from the particles causes a significant rocket force, which we proposeas the best method to account for the observed spatial distribution. Solar radiation is aplausible alternative, but only if the particles are very low density aggregates. If we treatthe particles as mini-nuclei, we estimate they account for < −
80% of the comet’s totalwater production rate (within 20.6 km). Dark dusty particles, however, are not favored basedon mass arguments. The water production rate from bright icy particles is constrained withan upper limit of 0.1 to 0.5% of the total water production rate of the comet. If indeedicy with a high albedo, these particles do not appear to account for the comet’s large waterproduction rate.
Erratum:
We have corrected the radii and masses of the large particles of comet103P/Hartley 2 and present revised conclusions in the attached erratum. ∗ Corresponding author
Email addresses: [email protected] (Michael S. Kelley), [email protected] (Don J.Lindler), [email protected] (Dennis Bodewits), [email protected] (Michael F. A’Hearn),
[email protected] (Carey M. Lisse), [email protected] (Ludmilla Kolokolova), [email protected] (Brendan Hermalyn) retired Kelley et al. 2013, Icarus 222, 634-652 a r X i v : . [ a s t r o - ph . E P ] S e p . Introduction Comet 103P/Hartley 2 (hereafter, 103P or Hartley 2) is a hyperactive comet. The ac-tivities of most comets are consistent with surfaces that are over 90% inert, i.e., activity isrestricted to a few localized sources. In contrast, the gas production rates of the hyperactivecomets suggest activity over ∼ Q H O ≈ × molec s − (Combi et al. 2011). The sublimation rate of an isothermal nucleus with a 4% Bond albedoat 1.03 AU (the 1997 perihelion distance of Hartley 2) is 2 . × molec cm s − (Cowanand A’Hearn 1979). Thus, approximately 10 km of surface area would be required to matchthe measured water production in 1997. However, constraints on the size of the nucleus byGroussin et al. (2004) and Lisse et al. (2009) suggested the total surface area was near 4–6 km , yielding an active fraction (area based on Q H O /area of the nucleus) near 1.7–2.5. Inorder to account for the high active fraction, Lisse et al. (2009) proposed that the coma con-tained a population of icy grains, which increased the surface area available for sublimation,and therefore allowed for a relatively high water production rate.The Deep Impact spacecraft flew by Comet 103P on 4 November 2010. The minimumflyby distance was 694 km, occurring at 13:59:47.31 UTC with a relative speed of 12.3 km s − (A’Hearn et al. 2011). Rather than just being the fifth comet nucleus to be imaged up closeby a spacecraft, Comet Hartley 2 might also be considered an archetype of the hyperactivecomets. The flyby images verified the small surface area of the nucleus (5.24 km ; Thomaset al. this issue), and Deep Impact’s
IR spectrometer revealed a coma of icy grains (A’Hearnet al. 2011). Knight and Schleicher (this issue) discovered tail of OH and NH, evident on10 km scales, which they conclude are derived from small grains of ice that were accelerateddown the tail before completely sublimating. Bonev et al. (this issue) observed enhancedheating of the water gas 20–150 km down the tail, which they also attribute to sublimatingice grains in the tail. The icy coma hypothesis of Lisse et al. (2009) seems to be qualitativelyvalid. What remains is a quantitative verification that the ice grains in the coma are asignificant source of water gas.A’Hearn et al. (2011) also discovered bright point sources distributed around the nucleusof Hartley 2 in visible wavelength images. The fluxes of these sources are consistent with apopulation of objects centimeter-sized or larger. A’Hearn et al. (2011) could not determine ifthese large particles were icy or refractory. In the following paper, we will attempt to discernthe nature of these point sources and if they might be the cause of Hartley 2’s hyperactivity.The point sources around Hartley 2 are found in visible wavelength images taken withboth the High Resolution Instrument (HRI) and the Medium Resolution Instrument (MRI).Their spatial distributions and densities vary, but they are not background stars; celestialsources are significantly streaked in images taken while the spacecraft-comet distance, ∆,was less than 10 km. That the HRI detected these point sources is significant. The HRIvisible camera (HRIVIS) is not in optimal focus, and has an accordingly large and doughnut-shaped point spread function (PSF) (Lindler et al. 2007). Thus, unresolved sources are easilydiscriminated from cosmic ray impacts and hot pixels. None of the point sources appear tobe resolved, therefore they must be smaller than the FWHM of the HRIVIS deconvolvedPSF (approximately 3 m at ∆ = 700 km).In addition to point sources, streaks of various lengths are present in many of the images.2he lengths of the streaks are correlated with exposure time (longer exposures produce longerstreaks), and inversely proportional to instrument pixel scale (streaks in HRIVIS images arelonger than in MRI). The streaks are parallel to each other, and the orientation is consistentwith the spacecraft’s motion as it tracks 103P’s nucleus. Stars and other distant objectsproduce streaks of uniform length and orientation, but we find a variety of streak lengthsin a single image, down to a few pixels long, which indicates they are spatially close to thenucleus and not celestial sources.Altogether, the above observations indicate that the nucleus of Comet 103P is surroundedby a coma of thousands of large particles. We will use the term “particle” to refer tothe observed point sources in this paper. A’Hearn et al. (2011) demonstrated that theseparticles are centimeter-sized or larger. Large particles are not unique to this comet, butthese observations are unique to large particles. These are the first images of sub-meterparticles from a comet seen as individual objects. Prior images in visible/infrared light(e.g., comet dust trails; Sykes and Walker 1992, Ishiguro et al. 2002, Reach et al. 2007)and with radar (Harmon et al. 2004) probed collections of many large particles. The onlyother observations of large individual particles from comets that we can think of are thoseof meteors (from meteor streams associated with particular comets) and milligram particleimpacts on spacecraft. Although the latter examples may only be of order 0.1–1 mm in size.In the following paper, we present our methods for detecting and measuring the photo-metric properties of the large particles in the coma of Comet 103P. We will convert theirfluxes into sizes, discuss their spatial distribution, attempt to constrain their composition,and compare them to large particles observed in radar observations of the comet. Assumingthey are icy, we also constrain their contribution to the comet’s total water production rate.
2. Observations
Images of Comet 103p Were taken with
Deep Impact ’s Medium Resolution Instrumentand High Resolution Instrument CCD cameras. Both cameras have 1024 × (cid:48)(cid:48) pixel − (2 µ rad), and the MRI pixel scale is 2.06 (cid:48)(cid:48) pixel − (10 µ rad). The instruments are described by Hampton et al. (2005), and their calibrationby Klaasen et al. (2008, in prep.). The MRI and HRIVIS data are available in the NASAPlanetary Data System (PDS) archive (McLaughlin et al. 2011a,b). Images taken with theMRI are labeled with the prefix mv, and for the HRIVIS the prefix is hv. We keep thisconvention throughout the paper. We used a pre-PDS data set, but the only significantdifference is the absolute flux calibration constant, which we account for in our photometry.Optimal focus for the HRIVIS instrument occurs about 6 mm before the focal plane.For this instrument, it is often beneficial to work with spatially restored (i.e., deconvolved)images. We follow the method of Lindler et al. (2007), using the HRIVIS PSF from theEPOXI mission (Barry et al. 2010), to restore raw HRIVIS images to near diffraction-limitedresolution. The HRIVIS images are deconvolved (Fig. 6b) with the Richardson-Lucy (R-L)method modified to handle non-Poisson CCD readout noise (Snyder et al. 1993, Lindler et al.2007). Note that R-L restoration methods conserve flux.The large particles are easily seen in MRI and HRIVIS images at ∆ (cid:46) km. Outsideof this range, the particles become increasingly faint, overwhelmed by the diffuse coma’ssurface brightness, and confused with stars, which are unstreaked point sources when the3pacecraft-comet distance is large. In a manual search, the earliest image in which wehave identified particles is HRIVIS image hv5000096, taken 22,400 km from the nucleus(CA-30 min, 1500.5 ms exposure time). The first MRI image in which we have positivelyidentified particles is mv5002025 (8580 km, CA-11 min, 500.5 ms).Most images at ∆ < km use the CLEAR1 filter. For sources with solar-like spectra,the CLEAR1 filters have effective wavelengths of λ e = 0 . µ m for HRIVIS, and 0.610 µ m forMRI, and a FWHM of ≈ . µ m. The absolute calibration uncertainties for the CLEAR1filters are 5% for HRIVIS, and 10% for MRI (Klaasen et al. 2008). Large particles havenot yet been identified in the HRI infrared spectrometer (HRIIR) scans (Protopapa et al.2011), but they may be discovered in the future when we can predict the location of thebrightest particles in IR data using their MRI and HRIVIS derived 3D positions and velocities(Hermalyn et al. this issue).A summary of all images used in this paper is presented in Table 1. Note that the HRIVISimages used in this paper are only a small sub-set of the HRIVIS particle observations. Wehave limited our investigation to the closest-approach images because of the great amountof time that must be taken to analyze the HRIVIS data. We also do not use any of the50.5 ms MRI images in the PDS archive. These images were taken along with the 500.5 msimages, but few particles are found in these images due to the decreased sensitivity and greatdistances from the nucleus.
3. Detection, photometry, and completeness
We automatically detect and measure all point sources throughout all MRI images takenwithin ∆ < . σ above the local background: 1 σ per pixel ≈ . × − ,0 . × − , and 0 . × − W m − µ m − sr − for 41, 121, and 501 ms frames, respectively.Twenty-five isolated and bright point sources were selected by hand from image mv5004056to estimate the MRI PSF. The sources were fit by DAOPHOT with a variety of functions,and the best-fit PSF used a Moffat function ( β parameter of 1.5; Moffat 1969) with a look-up table of empirical residuals. We compared DAOPHOT-derived fluxes of bright isolatedparticles to those measured with aperture photometry. The values agreed, with a meanerror of ∆ F λ /F λ = − . F λ is the difference between the DAOPHOT flux andthe aperture flux ( F λ, DAOPHOT − F λ, aper ).DAOPHOT iteratively estimates the background as it fits each point source and groups ofpoint sources, but we found—especially for detections closer to the nucleus—that subtractingan initial estimate of the background improved DAOPHOT’s ability to find and measure4ources. Furthermore, the MRI CCD has an instrumental background that varies row-by-row at the 1–4 DN level (the MRI and HRIVIS calibration constants for CLEAR1 are3 . × − and 1 . × − W m − µ m − sr − DN − s, respectively). The EPOXI pipelineincludes a routine to detect and remove this background, but it is not executed on imagesdominated by coma, such as those in our analysis. We derive our background estimate foreach image using two median filters (i.e., low-pass filters). We first apply a 3 ×
25 pixel (row × column) filter that defines the background stripes, as well as much of the coma. A second25 × ×
11 pixeluniform structuring element, which effectively generates an image of peak-to-peak values ina moving 11 ×
11 pixel box (Fig. 1c). We threshold the gradient image at a value of 0.1W m − µ m − sr − , then dilate the mask by 30 pixels to fill small gaps between features, andcover some of the nearby jets (Fig. 1d).DAOPHOT will occasionally fit streaks in the MRI images with PSFs. Most of thesefits have poor residuals, and are removed before the final analysis (described below). Butwe found several streaks persisting into the final analysis in images taken at large distancesfrom the comet (∆ (cid:38) F λ, min ).The completeness tests can be used to derive the photometric uncertainties. As anexample, we plot the input flux versus the DAOPHOT flux error for all artificial particlesadded to image mv5004046 (Fig 3). In this image, the mean flux error ∆ F λ /F λ = − −
10% to − σ ), we find the mean ∆ F λ /F λ ranges from − .
5% to − . igure 1: The process by which we remove the background and mask the nucleus in MRI images. (a) Afully calibrated image, mv5004046. (b) The median-filter subtracted image. (c) Image of the morphologicalgradient of image b. (d) The final mask, defined where image c > . − µ m − sr − . The mask hasbeen dilated by 30 pixels to fill gaps and cover some additional coma. The image scale and orientation aregiven in panel d. The projected vectors are comet-Sun ( (cid:12) ), comet-Earth ( ⊕ ), and spacecraft velocity ( v DI ). igure 2: Example point source detection completeness functions. All images were taken within ∆ =694 −
704 km, except mv5004009 (501 ms) taken at ∆ = 1858 km. The mv5004046 sub-frame is chosen tomatch the same area as covered in hv5004031. The HRIVIS test particles are selected from one of 9 specificfluxes (marked by squares and circles), whereas the MRI test particles are selected at random from a rangeof fluxes. The lowest of the HRIVIS bins is 10,000 e − . | ∆ F/F | > χ > . × σ from the background. For the 41 ms images 62 −
78% and 0 −
2% of the detected sourceswere rejected by the two criteria, respectively. For the 121 ms images the rejection rates were5 −
32% and 2 − − − > .
025 pixel − ). Thesesources are primarily rejected by the χ criterion. Regions with a rejection rate >
20% in a25 ×
25 pixel area will be masked from our analyses.
The analysis of the HRIVIS images required an approach independent from the MRIanalysis. Figure 6a shows an HRIVIS image of the large particles and Fig. 6b is the restored(i.e., deconvolved) image. The broad defocused PSF and the longer trails resulting fromparallax motion make PSF fitting even more difficult than in the MRI images. Our approachis to instead detect and measure a particle’s brightness in restored images. In summary, we:(1) iteratively remove the background from each image; (2) detect point sources and streaksin the restored images; (3) measure particle fluxes; and, (4) measure the completeness of ourparticle detection scheme. Below we detail our methods.We use an iterative technique to remove the background before final object detectionand photometry. The initial background is set to the deconvolved image filtered with a51 ×
51 pixel median filter. We then fit a smooth surface (using cubic splines) to the filteredimage to obtain our background image. The objects within the image will bias the medianfilter and the resulting image overestimates the background. To minimize this bias, we repeatthe process after ignoring all pixels more than 400 e − above the background (object pixels)during median filtering (1 e − s − = 4 . × − W m − µ m − sr; Klaasen et al. in prep.).We repeat this process three times to obtain and remove our final background (Fig. 6c).We detect the objects in the image by creating a mask of all pixels more than 1,000 e − above the background level (Fig. 6d). We selected the 1,000 e − threshold by visual inspectionof the restored image to avoid detection of restoration artifacts that result from “ringing”around the brighter objects (Lindler et al. this issue). We define contiguous masked pixelsas part of the same object. Visual inspection of this mask shows many instances where atrailed particle near the detection threshold is split into multiple objects because of noise8 igure 3: Input flux, F λ , plotted versus DAOPHOT error, ∆ F λ /F λ , for all artificial particles inserted inimage mv5004046. Most fluxes are measured to within 8% (1 σ ) of their true value, with a slight skew towardnegative errors. igure 4: (a) A detail of image mv5004046, after background coma subtraction. (b) The same image afterbeing cleaned of point sources detected and fit by DAOPHOT. In both images, particles with large residualsor low signal-to-noise flux ratios have been circled. Similar to Figure 1, the image scale and orientation areshown. The square in the lower right image shows the field of view of panels a and b with respect to thenucleus. igure 5: The same as Fig. 4, but for a region of high point source density in image mv5004029. igure 6: (a) Original HRIVIS image hv5004030. (b) Image deconvolved with the R-L method. (c) Image bafter background subtraction. (d) Mask of pixels greater than 1000 e − above the background. (e) Mask ofdetected sources. (f) Mask of the sources used to compute the flux distribution slope (here, length 5 pixelsor smaller and total flux greater than 20,000 e − ). − ( F λ = 1 . × − and 4 . × − W m − µ m − for exposure times of 126 and376 ms). On average, the effective aperture for point sources is approximately a 3 pixelradius circle.As done with the MRI analysis, we measure the completeness by inserting artificialobjects in the images of varying brightness and parallax length. These artificial objects areadded to the raw images before deconvolution. The completeness test also gives additionalevidence that our measured fluxes are linear with object brightness in the range of brightnessbeing considered. For the longer trails, the completeness decreased rapidly with flux. Wetherefore decided to ignore particles with a length more than 5 pixels in the restored image.In effect, this criterion limits the line-of-sight distance of the particles from the nucleus to < − < − F > ,
000 e − .
4. Particle fluxes
We measure the fluxes of all particles and compute their flux distributions, dn/dF . Theparticle fluxes approximately follow a power-law distribution. Therefore, we fit each fluxdistribution with a function of the form dn/dF ∝ F α using the method by Clauset et al.(2009), which is based on Kolmogorov-Smirnov tests and maximum likelihood fitting. Werestricted the flux range to particles brighter than our 80% completeness estimate. Thisrestriction increases the statistical errors in the fits but reduces possible bias errors resultingfrom photometry contamination from neighboring particles and residual background. More-over, the fitting method does not include a correction for completeness, so restricting the13uxes helps mitigate the effect of incomplete counting; the maximum completeness correc-tion over our fit range is 20%, which is insignificant in comparison to the strong power-lawslopes (see Fig. S6 of A’Hearn et al. 2011). Our results are listed in Table 2 and summarizedin Table 3. In Figs. 7 and 8, we plot the final flux distributions and their best-fit trends.The fluxes are plotted as a function of F λ, ≡ F λ ∆ / so that they may be directlycompared to each other (variation of the phase angle is small, ranging from 79 to 92 ◦ , and istherefore not considered). The error-weighted mean slopes for the MRI and HRIVIS framesare − . ± .
02 and − . ± .
03, respectively.The MRI and HRIVIS slopes are significantly different from each other. To verify thisdifference, we also computed the flux distributions from MRI sub-frames of images chosenclose in time to the HRIVIS images. The relative position of the HRIVIS and MRI fieldsare constant with time, therefore measuring the flux distribution in the same field of view isstraightforward. The comparison, however, is not exact because the frame-to-frame parallaxof individual particles can be significant (Hermalyn et al. this issue). After bad-PSF rejectionand completeness tests only ≈
130 particles could be used to determine the flux distributionsin the MRI sub-frames. This is a factor of 3–10 fewer than the number of particles in theHRIVIS images. The difference primarily relies on the larger aperture of the HRI primary,which allows for a fainter point source detection limit due to the finer resolution and greaterlight gathering area. Our best-fit parameters for the MRI sub-frames are listed in Table 2 andsummarized in Table 3. The uncertainties in the power-law slopes are quite large, rangingfrom 0.18 to 0.31, owing to the paucity of particles available in the MRI sub-frame images.The weighted mean slope is − . ± .
07, which is shallower than the mean MRI full-frameslope, but they agree at the 2 σ level.Taken together, the mean MRI full-frame, MRI sub-frame, and HRIVIS best-fit power-law slopes show evidence that the flux distribution steepens with increasing flux. Inspectionof the MRI full-frame distributions in Fig. 7 and the Kolmogorov-Smirnov probability ( P KS )values reveals that a power-law size distribution is not a good fit to most of the flux dis-tributions. This poor fit is especially true for the 121 ms frames at ∆ <
900 km wherethere is a significant curvature in the log-flux distributions of these images. The curvatureis non-existent or not prevalent in the shorter exposure MRI data, the distant MRI data(∆ > F λ, > − W m − µ m − , whereas the 121 ms MRI distributions are welldefined up to F λ, ≈ × − W m − µ m − . This difference is simply a matter of thelarger solid angle observed by the MRI camera as compared to the HRIVIS camera; a largersolid angle allows for more of the brightest particles to be detected. Second, the HRIVISdistributions are valid to lower fluxes than the MRI distributions (column F λ, min in Table 3),because the larger primary of the HRI allows for fainter particles to be accurately measured.Taken together, these differences demonstrate that the two instruments are measuring twodifferent particle flux regimes. In Fig. 9, we plot our best-fit power-law slopes versus theminimum flux used in each fit. A trend is evident; steeper slopes are correlated with larger F λ, min , but due to the different techniques, fit uncertainties, and fields of view involved,the correlation is not conclusive. We remind the reader that the correlation is not due toincomplete photometry because we restricted our data sets to fluxes where the completenessis better than 80%. Given this restriction, none of our completeness corrections are strong14 igure 7: Flux distributions ( dn/d log F ) and their best-fit power-law functions for all full-frame MRI datasets listed in Table 2. The flux distributions have been corrected for completeness, and their error bars arebased on Poisson statistics. The flux distributions are grouped by their integration times, and sorted bytime with the pre-closest approach images at the bottom. Each i -th flux distribution ( i = 0 , , . . . ) has beenoffset along the y-axis by 10 i , and labeled with ∆ for clarity. Additionally, the fluxes have been scaled by(∆ / so that they are directly comparable to each other. The gray × symbols mark particles not usedin the fit. igure 8: Same as Fig. 7, but for all HRIVIS images listed in Table 2. igure 9: Best-fit power-law slope versus minimum flux used in the fit for all images listed in Table 2. enough to have a significant effect on the best-fit slopes.In the absence of any additional information on the flux distribution we will proceedassuming that the differences between the MRI and HRIVIS flux distributions reflect the trueflux distribution of the entire particle population. As alternatives to a power-law distribution,we also considered a power-law with an exponential cut off, and a double power-law. Thefunctional forms are dndF ∝ F β exp ( − F/γ ) , (1) dndF ∝ Fγ β (cid:18) Fγ (cid:19) β − β , (2)where β i are power-law slopes, and γ is the flux at which the exponential decay reachesa factor of exp ( −
1) = 0 .
37 , and γ is the turn-over flux from the low- to the high-fluxslopes (both γ parameters are specified for ∆ = 700 km). The double power-law fits did notconverge on a single solution. The power-law with exponential cut-off function, however, isa better fit to the closest-approach data (∆ < − . ± .
07 and (5 . ± . × − W m − µ m − . Unfortunately, these best-fit valuesare poor fits at larger distances where the flux distribution better agrees with a power-law(Fig. 10). We suggest that the 121 ms distributions have a systematic effect that causesunder-counting of the largest flux bins. Again, we note that our completeness correctionsare not strong enough to significantly affect our fits, and that the distributions as shown inFigs. 7 and 10 have been corrected for completeness, yet the curvature remains in the closestapproach data.To test the hypothesis that the power-law slope is affected by the source density, wegenerated completely synthetic data sets with DAOPHOT, based on our MRI PSF. Wecreated 8 images each with the same power-law flux distribution ( dn/dF ∝ F − . ) andmaximum point source flux ( F λ, max = 1 × − W m − µ m − ), but F λ, min varied from5 × − down to 1 . × − W m − µ m − . We inserted the same number of point sources perlogarithmic flux bin, but because each image’s flux distribution spanned a different range, thetotal point source density varied from 4 × − pixel − up to 2.6 pixel − . We then measuredeach image’s flux distribution with DAOPHOT in a manner similar to our MRI images.Once the input column density reached 0.08 pixel − , our best-fit slopes steepened from theinput − . − . < α = − . ± .
02 and − . ± .
05, respectively. They agree despite the wide range in thenumber of particles fit (100–1000) and spacecraft-comet distances (900–5000 km, althoughthe best constraints are within 3000 km). Therefore, we consider these results to be robustand representative of the true flux distribution for F λ, (cid:38) × − W m − µ m − . Theshallower HRIVIS slopes appear valid for 1 × − (cid:46) F λ (cid:46) × − . We see no reason totrust one data set over the other, or to assume that a single power-law would be valid for allfluxes measured. Therefore, we will use a broken power-law for the remainder of the paper: dndF ∝ F α (cid:40) α = − .
85 for F λ, < . × − W m − µ m − α = − .
80 for F λ, ≥ . × − W m − µ m − , (3)where F λ, = 7 . × − is the break in the power-law, − . ± .
02 is the average HRIVISslope, and − . ± .
02 is the average MRI slope, measured from images at ∆ >
900 kmto avoid the apparent crowding effects at closest approach. The break was derived by least-squares fitting the combined MRI and HRIVIS data sets. The broken power-law and thecombined HRIVIS and MRI flux distributions are presented in Fig. 11. This model is a goodmatch to the ensemble (HRIVIS + MRI) data set.
Due to the great numbers of large particles, we can only accurately measure the totalscattered flux from the brightest particles in each image. Our best estimate of the frac-tion of the coma flux attributable to large particles is 0.11% for fluxes ranging 0 . − . × − W cm − µ m − (row labeled “MRI (distant)” in Table 3). This estimate was derivedfrom images with relatively little point source crowding (∆ > igure 10: Best-fit power-law with exponential cut off, derived from 121 ms MRI images taken within∆ = 1000 km, compared to selected flux distributions, labeled with “∆ (exposure time)” in km and ms. Thehistograms have been offset by a factor of 10 i for clarity. igure 11: Combined HRIVIS and MRI flux distributions ( F λ ≥ F λ, min ), scaled to N ( F λ = 7 . × − ) = 1.For the MRI data, only full-frame distributions from ∆ >
900 km are shown. The solid line is our best-fitbroken power-law (Eq. 3).
N >
300 particles). In order to build a complete cen-sus of the particles, we need to extrapolate the results from that limited range down tothe faint end of the distribution. We find that the brightest particles observed throughoutthe closest approach images are consistently near F λ, = 4 . × − W cm − µ m − . Onthe other end, the faintest particles we can manually find and measure have fluxes of order F λ, ∼ − W cm − µ m − (Fig. 12). If the very steep flux distribution we derived in § F λ, = 1 × − down to 1 × − W cm − µ m − , there should be severalmillion particles per image. However, at such great pixel densities (1 particle per 17 pixelarea) it should be very difficult to find faint isolated particles (the core of the HRIVIS nativePSF has an area of 113 pixels). Furthermore, if we let the flux distribution continue downto 1 × − W cm − µ m − , the large particles account for 100% of the coma flux, leaving noroom for any fainter particles (millimeter sized and smaller). Therefore, the flux distributionmust change or be truncated at fluxes fainter than F λ, = 1 × − W cm − µ m − . Theuncertainty in the lower flux limit is a major source of error in all estimates of the totalnumber, flux, cross section, and similar derived quantities. We list the results from our ex-trapolations in Table 4. We estimate that the large particles account for 2–14% of the totalcoma flux near the nucleus.If we take the the first image in which particles are easily seen, mv5002051 at ∆ = 5148 km(Fig. 13), we can estimate the total number of particles within a 20.6 km radius (the largestcircular aperture centered on the nucleus that fits within the image). We find a total comaflux of (2 . ± . × − W m − µ m − (includes the bright jets, but not the nucleus) andassume that 2–14% of that flux is attributable to large particles. The results are presentedin Table 4 and will be used below when we estimate the cross section, mass, and waterproduction rate of the large particles.Up to now, we have not addressed streaked particles in our MRI images. If a significantfraction of the flux in our images at ∆ ≈ ≥ . r − profile, the fraction of smeared particles reduces to effectively zero.
5. Particle size and composition
In the absence of any compositional information on these large particles, we assume twocases to demonstrate their likely range of sizes. First, we will consider that the particles arerefractory, and photometrically behave like comet nuclei (the “dusty” case). Then, we willconsider that the particles are icy, and behave like the icy satellite Europa (the “icy” case).We stress that these two cases are examples only. They may not reflect the true natureof the particles, but they do yield useful limits on the particle sizes. We will show thatthe particles are much larger than the 0.1–100 µ m sizes typically considered in comet dust.We purposefully avoid interpreting the large particles with phase functions that have beenderived for comet comae. Light scattered by comet dust comae is expected to be dominated21 igure 12: Two sub-frames of HRIVIS image hv5004031 (∆ = 694 km) showing the crowded field of particles:a) an MRI context image (mv5004046) with the approximate HRIVIS field of view outlined with a box (imageis log scaled from 0.001 to 1.0 W m − µ m − sr − ); b) HRIVIS image, sub-frames c/d and e/f are outlined andlabeled; c) HRIVIS sub-frame; d) the same as c, but deconvolved to enhance the spatial resolution; e) anotherHRIVIS sub-frame; f) panel e, deconvolved. Two particles with fluxes near 1 − × − W m − µ m − havebeen circled. The HRIVIS field of view is 1.4 km, and the sub-frames are 0.42 km. igure 13: Image mv5002051, background coma removed. Point sources identified by DAOPHOT with fluxesgreater larger than the image’s 80% completeness limit are circled to show the extent of the large particlecoma (particles below this limit can still be seen by eye). An image of the nucleus has been inserted intothe central masked region. The box marks the location of the inset image. Stars, most of which have beenautomatically masked, are streaked over 13 pixels in this image.
23y dust grains in the sub-micrometer to micrometer size range (Kolokolova et al. 2004),which scatter optical light differently than centimeter-sized particles.First we take the case in which the particles photometrically behave like comet nuclei,i.e., they have a very low albedo, and have a phase angle behavior like a macroscopic objectand not like small dust grains. We refer to this case as the “dusty case,” but, just likecomet nuclei, these model particles may be internally icy. We adopt the geometric albedo( A p = 0 . − . . θ mag for θ in units of degrees) ofHartley 2’s nucleus (Li et al. this issue). The geometric albedo is defined as the ratio ofthe energy scattered from the object toward a phase angle of 0 ◦ to that scattered from awhite Lambertian disk with the same cross section (cf. Hanner et al. 1981 for this and otherrelevant albedo definitions).In Section 4.1, we found that individual particle fluxes range from F λ, = 1 × − to4 . × − W m − µ m − , where F λ, is the particle flux normalized to a distance of 700 km.To convert between flux and cross section, we use the formula F λ = A p Φ( θ ) σ S λ, (cid:12) r h (4)where F λ is the particle flux in units of W m − µ m − , σ is the cross-sectional area of theparticle (cm ), S λ, (cid:12) is the solar flux density at 1 AU (W m − µ m − ), r h is the heliocentricdistance of the particle (1.064 AU), and ∆ is the spacecraft-particle distance (cm). For theHRIVIS and MRI CLEAR1 filters, we use a solar flux density of 1471 and 1435 W m − µ m − ,respectively (Klaasen et al. in prep.). To convert from cross section to effective radius,we assume a spherical geometry: σ = πa . Altogether, assuming a comet nucleus-likephotometric behavior, the effective radii of the particles range from 10 to 221 cm. The largestof these particles (4 m diameter) are just over the resolution of the HRIVIS reconstructedframes (about 3 m at a distance of 700 m). We consider these sizes to be an upper limit tothe true particle sizes. For a lower-limit estimate, we consider the case where the particlescattering function is similar to the icy satellite Europa: A p = 0 .
67 and Φ = 1 . − . θ +2 . × − θ (Buratti and Veverka 1983, Grundy et al. 2007). At a phase angle of 80 ◦ ,Europa’s phase function is 0.34. For this parameter set the effective radii are 0.8 to 17 cm.Following Eq. 4, where flux is proportional to cross-sectional area, we can compute thetotal observed particle cross section and add it to Table 4. We have assumed our icy particlecase, using the photometric parameters of Europa. To instead use our dusty case, multiplythe cross sections in the table by 158.Our flux distributions imply a very steep size distribution. Assuming a spherical geome-try, the differential size distribution is dnda = 2 N F α +11 a α +1 , (5)where F is the flux from a 1 cm radius particle in units of W m − µ m − , and particle radius a is in units of cm. The constant N is the solution to the equation: N = N (cid:90) F λ, max F λ, min dndF dF, (6)24sing the appropriate values from Table 4. Our low-flux power-law slope ( α = − . a − . . This slope is steeper than the manysize distribution estimates of small through large dust grains ( a ∼ µ m to 1 mm with slopesnear − −
3) based on grain thermal emission (Lisse et al. 1998, Harker et al. 2002, 2011),grain dynamics (Fulle 2004, Reach et al. 2007, Kelley et al. 2008, Vaubaillon and Reach 2010),and dust flux monitors on spacecraft (McDonnell et al. 1987, Green et al. 2004). Specificallyfor Hartley 2, Bauer et al. (2011) estimate the comet’s overall size distribution to follow apower-law slope of − . ± .
3, derived by comparing R -band and WISE 12 and 22 µ m fluxes.Epifani et al. (2001) fit an ISOCAM 15 µ m image of the comet taken about 10 days afterperihelion with a dust dynamical model. They report a time-averaged power-law slope of − . ± .
1, but inspection of their Fig. 10 suggests − . − . − .
7, but thereis no requirement that they be the same.To better understand the size, and thereby the composition, of the large particles, weinvestigate the largest particle that may be lifted from the nucleus, a crit . This parameter isestimated by comparing the force of gravity to the gas drag force at the surface of the comet.Meech and Svore˘n (2004) integrated the equation of motion for spherical particles ejectedfrom a spherical nucleus and found a crit = 9 µm H Qv th π ρ p ρ N R N G , (7)where µ is the atomic weight of the gas (amu), m H is the mass of hydrogen (g), Q is the gasproduction rate (s − ), v th is the mean thermal expansion speed of the gas (cm s − ), ρ p is thedensity of the particle (g cm − ), ρ N is the density of the nucleus (g cm − ), R N is the radius ofthe nucleus (cm), and G is the gravitational constant (cm g − s − ). With the shape modelof Comet Hartley 2, Thomas et al. (this issue) estimate the surface gravity of the nucleus tobe a N = 0 . − . − , which includes the rotation state of the nucleus. Therefore,we re-write the equation from Meech and Svore˘n (2004) to use the gravitational accelerationat the nucleus a crit = 3 µm H Qv th πρ p a N R N . (8)The bulk material density for dust is ∼ − and for ice is 1.0 g cm − . For the dustycase, we will consider porous aggregates of dust with a total density of 0.3 g cm − (i.e., 90%vacuum). For the icy case, we will at first assume 1.0 g cm − , but later consider porousaggregates with ρ p = 0 . − .The total water production rate of Comet Hartley 2 near closest approach has beenestimated via several methods to be Q (H O) ≈ . − . × s − (A’Hearn et al. 2011,Combi et al. 2011, Dello Russo et al. 2011, Meech et al. 2011, Mumma et al. 2011, Knightand Schleicher this issue). Yet, the coma contains a significant amount of water ice (A’Hearnet al. 2011, Protopapa et al. 2011), which could be supplying a large fraction of the watervapor around the comet. Moreover, the water production rate is not uniformly distributedover the surface (A’Hearn et al. 2011). We can account for these observations by multiplyingthe water production rate by the ratio f surface /f active , where f surface is the fraction of thewater vapor produced at the surface of the nucleus, and f active is the areal fraction of surface25hat is active. For illustrative purposes, we will assume that 1% of the water vapor isproduced from 10% of the surface. The remaining 99% of the water vapor sublimates fromthe icy grain halo.For a nucleus water production rate of Q (H O) f surface /f active = 10 s − , v th = 0 . − ,Hartley 2’s mean radius (0.58 km), mean surface gravity, and assuming a particle density ofpure ice ( ρ p = 1 g cm − ), we find a crit = 8 cm. If we instead take CO as the driving gas,with a production rate of 10 s − (A’Hearn et al. 2011), f surface = 100%, and f active = 10%,then a crit becomes 200 cm for solid ice spheres. Based on this exercise, the icy model sizeestimates, a ≤
17 cm, are reasonable.If instead of icy particles, we assume nucleus-like particles with a density of 0.3 g cm − ,our a crit estimates increase to 28 cm (H O) and 670 cm (CO ). Compared to our sizeestimates of a ≤
210 cm, dark, dusty particles are plausible if CO is the driving gas; watercan be made consistent if f surface /f active is increased to 1.0.Aggregate particles are more easily lifted by gas drag due to their larger surface areaper mass. Nakamura and Hidaka (1998) found that the drag force for an aggregate isapproximately the same as the drag force on an area-equivalent sphere (with an error of lessthan 40% in the large aggregate limit). Since our particle radii are based on the observedflux, which is proportional to the cross-sectional area, our radii are already defined by area-equivalent spheres. Therefore, by revising our particle density we can use Eq. 8 to estimate a crit for aggregates.We assembled model particles using a ballistic particle-cluster aggregate (BPCA) method(Meakin 1984). As the size of the aggregate grows beyond a few thousand monomers thedensity asymptotically approaches 10% of the bulk material density. Thus, for a refractorymaterial with a bulk density near 3 g cm − , a centimeter-sized BPCA particle would have adensity of 0.3 g cm − . The a crit estimates will be the same as in our nucleus-like case above.Treating the large particles as aggregates rather than solid spheres better agrees with theHRIIR spectra of the coma. A’Hearn et al. (2011) and Protopapa et al. (2011) studied thewater ice absorption features and found that they are most consistent with icy aggregateswith monomer radii ≈ µ m. An icy BPCA particle would have a density of 0.1 g cm − . Thelowered density for icy aggregates increases a crit by a factor of 10, giving us a healthy marginfor launching large icy particles off the surface of the nucleus, even where water sublimationis driving the activity.In summary, the dusty particle case produces very large particle estimates (up to 2 min radius) that are just at the resolution limit of the HRIVIS instrument. Gas expansionfrom CO is sufficient to lift these large particles from the surface of the comet if they havea comet-like density of ≈ . − . The water-ice case produces particle estimates up to ≈
20 cm in radius, which are easily lifted from the nucleus by water or CO expansion.
6. Spatial distribution and origin
Mapping the spatial distribution of the particles gives clues to their origins and dynamics.In Figs. 14–17, we plot the column density of particles and total coma surface brightnesscontours for all 501 and 121 ms MRI images listed in Table 2. The column density imageswere derived from our final photometry lists for F λ > F λ, min , binned onto a 38 ×
38 pixelgrid. Inspection of the figures reveals that the coma and the particles have different spatial26istributions. The particles are biased to the anti-sunward direction on scales > − Rotation of the nucleus has a direct impact on the spatial distribution of particles. Toestimate this effect, we must first recognize that the particle outflow speeds are low. Inorder for the particles to be seen as point sources in HRIVIS images at closest approach,their speeds must be lower than ≈ / .
376 s = 8 m s − . A lower constraint is computed byHermalyn et al. (this issue) based on the 3D positions of the particles. They find 0.5–2 m s − to be more typical (note that these speeds are not necessarily radial). At such low speeds,the large particles take (cid:38) s to reach 2 km from the nucleus. In contrast, small dustgrains move much more quickly with outflow speeds expected to be of order 100 m s − . Thefine dust reaches 2 km in as little as 20 s.The positions of the major jets are governed by the rotation of the long axis about theangular momentum vector, with a period of 18.4 h near closest approach; the long axis isinclined to the angular momentum vector by 81 ◦ (Belton et al. this issue). Taking 1 m s − as the particle outflow speed from the surface, the nucleus will have rotated 10 ◦ by the timethe particles have traveled 2 km. So, the rotation state has a minor consequence on thedistribution of particles at 2 km, and their spatial distribution should be closely related totheir source regions. This observation and the assumption of radial motion suggests thatthe strong sunward jets are not the primary source of the large particles, but instead theyare ejected from along the long-axis of the nucleus. The jets pointed towards the bottomof the images in Fig. 15 would be the next likely source region for particles. There is alsoa large population of particles towards the top of Figs. 14 and 15, but they do not have anapparent source region (i.e., this side of the nucleus does not appear to be as active as theother regions). However, we note that not all potential sources are apparent in the MRIimages. For example, the apparent water jet seen in Fig. 5 of A’Hearn et al. (2011), whichpoints to the top right of Figs. 16 and 17, has no clear optical counterpart but may alsocontribute to the large particle production (n.b., this water jet does not appear to containice, and therefore is unlikely to be a source of large icy particles).Taking a lower ejection speed of 1 cm s − , the nucleus can rotate three times beforeparticles travel 2 km. Thus on this length scale, and in the absence of any other perturbingforces, the particles would have a spatial distribution correlated with the activity profile ofthe nucleus. Because the CO and H O gas production rates and the optical light curve peakwhen the small end is pointed towards the Sun (A’Hearn et al. 2011), in the absence of otherforces the large particle density should peak in the solar direction, which is not observed.27 igure 14: Contours of the total coma surface brightness superimposed over particle column density for 501and 121 ms MRI images mv5002051 through mv5004025. The contours are spaced at factor of 2 intervals,and the dashed line is 0.008 W m − µ m − sr − . The coma image was smoothed with a Gaussian kernelfunction (7 pixel FWHM) before the contours were created. Only particles with F λ > F λ, min are considered,and the particle bins are 38 ×
38 pixels in size. The region closest to the nucleus is masked from the analysis,and has been replaced with each epoch’s image of the nucleus and inner-most coma. In all images, theprojected velocity of the spacecraft is toward the bottom, and the sunward direction is approximately to theright. igure 15: Same as Fig. 14, but for images mv5004029 through mv5004053. igure 16: Same as Fig. 14, but for images mv5004056 through mv5006020. igure 17: Same as Fig. 14, but for images mv5006024 through mv5006046. .2. Solar radiation pressure In principle, solar radiation pressure could redistribute large particles into the anti-sunward direction. The acceleration from radiation, a rad , in units of cm s − is (Burns et al.1979) a rad = Q pr S (cid:12) σcmr h , (9)where Q pr is the radiation pressure efficiency factor, S (cid:12) is the integrated flux density (1 . × erg cm − s − at 1 AU), σ is the geometrical cross section of the particle (cm ), c is thespeed of light (cm s − ), m is the mass of the particle in question, and r h is the heliocentricdistance (AU). We have already dropped all velocity dependent terms from a rad (cf. Burnset al. 1979). It is common to express radiation pressure with the parameter β defined as theratio of the force of solar radiation pressure to the gravitational force from the Sun, β ≡ F rad F grav = 5 . × − Q pr ρ p a . (10)The radiation pressure efficiency is 1 for perfectly absorbing, isotropically emitting spheres.For our particles, we again take the icy particle case and derive Q pr from the phase func-tion and geometric albedo (van de Hulst 1957): Q pr ≈ − A B cos α , where A B is the Bondalbedo, and cos α describes the anisotropy of the scattered light, where α = 180 − θ isthe scattering angle. The relationship between the Bond albedo and geometric albedo is A B = A p (cid:82) π Φ( α ) sin αdα (Hanner et al. 1981). Altogether, we compute Q pr = 1 .
58. For themass, we again assume two cases for particles with a = 10 cm: (1) solid spheres with thedensity of ice, 1 g cm − ; and (2) compact aggregates of ice (1 µ m radius monomers) with anoverall particle density of 0.1 g cm − . For the latter case, we note that detailed calculationswill be required to understand how albedo and the anisotropy of scattering are affected bythe complex aggregate shape, and we reserve this investigation for future work. We computeaccelerations of 4 . × − and 0 . × − cm s − for 10 cm solid and BPCA aggregate iceparticles, equivalent to β = 9 × − , and 1 × − . For comparison, comet dust trails arecomprised of grains with β (cid:46) − (Sykes and Walker 1992, Reach et al. 2007).In Figs. 14–17, particles are found out to the image edges in the sunward direction.However, the sunward/anti-sunward asymmetry is clear on 2–4 km length scales. In the restframe of the comet, the relationship between turnaround distance ( d ), ejection speed ( v ej ),and acceleration from radiation pressure is d = v ej / /a rad . Solving for ejection speed v ej yields v ej = 3 dQ pr S (cid:12) cr h ρ p a = 6 . × − dQ pr ρ p a , (11)for v ej in units of cm s − , d in cm, ρ p in g cm − , and a in cm. For our icy particle case v ej = 4 . − . ρa ) − / cm s − (4–6 cm s − for a 10 cm aggregate, and 1–2 cm s − for solid ice).If particles are truly ejected at these speeds, solar radiation pressure would take (cid:38) ρa s toaccelerate the particles to the (cid:38)
50 cm s − speeds measured by Hermalyn et al. (this issue).This result also implies that the dominant velocity component for particles more than a fewkilometers from the nucleus would be distinctly in the anti-solar direction, yet only a weakasymmetry in the velocity is observed (Hermalyn et al. this issue). Therefore, radiationpressure does not govern the dynamics for our icy particle cases. The same conclusion is32eached for the dusty particle case with a nucleus-like density ( Q pr = 1 . ρ p = 0 . − ): v ej = 6 . − . a − / cm s − or 0.6–0.9 cm s − for a 100 cm particle.If we instead assume an ejection speed of 1 m s − for a 10 cm particle, their densitiesmust be ρ p = 2 − × − g cm − in order to be turned around by d = 2 − The rocket effect is the acceleration of the particles due to the sublimation of water ice.For a spherical geometry with radial outflow, a rocket = 3 µm H Zv th f ice ρ p a = 1 . × − Zρ p a , (12)where Z is the sublimation rate of the particle (molec cm − s − ), µ is the molecular weightof the sublimating ice (18 u for water), m H is the mass of hydrogen (g), and f ice is the icefraction of the particle (all remaining parameters are in cgs units). Following Reach et al.(2009), we define the dimensionless rocket effect parameter α as the ratio of the force fromthe sublimation mass-loss, F rocket , to the force of gravity from the Sun, F grav : α ≡ F rocket F grav = 2 . × µm H Zv th f ice GM (cid:12) ρ p a , (13)where G is the gravitational constant (cm g − s − ), and M (cid:12) is the mass of the Sun (g). Therocket effect parameter should have a heliocentric distance dependency, since the product Zv th does not necessarily vary as r − h , but Eq. 13 will serve as a good approximation forsmall ∆ r h . The α parameter is analogous to the β parameter for dust; both parametersquantify a force directed away from the Sun in fractions of the solar gravitational force.Assuming no losses from scattering or thermal emission, the maximum water sublima-tion rate from ice ( Z max ) can be computed from the solar flux density, particle absorptionefficiency ( Q abs ), and the latent heat of sublimation for water ice ( L ), Z max = S (cid:12) Q abs N A r h L (14)where N A is Avogadro’s number, and L ranges from 5 . × to 5 . × erg mol − fortemperatures from 100 to 300 K (Murphy and Koop 2005). For large compact aggregates(i.e., porous spheres) and solid particles Q abs ≈
1, but it is larger for fluffy aggregates sincethey have light scattering properties more like a collection of monomers, rather than a singlesolid particle (Kolokolova et al. 2007). This last comment aside, Z max becomes 1 . × molec cm − s − for T = 300 K. To account for scattering, Z max will scale with 1 − A B ,where A B is the Bond albedo. In § A B = 0 .
84 for our icy particle model.Scattering reduces the Z max of the icy model to Z max,icy = 2 . × molec cm − s − . Forour dusty model, A B = 0 .
013 and Z max,dusty ≈ Z max .For comparison, consider the model of Beer et al. (2006) for the sublimation rate of icyparticles. They employ spherical grains, heated by absorption of sunlight and cooled by33hermal emission and sublimation, and computed the absorption and emission efficiencieswith Mie scattering and effective medium theory considering both pure ice grains, and “dirty-ice” grains, i.e., ice mixed with a generic absorber (dust). Mixing dust with the ice has asignificant effect on the ice equilibrium temperature, but the effect is not a strong functionof the ice-to-dust mass ratio (they tested m ice /m dust = 0 . r h = 1 .
09 AU are the best examples for our scenario. For a > .
01 cm, pure ice grains have lifetimes of τ pure = 1 × a s for particle radius a measured in cm. Dirty-ice grains have a much shorter lifetime: τ dirty = 1 × a s. Sincethe large particles spend most of their lifetime larger than 0.01 cm, we can transform theirlifetimes into sublimation rates: Z = 4 ρ p a µm H τ f ice . (15)The Beer et al. (2006) model sublimation rates for large particles are Z dirty = 4 × molec cm − s − , and Z pure = 4 × molec cm − s − . These values are comparable to orless than our maximum sublimation rates.Taking Z max,icy , a 10 cm radius particle is accelerated at a rate a rocket = 0 .
024 cm s − away from the Sun, yielding a rocket parameter α = 0 . Z = Z max , we compute α = 0 .
27. Unlike radiation pressure, therocket effect is potentially very strong.Acceleration from sublimation will distribute the particles in the anti-solar direction.Following our method for radiation pressure, we can constrain the particle ejection speedwith the implied sublimation rate v ej = 2 . × − Zdρ p a . (16)Again, adopting 2–4 km as our typical turnaround distance, and taking Z = Z max,icy , we findejection speeds of 310 − a − / cm s − for our solid ice case (990 − a − / cm s − foricy aggregates). These ejection speeds are higher than the instantaneous speeds measuredby Hermalyn et al. (this issue), but the two speeds do not need to agree since one is at/nearthe surface and the other is out in the coma. For Z = Z max , the resulting ejection speedsare increased: v ej = 780 − a − / cm s − (solid ice) and 2500 − a − / cm s − (icyaggregates). However, in the above analysis we have assumed that only the sunlit hemisphereis sublimating. By distributing the sublimation across more of the surface, we can decreasethe implied ejection speeds to 10–100 cm s − , similar to the speed measured in the coma byHermalyn et al. (this issue). Therefore, we conclude that a sublimation rate excess on thesunlit hemisphere of order 10 molec cm − s − readily describes the sunward/anti-sunwardparticle asymmetry. Detailed simulations will be needed to fully account for the observeddistribution of particle velocities.
7. Mass and water production rate
Table 4 includes the total particle mass, based on our icy photometric model and a particledensity of 1 g cm − . For the dusty model and ρ p = 0 . − , the masses are increased by34 factor of 672. With our preferred flux lower limit of 0.1–1 . × − , the particle massescorrespond to 0 . − . M N (icy), 23 − M N (dusty), where M N = 2 . × g is the massof the nucleus, assuming a 0.3 g cm − density (Thomas et al. this issue). It is clear that thedusty case is impossible. The only way we can reduce the estimated total mass for thesedark particles is by decreasing their densities to well below 10 − g cm − . Therefore, we donot favor the dusty case, but it does remain as a possible interpretation. Given that the totalmass lost from the comet per orbit is of order 1% of the nucleus (Thomas et al. this issue),even the solid ice particles may be too massive. Porous ice particles (e.g., ρ p = 0 . − )should be considered the most likely case.Icy particles will begin sublimating as soon as they are released from the nucleus andwarmed by insolation. In § molec cm − s − on theparticle’s sunlit hemisphere can describe the observed sunward/anti-sunward asymmetryin particle column density. We also computed the maximum sublimation rate, based onenergy balance between absorbed solar radiation and sublimation. We apply this lattervalue, Z max,icy to all of the large particles and compute water production rates. The totallarge particle water production rate within a 20.6 km radius aperture is limited to < . − × molec s − , which is < . − .
5% of the total water production rate of the comet( ≈ × molec s − ). We can change the water production rate by assuming a differentphotometric model as Q ∝ (1 − A B )( A p Φ) − . For our nucleus-like case, we find Q (H O) < −
80% of the total water production rate, but, unless the particles are fluffy aggregateswith ρ (cid:46) − g cm − , we rule out this case based on their mass. The icy particle case yieldsour best estimate of the water production rate, Q (H O) < (1 − × − ) Q total .
8. Comparison to other observations
Harmon et al. (2011) observed comet Hartley 2 with Arecibo S-band ( λ = 12 . Deep Impact ’s closest-approach. Intheir average Doppler spectrum, they observe a strong grain-coma echo, with a characteristicradial velocity dispersion of 4 m s − . The velocity distribution is asymmetric, with a rangeof ≈ −
50 to +13 m s − with respect to the nucleus (negative velocities are away from theEarth). The coma has a strongly depolarized echo, which indicates the radii of the largestparticles are well above the Rayleigh limit of λ/ π = 2 cm. Harmon et al. (2011) suggest thatthere exists a significant population of particles with decimeter sizes or larger. We explorethe possibility that the radar observations may be the large particles imaged by Deep Impact .The average radar cross section of the coma was 0.89 km . From our MRI observations,we derived a total icy particle cross section of σ = 4 × − to 3 × − km within 20.6 kmfrom the nucleus. Our cross section is more than two orders of magnitude smaller thanthe radar observed cross section. However, their beam is much larger than what the MRIcan image when individual particles are detectable. We have found particles out to 40 kmin Fig. 13, but the Arecibo beam size is 32,000 km at the distance of the comet. Withspeeds of order 4 m s − , and lifetimes of order 10 s (dirty ice), the large icy particle comawould extend out to ∼
400 km. This estimate implies our census of the large particles isincomplete by about a factor of < / . (cid:46) .
01 km , which still remains inconsistent with the radar results.35ssuming for the moment that the radar cross section properly reflects the total icyparticle population we compute an upper limit to the total water production rate of < × molec s − . The SWAN instrument on the SOHO satellite observes Ly α emissionover fields of view much larger than the radar beam size (10 km pixel − ). These observationsare a good point of reference for a total water production rate that is sure to include waterproduced by any large particles observed with Arecibo. Combi et al. (2011) measured a waterproduction rate of 6–9 × molec s − near perihelion (Combi et al. 2011). By this estimate,it seems that the large particles could account for a substantial fraction of the total waterproduction rate of the comet. Note, however, that the SWAN-based water production ratesare on par with those observed in 4000 km apertures and smaller. Based on the aperturesizes and water production rates listed in Table 5, most of the water is produced close to thenucleus, perhaps within a few tens or hundreds of kilometers, suggesting that the Areciboobserved particles have a low water sublimation rate, if any.A dusty large particle population yields a cross section of σ = 0 . − . ( § ∼ − −
250 km from the comet. There is no indication in Figs. 14 and 17 that thelarge particle coma is truncated on this length scale. Moreover, dusty particles may not havesublimating ice. Instead, they could fragment into finer particles. If the large particles aredusty, they must fragment on (cid:46) − −
250 km length scales in order to keep their total crosssection less than or equal to the observed radar cross section. Just based on the observedcross sections, we consider the dusty case to be less likely than the icy case, but still find theicy case to be lacking. Of course, a coma of both icy and dusty particles is possible. Moreinformation on the composition and light scattering properties, including radar wavelengths,will be needed to reconcile the radar and MRI observations.
9. Summary
Comet Hartley 2 is surrounded by a coma of large particles with radii (cid:38)
Deep Impact , we measured their total flux and flux distribution, basedon photometry of individual particles. The flux distribution of these particles implies avery steep size distribution with power-law slopes ranging from − . − .
7. We estimatethat the particles account for 2–14% of the total flux from the near-nucleus coma. Thespatial distribution of the particles is biased to the anti-sunward direction, as observed bythe spacecraft both pre- and post-closest approach. Radial expansion from the active areas ofthe rotating nucleus does not explain the observed spatial distribution, even if the ejectionspeeds are very low ( ∼ − ). Radiation pressure from sunlight cannot redistributethem into the anti-sunward direction on small enough length scales unless the particles haveextremely low densities ( ∼ − g cm − ) or low radial ejection velocities ( (cid:46)
10 cm s − ). Lowejection velocities suggest there should be a strong anti-sunward velocity component in thecoma, but this does not agree with the velocity distribution observed by Hermalyn et al.(this issue).We examined two possible particle compositions. Our models were based on the photo-metric properties of the nucleus of Hartley 2 (dusty case: low albedo, 0.3 g cm − ) and theJovian satellite Europa (icy case: high albedo, 0.1–1.0 g cm − ), and serve as approximatelimiting cases. 36he dusty case produces particle size estimates ranging from 10 cm to 2 m in radius,the largest of which is just at the limiting resolution of Deep Impact ’s HRIVIS camera atclosest approach. Such large particles may be lifted off the nucleus by gas drag if CO is thedriving gas. Water is a plausible alternative if the water production rate from the nucleusis at least 10 molec s − . Based on the dusty model, the total large particle cross sectionwithin 20.6 km from the nucleus is 0.07-0.5 km , similar to the 0.89 km radar cross sectionobserved by Harmon et al. (2011). If these particles are mini-nuclei, we estimate they accountfor 16–80% of the comet’s total water production rate (within 20.6 km). However, we canall but rule out the dusty case based on total mass estimates of the large particles, whichare in excess of 10 nucleus masses for densities of 0.3 g cm − . Densities (cid:46) − g cm − arerequired to reduce the total mass of the particles to a few percent of the nucleus, which isneeded to keep the particle mass less than the total mass lost from the comet per orbit (2%during the 2010 apparition; Thomas et al. this issue).The icy case produces particle size estimates ranging from 1 to 20 cm in radius. Theseparticles are easily lifted by water and CO gas drag. Icy particles would sublimate as soon asthey are heated by sunlight. If the particles have a net sublimation on their sunlit sides, theywould feel a rocket force that could easily distribute the particles into the anti-sunward direc-tion. The sublimation rate excess required for this redistribution is (cid:46) molec cm − s − ,where the exact value depends on the particle ejection velocities. The cross section of icyparticles within 20.6 km is much smaller than the observed radar cross section by two tothree orders of magnitude. The water production rate of the Deep Impact observed particlesis limited to < . − .
5% of the comet’s total water production rate. The total mass ofthe particles for a density of 1 g cm − is 3–10% the total mass of the nucleus. Thus, porousaggregates with ρ p (cid:46) . − should be considered more likely than solid ice particles.We consider the icy case to be more likely than the dusty case for three reasons: (1) theicy particles are more easily lifted by gas drag; (2) we can account for the sunward/anti-sunward asymmetry in the particle distribution if ice is sublimating on their sunlit sides;and (3) the total large particle mass for the dusty case is much greater than the total massof the nucleus. However, several details are needed in order to test our hypothesis. We needan improved icy particle model that treats the large particles more like macroscopic objectsto better understand their water production rates (if icy) and light scattering properties(dusty or icy). We need to better constrain the particle densities, which may rule out thedusty case. A hydrodynamic analysis of the near-nucleus coma, especially around the highlyactive small end, would improve our knowledge of the dynamics of large particles. A betterunderstanding of the radar coma that includes grains smaller than λ/ π could help resolvethe discrepancy between our icy particle cross section and the observed radar coma crosssection.If indeed icy with a high albedo, the large particles do not appear to be the source of thecomet’s enhanced water production rate; although, as discussed above, there is much workthat can be done to refine this conclusion. We suspect that the small icy grains in the jets,as observed in Deep Impact
IR spectra (A’Hearn et al. 2011, Protopapa et al. 2011), are asignificant source of water, and the primary cause of the hyperactivity of Comet Hartley 2.37 cknowledgments
The authors thank Lev Nagdimunov (UMD) for assistance in computing cluster aggregateporosities, Adam Ginsberg for providing the power-law fitting code, and Bj¨orn Davidssonand an anonymous referee for helpful comments that improved this manuscript.This work was supported by NASA’s Discovery Program contract NNM07AA99C to theUniversity of Maryland and task order NMO711002 to the Jet Propulsion Laboratory.This research made use of the PyRAF and PyFITS software packages available at . PyRAF and PyFITS are products ofthe Space Telescope Science Institute, which is operated by AURA for NASA.
References
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Icarus.Tody, D., 1993. IRAF in the Nineties. In: R. J. Hanisch, R. J. V. Brissenden, & J. Barnes(Ed.), Astronomical Data Analysis Software and Systems II. Vol. 52 of AstronomicalSociety of the Pacific Conference Series. p. 173.van de Hulst, H. C., 1957. Light Scattering by Small Particles. John Wiley and Sons, NewYork.Vaubaillon, J. J., Reach, W. T., 2010. Spitzer Space Telescope Observations and the ParticleSize Distribution of Comet 73P/Schwassmann-Wachmann 3. Astron. J. 139, 1491–1498.41 able 1: HRIVIS and MRI images considered in this paper. Table columns are: t − t enc , time relative toencounter; Exp., exposure time; ∆, spacecraft-comet distance; Scale, pixel scale at the distance of the comet; φ , phase (Sun-comet-spacecraft) angle. Image t − t enc Exp. ∆ Scale φ (s) (ms) (km) (m) ( ◦ )HRIVIShv5004024 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − able 2: Online only table.
All single power-law flux distributions derived from HRIVIS and MRIimages. Table columns: F λ , the minimum, maximum, and total particle fluxes in the fit; N , numberof particles in the fit; α , best-fit power-law slope and uncertainty; P KS , the Kolmogorov-Smirnovprobability that the distribution is a power-law; F T /F coma , the fraction of the total coma fluxattributable to point sources with F λ > F λ, min in MRI full-frame data, excluding regions near thenucleus.Image ∆ F λ a N α P KS F T /F coma min max total(km) (10 − W m − µ m − ) (%)HRIVIShv5004024 915 1.8 20.5 1853 641 -3.06 ± · · · hv5004025 847 1.5 31.6 3128 986 -2.88 ± < · · · hv5004027 781 1.4 29.0 1388 430 -2.86 ± · · · hv5004028 732 1.5 33.1 2058 685 -2.91 ± · · · hv5004030 704 0.9 35.5 1222 613 -2.67 ± · · · hv5004031 694 1.5 40.6 1252 419 -2.70 ± · · · hv5004033 704 0.9 34.3 918 379 -2.67 ± · · · hv5004034 735 1.5 31.2 1017 374 -2.93 ± · · · hv5004036 787 0.9 23.1 902 496 -2.94 ± · · · hv5004037 851 1.5 21.2 1005 425 -3.08 ± · · · MRI sub-framemv5004029 964 4.1 14.1 692 ± ± · · · mv5004031 885 4.5 13.4 660 ± ± · · · mv6000001 815 4.2 11.4 770 ± ± · · · mv5004041 760 3.9 16.6 836 ± ± · · · mv5004044 719 3.8 18.8 933 ± ± · · · mv5004046 697 3.6 12.9 631 ± ± · · · mv5004051 696 3.0 13.8 729 ± ± · · · mv5004053 715 3.1 21.3 553 ± ± · · · mv5004056 753 2.8 11.1 637 ± ± · · · mv5004058 807 3.0 9.0 437 ± ± · · · MRI 41 msmv5004027 1006 5.5 16.4 1066 ± ± ± ± ±
12 189 -3.76 ± ±
13 252 -4.04 ± ±
15 321 -4.02 ± ±
15 331 -3.96 ± ±
17 390 -3.68 ± ±
16 374 -3.74 ± ±
15 309 -3.59 ± ±
13 208 -3.60 ± ±
13 235 -3.78 ± ±
10 188 -4.08 ± ± ± ±
13 838 -3.95 ± ±
16 1191 -3.87 ± ±
20 1291 -3.88 ± ±
23 1501 -3.87 ± < Continued on next page ontinued Image ∆ F λ a N α P KS F T /F coma min max total(km) (10 − W m − µ m − ) (%)mv6000001 815 4.1 19.7 7337 ±
22 1262 -3.95 ± ±
26 1468 -3.91 ± ±
26 1770 -3.60 ± < ±
25 1391 -3.76 ± ±
26 1769 -3.65 ± < ±
26 1769 -3.57 ± < ±
26 1806 -3.58 ± < ±
27 1937 -3.65 ± < ±
24 1681 -3.65 ± < ±
21 1771 -3.64 ± ±
17 1426 -3.82 ± ±
11 965 -3.87 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± able 3: Summary of single power-law fits to individual HRIVIS and MRI flux distributions listed in Table 2.Table columns: F λ , mean minimum and maximum particle fluxes; N , mean number of particles; α , minimum,maximum, mean, and standard deviation of the flux distribution power-law slopes; P KS , mean Kolmogorov-Smirnov probability that the distributions are power-laws; F T /F coma , mean fraction of the total coma fluxattributable to point sources with F λ > F λ, min in MRI full-frame data, excluding regions near the nucleus.Instrument ∆ F λ N α P KS F T /F coma min max min max min max mean σ (km) ( − W m − µ m − ) (%)HRI 694 915 1.3 30.0 545 -3.08 -2.67 -2.87 0.14 0.125 · · · MRI sub-frame 696 964 3.6 14.2 127 -4.31 -3.22 -3.70 0.35 0.341 · · ·
MRI 41 ms 694 1006 5.5 20.2 245 -4.39 -3.59 -3.90 0.22 0.384 0.0023MRI 121 ms 696 1303 3.1 17.0 1489 -3.95 -3.57 -3.76 0.14 0.021 0.0059MRI 501 ms 1858 5148 0.7 3.5 531 -4.85 -3.59 -4.06 0.43 0.330 0.0008MRI (distant) a a Averages of the the four 501 ms MRI images with ∆ > N >
300 particles, with which weuse to define the fraction of the coma flux attributable to large particles. For this row only, F λ has beencorrected to a distance of ∆ = 700 km before averaging.Table 4: Flux range ( F λ, min , F λ, max ), radius range ( a icy ), total number ( N ), flux ( F λ,T ), fraction of comaflux in particles ( F T /F coma ), cross section ( σ icy ), mass ( M icy ), and water production rate ( Q (H O)) for thelarge particles measured at ∆ = 2100 − φ = 86 ◦ ) and a densityof 1.0 g cm − . We also extrapolate these quantities down to 10 − W m − µ m − , and to our total comameasurement for a 20.6 km aperture at ∆ = 5148 km. Uncertainties are derived from the power-law slopeuncertainties ( ± .
02) and are symmetric in log space. F λ, mina F λ, maxa a icy log N log F λ,T a F T F coma log σ icy log M icy log Q max × − (W m − µ m − ) (cm) (W m − µ m − ) (%) (cm ) (g) (s − )∆ ≈ b b b -8.06 b , c b ± ± ± . ± ± ± e
45 0.8 16.5 6.76 ± ± ± . ± ± ± ± ± +12 . − . ± ± ± ± ± ± . d ± ± ± e
45 0.8 16.5 9.02 ± ± ± . d ± ± ± a Fluxes have been corrected to a distance of 700 km. b Measured values. c The instrumental and calibration uncertainties are 2 × − W m − µ m − and 10%, respectively. d Assumed value. e Our preferred lower flux limit is between 0.1 and 1.0 × − W m − µ m − . able 5: Summary of comet Hartley 2 water production rates observed near perihelion, and derived in thisstudy. Date(s) Aper. a Q (H O) Notes(UT, 2010) (km) (10 molec s − )18 Oct 9 . × ± ± ± . × (cid:46)
2. Large particles only, Harmon et al. 2011 + this work31 Oct b ± . × ± ± (cid:46) a Slit half-width or aperture radius. We have assumed the 0.43 (cid:48)(cid:48) slit for the Mumma et al. (2011) observations,and a 3 pixel radius for the Combi et al. (2011) observations. b Mean and standard deviation of 5 measurements taken on 31 Oct. rratum to “A distribution of large particles in the coma of Comet103P/Hartley 2”: [Icarus 222, 634–652 (2013)] Michael S. P. Kelley a,* , Don J. Lindler b , Dennis Bodewits a , Michael F. A’Hearn a , Carey M.Lisse c , Ludmilla Kolokolova a , Jochen Kissel d,1 , Brendan Hermalyn e a Department of Astronomy, University of Maryland, College Park, MD 20742-2421, USA b Sigma Space Corporation, 4600 Forbes Boulevard, Lanham, MD 20706, USA c Johns Hopkins University–Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723,USA d Max-Planck-Institut f¨ur Sonnensystemforschung, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany e NASA Ames Research Center / SETI Institute, MS 245-3 BLDG245, Moffett Field, CA 94035
1. Introduction
Comet 103P/Hartley 2 has a high water production rate to surface area ratio, suggestingthe nucleus is nearly 100% active. In contrast, images of the nucleus from the
Deep ImpactFlyby spacecraft show strong localized activity, with an inner-coma enriched in CO gas andwater-ice grains. Rather than being produced solely by water ice sublimation at the nucleus,the hyperactivity of Hartley 2 may be due to water-ice sublimation in the coma. However, thecontribution of coma grains is poorly constrained, leaving the icy-grain hypothesis unproven(A’Hearn et al. 2011, Protopapa et al. 2014).Images from the Deep Impact Flyby spacecraft show thousands of point sources surround-ing the nucleus of comet Hartley 2 (A’Hearn et al. 2011). The point sources are individualparticles ejected by the comet. We measured their brightnesses and summarized their sizes,total mass, and spatial distribution (Kelley et al. 2013). The photometric properties (albedo,phase function) of the particles are unknown, therefore we adopted two models as approxi-mate limiting cases: 1) bright, icy particles with photometric properties similar to the Joviansatellite Europa; and 2) dark, nucleus-like particles with properties similar to the nucleus ofHartley 2. For the bright, icy case, we reported the largest particle had a radius of 20 cm andthe total population mass within 21 km of the nucleus was up to 3 to 10% of the nucleus mass(assuming a density of 1 g cm − ). For the dark, nucleus-like case, the largest particle was2 m in radius and the total population up to 230 to 700% of the nucleus mass (0.3 g cm − ).Based on this mass calculation, we ruled out the nucleus-like case and favored the icy case.Moreover, the icy case produces particles too small to account for the comet’s hyperactivity.We have found three errors in our calculations that affected our radius and mass estimates.We regret the errors, as they have significant consequences in our analysis of the particles.We provide updated calculations and interpretations in Sections 2 and 3. In addition, weprovide a few minor clarifications and corrections to our original paper in Section 4. retired Kelley et al. 2015, Icarus, in press . Radius and mass corrections
We have found an error that affects the computed particle radii (and dependent quanti-ties), and two additional errors that affect population masses. First, Eq. 4 of Kelley et al.(2013) is missing a factor of π in the denominator. The correct equation is F λ = A p Φ( θ ) σS λ, (cid:12) πr h ∆ (4)where F λ is the particle flux density, A p is the geometric albedo, Φ( θ ) is the phase functionevaluated at the phase angle θ , σ is the cross-sectional area of the particle, S λ, (cid:12) is the solarflux density at 1 AU, r h is the heliocentric distance of the particle, and ∆ is the spacecraft-particle distance. To account for this factor of π , all parameters derived from particle fluxesmust be scaled as follows: radius by a factor of √ π , cross section and water production rateby a factor of π , and mass by a factor of π / .Second, our computations of total population mass (Table 4) erroneously used 1000 g cm − ,rather than the 1 g cm − quoted in the text. In addition, our analytical formula for integrat-ing the total mass of a population of grains given a power-law flux distribution was missinga factor of 2. The formula was not given in the paper, but for completeness the correctedformula is M = 8 πρ N F α − (cid:90) a max a min a α +2 da, where M is the total population mass, ρ is the particle mass density, N is the flux distributionnormalization constant, F is the flux density from a 1-cm radius particle (via Eq. 4), a is theparticle radius, a min , a max are the limits of the integration (corresponding to the estimatedflux density limits), and α is the power-law index of the differential flux distribution (Eq. 3).Thus, total population masses must be scaled by a factor of 1 / π / scale factor from above (the total scale factor is 0.011). Other quantities that depend on individual particle mass (e.g., β in Section 6.2, α and v ej in Section 6.3) are unaffected.We present a revised Table 4, with corrected radii, cross sections, mass, and maximumwater production rates. In addition, all calculations are now based on the normalizationconstant N (Eq. 6), whereas previously some calculations were normalized via the totalobserved particle flux. This change increases most flux-based quantities by 13%. Overall,our analysis is significantly affected by the revised Table 4.The solid water-ice case (see Section 5 of Kelley et al. 2013 for definitions of our icy anddusty cases) produces particle estimates up to 30 cm in radius with an estimated populationmass (within a 20.6 km aperture) of 0 . − × g, or up to 0.02% of the nucleus mass. Thismass is two orders of magnitude lower than the ∼
2% of the nucleus that was lost in 2010(Thomas et al. 2013). Porous ice particles are no longer necessary to reduce the estimatedpopulation mass below the orbital mass loss of the comet. The total water production rateestimate is Q max = 0 . − × s − , or 0 . − .
2% of the comet’s total water productionrate, insufficient to account for the comet’s hyperactivity.To convert from the icy case to the dusty case multiply radii by 12.6, cross sections by158, masses by 597, and Q max by 1007. The dusty particle case produces particles up to4 m in radius and a total population mass of 0 . − × g, or 0.6–14% of the mass of thenucleus, potentially exceeding the estimated orbital mass-loss rate of the comet. If the dusty2articles behave like mini-comet nuclei, their total water production rate may be as largeas Q max = 0 . − × s − , or 30–200% of that of the comet during the encounter. Ourupdated cross sections, 0.2–2 km , are comparable to and slightly exceed the cross sectionmeasured by radar observations (0.89 km ; Harmon et al. 2011).The new radius estimates produce particles up to 8 m diameter. Such particles should beresolved in the High-Resolution Instrument (HRI) visible images, which has a reconstructedresolution of approximately 3 m at closest approach. However, our original examination ofthe data revealed no resolved sources, suggesting a maximum size of approximately 3 m. Thesmaller size could be accommodated through an increase in the model albedo (from 0.049to 0.31) or by reducing the phase function coefficient (from 0.046 to 0.023 mag per degree;see Section 5 of Kelley et al. 2013). Both cases reduce the total population mass to <
1% ofthe nucleus mass and maximum water production rate to 40% of the total comet productionrate. We save a more thorough search for resolved HRI particles for future work.
3. Interpretation
Given the above revisions, the dusty case may solve the apparent hyperactivity of thecomet. However, if these particles are long lived (i.e., not fragmenting) and moving awaywith radial velocities of order 0.1 m s − (Hermalyn et al. 2013), then their lifetimes in a21 km radius aperture are of order 60 hours, implying a significantly large mass-loss rateof ∼ kg s − for the assumed particle density of 0.3 g cm − . This mass-loss rate mustbe increased by an order of magnitude if we instead consider the radar observed velocitydispersion of 4 m s − (Harmon et al. 2011). These estimates significantly exceed the comet’stotal water and carbon dioxide production rates ( ∼
300 kg s − and ∼
160 kg s − , respectively;A’Hearn et al. 2011) near the time of the flyby. Therefore, understanding the large particledynamics is critical to determining the erosion rate of the comet, and if the dusty caseremains valid. In parallel, a definitive upper limit to particle sizes should be derived from amore detailed search for resolved particles in HRI images.We have outlined two dramatically different scenarios for the physical properties of thelarge particles of Hartley 2. Intermediate sets of photometric and compositional propertiesare possible, and such cases cannot be ruled out. We do consider the dusty case (low albedo,0.3 g cm − ) to be less likely due to the large and massive particles implied, but cannotconfidently rule it out at this time. However, note that higher albedos or different phasefunctions may be employed to produce a more physically consistent picture for the largeparticles.
4. Other corrections
Equation 5 is correctly referred to as the differential size distribution, but elsewhere thetext commonly omits the “differential” adjective. We reviewed our comparisons to otherestimates of the dust size distribution, and the nomenclature used in the literature is notvery specific, occasionally labeling differential size distributions as size distributions (just aswe did). Our best understanding of the investigations listed in the text is that most indeedreport the differential size distribution. Our conclusion, that the differential size distributionslope, − .
7, is steeper than other estimates, remains valid.3he normalization constant N is not needed in Eq. 6, it is already included in Eq. 5.The corrected Eq. 6 is N = (cid:90) F λ, max F λ, min dndF dF. (6)The calculations for the paper used the above formula.The nucleus mass listed in Section 7 was expressed in units of kg, but reported as g. Thenucleus mass is M N = 2 . × g (Thomas et al. 2013).Finally, the units of flux density in Section 4.2 should be W m − µ m − , and not W cm − µ m − as stated. Acknowledgments
We thank Katherine Kretke for identifying the mass discrepancy in Table 4, and MichaelBelton for motivating other clarifying comments.This work was supported by NASA (USA) through the Planetary Mission Data AnalysisProgram contract NNX12AQ64G to the University of Maryland.
References
A’Hearn, M. F., et al., 2011. EPOXI at Comet Hartley 2. Science 332, 1396–1400.Harmon, J. K., Nolan, M. C., Howell, E. S., Giorgini, J. D., Taylor, P. A., 2011. RadarObservations of Comet 103P/Hartley 2. Astrophys. J., Lett. 734, L2.Hermalyn, B., et al., 2013. The detection, localization, and dynamics of large icy particlessurrounding Comet 103P/Hartley 2. Icarus 222, 625–633.Kelley, M. S., et al., 2013. A distribution of large particles in the coma of Comet103P/Hartley 2. Icarus 222, 634–652.Protopapa, S., et al., 2014. Water ice and dust in the innermost coma of comet 103P/Hartley2. Icarus 238, 191–204.Thomas, P. C., et al., 2013. Shape, density, and geology of the nucleus of Comet 103P/Hartley2. Icarus 222, 550–558. 4 able 4: Flux range ( F λ,min , F λ,max ), radius range ( a icy ), total number ( N ), total flux ( F λ,T ), fraction ofcoma flux in particles ( F T /F coma ), cross section ( σ icy ), mass ( M icy ), and maximum water production rate( Q max (H O)) for the large particles measured at ∆ = 2100 − φ = 86 ◦ ) and a density of 1.0 g cm − . We also extrapolate these quantities down to 10 − W m − µ m − ,and to our total coma measurement for a 20.6 km aperture at ∆ = 5148 km. Uncertainties are derived fromthe power-law slope uncertainties ( ± .
02) and are symmetric in log space. F λ,min a F λ,max a a icy log N log F λ,T a F T F coma log σ icy log M icy log Q max × − (W m − µ m − ) (cm) (W m − µ m − ) (%) (cm ) (g) (s − )∆ ≈ b b ± b -8.01 ± c ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±
13 8.72 ± ± ± ± ± ± d ± ± ± ± ± ± d ± ± ± a Fluxes have been corrected to a distance of 700 km. b Measured values. c The instrumental and calibration uncertainties are 2 × − W m − µ m − and 10%, respectively. d Assumed value. e Our preferred lower flux limit is between 0.1 and 1.0 × − W m − µ m − ..