Icarus 221 (2012) 721–734

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Icarus

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The calculation of Afq and mass loss rate for comets

Uwe Fink a,⇑, Martin Rubin b

a Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, United Statesb Department of Atmospheric, Oceanic and Space Sciences, University of Michigan, Ann Arbor, MI 48109, United States

a r t i c l e i n f o

Article history:Received 21 May 2012Revised 1 August 2012Accepted 1 September 2012Available online 18 September 2012

Keywords:Comets, DustCometsComets, Coma

0019-1035/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.icarus.2012.09.001

⇑ Corresponding author.E-mail address: uwefink@lpl.arizona.edu (U. Fink)

a b s t r a c t

Ab initio calculations of Afq are presented using Mie scattering theory and a Direct Simulation Monte Car-lo (DSMC) dust outflow model in support of the Rosetta mission and its target 67P/Churyumov-Gera-simenko (CG). These calculations are performed for particle sizes ranging from 0.010 lm to 1.0 cm. Thepresent status of our knowledge of various differential particle size distributions is reviewed and a varietyof particle size distributions is used to explore their effect on Afq, and the dust mass production _m. A newsimple two parameter particle size distribution that curtails the effect of particles below 1 lm is devel-oped. The contributions of all particle sizes are summed to get a resulting overall Afq. The resultant Afqcould not easily be predicted a priori and turned out to be considerably more constraining regarding themass loss rate than expected. It is found that a proper calculation of Afq combined with a good Afq mea-surement can constrain the dust/gas ratio in the coma of comets as well as other methods presently avail-able. Phase curves of Afq versus scattering angle are calculated and produce good agreement withobservational data.

The major conclusions of our calculations are:

– The original definition of A in Afq is problematical and Afq should be: qscaðn; kÞ � pðgÞ � f � q.

Nevertheless, we keep the present nomenclature of Afq as a measured quantity for an ensemble of comaparticles.

– The ratio between Afq and the dust mass loss rate _m is dominated by the particle size distribution.– For most particle size distributions presently in use, small particles in the range from 0.10 to 1.0 lm

contribute a large fraction to Afq.– Simplifying the calculation of Afq by considering only large particles and approximating qsca does not

represent a realistic model. Mie scattering theory or if necessary, more complex scattering calcula-tions must be used.

– For the commonly used particle size distribution, dn/da � a�3.5 to a�4, there is a natural cut off in Afqcontribution for both small and large particles.

– The scattering phase function must be taken into account for each particle size; otherwise the contri-bution of large particles can be over-estimated by a factor of 10.

– Using an imaginary index of refraction of i = 0.10 does not produce sufficient backscattering to matchobservational data.

– A mixture of dark particles with i P 0.10 and brighter silicate particles with i 6 0.04 matches theobserved phase curves quite well.

– Using current observational constraints, we find the dust/gas mass-production ratio of CG at 1.3 AU isconfined to a range of 0.03–0.5 with a reasonably likely value around 0.1.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction ing of the quantities in Afq are stated as the albedo, and the filling

The concept of Afq as a measure of the solar radiation reflectedfrom the dust coma of a comet was introduced in a paper byA’Hearn et al. (1984) on observations of comet Bowell. The mean-

ll rights reserved.

.

factor of the grains within the instrument field of view having aprojected radius, q. The paper defines the albedo A, as the totallight reflected by the cometary particles to the total light removedfrom the solar flux by the cometary particles. This definition isproblematical and we shall show in Section 4 that for the singlescattering process in the coma of a comet this quantity shouldnot be called an albedo but should really be replaced by

722 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

qscaðkÞpðgÞ the scattering efficiency of a particle times its phasefunction. The filling factor, f, is defined as the total cross sectionof the grains within the field of view divided by the area of the fieldof view pq2. Despite this problem in the definition of Afq we keepthe present nomenclature and use the italic notation Afq through-out this paper, implying a measured quantity.

The use of Afq for measuring the continuum of a comet wasintroduced because for most cometary observations, the line ofsight column density falls off as 1/q. On the other hand the pro-jected observational area increases as q2. Thus the quantity Afq be-comes independent of the specific observational circumstances orthe instrument settings used, e.g. photometric aperture, or slitwidth and length for spectroscopic observations. Inter-comparisonof Afq values by different observers can readily be accomplished.The introduction of Afq for the measurement of the cometary dustcontinuum has been extremely useful and it is now almost univer-sally used.

While the concept of Afq is extremely useful, it has some limi-tations which we briefly outline. It requires spherical symmetry ofthe coma. It assumes that there is no production or destruction ofthe dust after it leaves the nucleus, and that the dust has a constantoutflow velocity. It fails in the range of 1–20 km or so from thecometary surface where the dust is still coupled to the gas andthe dust has not yet reached its terminal velocity. It also fails farfrom the nucleus at distances of the order of 105 km where solarradiation pressure pushes the dust into the well known cometarydust tail. It can be compromised if there is strong jet activity andthe coma is asymmetric. It probably is also not appropriate fordescribing the coma for comets receding from the Sun at largeheliocentric distances where most of the dust may be comprisedby left over large particles captured into Keplerian orbits.

Spatial brightness profiles versus q for 14 comets including CG(Baum et al., 1992) show that about half their observed comets fol-low a 1/q fall off quite closely, while the other half exhibit a some-what steeper fall-off. Some of the deviation in this data-set isclearly caused by noticeable activity variation of a particular co-met. The deviation from 1/q, however, is not severe and wouldnot invalidate the use of Afq as a first order description of the dustenvironment in the inner coma of a comet.

Before the introduction of Afq, the dust continuum was de-scribed by the product the ‘‘geometric’’ albedo times the total crosssectional area of the dust, pr (e.g. Newburn and Spinrad, 1985;Johnson et al., 1984). Since this product varies depending on thelocation of the observers field of view (often not properly specified)inter-comparison of the dust continuum measured by differentobservers was quite difficult. We note however that a modifiedform of this concept with proper definition of A will be necessaryfor the interpretation of spacecraft observations where a localizedregion of the continuum along a specific line of sight is probed. Thissubject will not be explored further in this paper although itsimplementation is implicitly included in the equations of Section 4.Such a formulation will remove some of the limitations describedin the above paragraph.

While the formula for measuring Afq is straightforward andfairly simple to apply to observed cometary brightness measure-ments (Eq. (5)), it has been difficult to interpret Afq in terms ofphysical properties of the dust. Particularly, the interpretation ofAfq in terms of a dust mass production rate similar to the interpre-tation of the gas emissions in terms of gas production rates has notbeen readily possible. In this paper we go through a more rigorousderivation and calculation of Afq which considers the particle sizedependent velocity, scattering efficiency and phase function. Wedo not limit ourselves to large particles for which approximationsto the scattering efficiencies and phase functions can be used. Weuse a variety of particle size distributions to explore their effect onthe resulting dust/gas ratios.

The impetus of the present work came from the Direct Simula-tion Monte Carlo (DSMC) calculations performed by the Universityof Michigan Group (Tenishev et al., 2008) in preparation for the Ro-setta mission to Comet 67P/Churyumov-Gerasimenko (also referredto as CG in this paper). The results on the dust environment havebeen published by Tenishev et al. (2011). The back coupling ofthe influence of the dust on the gas is included (also sometimes re-ferred to as mass-loading), and the dust velocities as a function ofparticle size and distance from the nucleus are properly accountedfor. This work provides dust densities and dust velocities for a vari-ety of dust production rates and heliocentric distances. Since thesecalculations represent rather exact simulations we use this data-set to make calculation of Afq.

We give a brief review of some of the presently available dataon particle size distributions in Section 3 We find in our calcula-tions that, using the most reasonable particle size distributions,smaller particles between 0.1 and 1.0 lm contribute the majorportion to Afq (see Section 4.1). Determinations of mass productionrates from the continuum under the assumption of large particlesonly, with a constant ‘‘Albedo’’ and a constant phase function ashas been done in the past (e.g. Newburn and Spinrad, 1985; Weileret al., 2003; Agarwal et al., 2007, 2010) is not a physically realisticmodel. The scattering cross section and phase function have to becalculated for each particle size, multiplied by the number of theseparticles given by the size distribution, and then summed up to ob-tain an overall Afq which can be compared to measurements.

It has long been felt that interpretation of Afq determined fromthe visible spectral range cannot lead to realistic mass loss rates. Toour surprise, however, we find that using our more rigorous calcu-lation of Afq, and recent advances in our understanding of particlevelocity calculations using DSMC methods (Section 2), a betterknowledge of the dust composition from the ISO and Spitzer space-craft IR data and thus the complex refractive index (see discussionin Section 4.1 and Table 2), and a critical review of the particle sizedistribution (Section 3), we can determine dust/gas ratios whichare as reliable as those determined by other means, for examplethe infrared emission of the dust coma (Hanner, 1983a,b). Ouranalysis can provide useful input to the Rosetta mission dust envi-ronment modeling.

2. The DSMC model: Dust velocities and densities

The major results of the Direct Simulation Monte Carlo (DSMC)simulations have been published in two papers, the first on numer-ical simulations of the gas environment in the coma of comet CG(Tenishev et al., 2008), and the second on simulations of the dustcoma (Tenishev et al., 2011). Numerical results of the simulationscan be obtained on the site ‘‘Inner Coma Environment Simulator’’(http://ices.engin.umich.edu). We summarize some major pointshere. The calculations are axio-symmetric around the x axis, andwere carried out as simulations for the Rosetta spacecraft targetCG. The nucleus of CG is assumed to be 2 km in radius, with onehemisphere illuminated by the Sun. For the nucleus a density of0.3 g/cm3 is used and for the dust 1.0 g/cm3. A variety of heliocen-tric distances was calculated but we concentrate here on the calcu-lations at a heliocentric distance of 1.29 AU, which is quitecommon for Earth based cometary observations. The gas produc-tion used is QH2O = 4.76 � 1027 molecules/s and QCO = 2.4 � 1026 -molecules/s, leading to a gas mass loss rate of 153 kg/s.

The DSMC calculations follow a large number of gas and dustparticles emanating from the nucleus and take into considerationfactors such as elastic and inelastic collisions between gas mole-cules and between gas molecules and dust particles, ballistic tra-jectories of dust particles falling back to the nucleus surface orbecoming part of the dust coma, interchange of internal molecular

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 723

vibrational and rotational energy of gas molecules with transla-tional energy during collisions. It thus provides quite a realistic pic-ture of the physical conditions of the dust and gas outflow from acometary nucleus without making approximations to the Boltz-mann equation as is done in the hydrodynamic flow approach.The accuracy of the results is limited only by the number of parti-cles that can easily be accommodated by large parallel processingmachines, by the realism of the physical processes and reactionsincluded in the simulations, and by the expertise and numericalcalculation skills of the authors. Outputs of the calculations arequantities such as gas and dust number densities as a function oflocation with respect to the nucleus as well as velocities and ki-netic and rotational temperatures. We show two sample outputsproduced by these calculations in Figs. 1 and 2. We will use theseas well as their parameterization to calculate Afq under variousassumptions in Section 5.

Fig. 1 displays the dust particle velocities at five distances fromthe nucleus center. The first distance of 2.1 km is just above the nu-cleus surface. The figure shows that by 10–20 km from the nucleuscenter (with the acceptance of some imprecision, even at 5 km, or3 km above the nucleus surface) the dust particles have decoupledfrom the gas and have reached terminal velocity. The figure alsoshows that at a distance of 20 km from the nucleus, except for par-ticles smaller than about 1 lm, a simple power law of v (m/s) = 0.1423 � a�0.5, where a is the dust particle radius in m, fitsthe data quite well. The deviation of the velocity from this lawfor particles below 1 lm can readily be corrected. We use a cor-rected version of the above velocity fit in our parameterizationfor calculating Afq in Section 5. Because the gas provides the accel-erating force, the dust particle velocity can never be greater thanthat of the gas.

Fig. 2 shows the dust density in the sunward direction as a func-tion of particle size at a distance of 10 km from the center of thenucleus. The DSMC model uses a particle size distribution ofdQda � a�4, equivalent to Qd(a) � a�3 where Qd(a) is the productionrate of particles with radius a at the nucleus surface. The lower

Fig. 1. Results of DSMC calculations versus particle size for dust particle velocities at fivhave decoupled from the gas by about 10–20 km from the nucleus and have reached ter20 km from the nucleus using the approximation v (m/s) = 0.1423a�0.5, a being the partparticles below 1 lm, for which we used a different fit (not shown).

limit of the DSMC particle size distribution is arbitrarily set at0.10 lm and the upper limit is 10.0 mm, the largest liftable dustsize under the assumptions of the DSMC gas production rate. Thedust particles are distributed in 30 logarithmically spaced size bins,six bins per decade and thus a factor of 1.4678 apart. We note thatthe dust mass loss rates and dust densities in the paper by Teni-shev et al. (2011) were rather arbitrarily picked and are thus a con-siderable factor too high. The dust densities plotted in Fig. 2 wereadjusted to produce reasonable agreement with measured Afqvalues.

For the range of dust and gas production rates of comet CG, thevelocities, dust particle size distributions and dust and gas produc-tion rates can readily be scaled to other particle density and parti-cle size distributions. It is not until the gas and dust productionrates are about a factor of 20 larger that non-linear effects comeinto play. In other words, the back-coupling of the dust on thegas phase is negligible.

The term back-coupling of the dust on the gas phase is oftencalled dust-mass loading, which however should not be confusedwith the mass loading (both dust and gas) of cometary ions inthe solar wind.

Five additional curves are plotted on Fig. 2. The first is a param-eterized fit to the DSMC density results using a simple sphericallysymmetric outflow model where the particle density is given bynðrÞ ¼ Qd

4pvd

1r2 where Qd(a) = 2.613 � 10�5 � a�3. This matches the

dust density of the DSMC model for 10.0 lm particles. The dustvelocity is taken from the parameterized form of Fig. 1. We notethat the slope of this curve goes as a�2.5, for grains larger than1 lm, since dQ

da ¼ a�4; QðaÞ � a�3; nðaÞ � a�3

vd� a�2:5. It is seen that

the parameterized version fits the DSMC model quite well. Weuse it to extend the particle sizes beyond the limits of the presentlyavailable DSMC model. We also use it to explore various other par-ticle size distributions without having to go through the extensivecalculations of the DSMC model.

The other four curves use the particle size distribution proposedby us (Eq. (3)) and by Hanner (Eq. (2)) which will be discussed in

e distances from the nucleus center. The nucleus radius is 2 km. The dust particlesminal velocity. For illustrative purposes we also show a simple fit to the velocity aticle radius in m. This simple expression fits the dust velocity quite fine except for

Fig. 2. Dust particle number density at 10 km from the nucleus center versus particle radius using the DSMC model scaled to reproduce observed Afq values for 67P/Churyumov-Gerasimenko. Also shown are number density calculations using different particle size distributions, a simple differential particle size distribution dQ/da � a�4,and four distributions having small particle cut-offs that curtail the effect of small particles on Afq. All particle densities have been normalized to equal number densities at aparticle size of 10.0 lm. We plot two of our proposed distributions (Eq. (3)) with ap = 0.095 lm and ap = 0.50 lm and compare them to two Hanner distributions (dashed)with ap = 0.095 lm and ap = 0.48 lm. Our simpler and continuous dust size distributions (Eq. (3)), and those proposed by Hanner (Eq. (2)) agree quite well. As described in thetext, we note that the slope of the number density at large particle sizes goes as a�2.5, since dQ

da ¼ a�4; nðaÞ � a�3

vd� a�2:5.

724 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

detail in Section 3. We plot two cases, a peak particle radius of0.095 lm, and a peak particle radius of �0.50 lm. Particle densi-ties for all cases were normalized to equal number at 10.0 lm.For all these curves we use a differential particle size exponent of4.0 for the large particles. It can be seen that for particles largerthan a few lm the distribution is practically indistinguishable froma simple power law. However the particle density falls off quite fastfor particle sizes below 1 lm particularly for the case with a peakparticle radius of 0.50 lm. This can have a significant effect on theresulting Afq values which will be considered in Section 5.

As already mentioned we do not consider the dust number den-sities of the DSMC model as absolute values but use them as scal-able numbers, since we are mostly interested in the ratio of thedust/gas. The asymmetries in the axio-symmetric (or 2D) modelclearly produce significant differences in quantities such as dustdensity, dust velocity, largest liftable size between the sunlit andshaded hemispheres. However, to first order the ratio we are inter-ested in will not be seriously affected. In addition, rotation of thenucleus (which is not part of the model), and lateral transport (inthe y and z directions) will have a tendency to make these quanti-ties more symmetric and homogeneous, and thus more closelyapproximate the coma of a comet as observed from Earth basedtelescopes. Essentially we approximate the 2D model with a spher-ically symmetric 1D model, using the 2D model as a guide to thevarious parameters.

Since we are concerned with the ratio of the dust/gas, we didnot correct for the effective average size, effective particle surfacearea, or effective average mass in each size bin for the various sizedistributions. For example for a flat size distribution the effectiveaverage size in each bin is 1.234 times the value of the lower sizewhile for the distribution dQ

da ¼ a�4 it is 1.175 times the lower binsize value. Finally we note that Qd is an effective dust productionrate that can flow out freely from the nucleus and not the surfaceproduction rate in the DSMC model since many of the large parti-

cles can fall back to the surface under the action of the comet’sgravity.

3. Particle size distribution

At present an accepted and well established particle size distri-bution for the dust in the cometary coma is not available. A com-mon and simple form of the differential particle size distributionfor particles with radius a is an exponential type given by

dQda¼ dn

da

� �¼ g0a�a or QðaÞ � a�aþ1 ð1Þ

Q(a) is the production rate of particles of size a. We assume herethat when various authors talk about the differential particle sizedistribution, ðdn

daÞ, they actually mean dQda , and not the particle density

distribution at some point in the coma which will have a differentdistribution as discussed earlier; but this is not always made clear.The exponent a is typically between 3 and 4, and g0 is a normaliza-tion constant. We discuss various determinations of this exponentbelow. For their DSMC dust model simulations, Tenishev et al.(2011), have used the above differential particle size distributionwith an exponent of 4.

A somewhat more complex expression for the particle size dis-tribution is used by Hanner (1983a,b) for the interpretation of ther-mal infrared emission from comets. The formula is

dQda¼ g0 1� a0

a

� �M a0

a

� �að2Þ

In this expression a0 is a minimum grain radius, which Hanner(1983a,b) has taken to be 0.10 lm. This expression has no particlesbelow a0 and results in a maximum for the particle size distributionat a particle radius of ap, where ap ¼ a0ðMþa

a Þ. For grains larger thanabout 10 � a0 the expression reverts to the simpler form of Eq. (1).

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 725

Hanner (1983a) used a peak radius of 0.58 lm with M = 20 anda = 4.2. The exponent of 4.2 was taken from a study of cometaryanti-tails by Sekanina (1979). More than 90% of the thermal emis-sion from cometary dust arises from grains less than�20 lm (Han-ner, 1983a), so that this type of data-set gives no independentinformation about the slope at large particle sizes. Hanner(1983a) also notes that a0 must be less than 0.20 lm but it is notpossible to eliminate = 0.02 lm on the basis of thermal infrareddata. We have plotted Hanner particle size distributions for peakparticle radii of 0.095 and 0.475 lm and a = 4.0 in Fig. 2, and illus-trate their effect on Afq later (cf. Fig. 6).

Instead of using the Hanner particle size distribution, we pro-pose a slightly simpler version:

dQda¼ g0e�

a0að Þ � a0

a

� �að3Þ

It has only two free parameters a and a0 (or ap). Since the maximumof the distribution is given by ap ¼ a0

a , ap can be substituted for a0 inEq. (3). As can be seen in Fig. 2, this distribution has essentially thesame shape as the Hanner distribution and falls off as precipitouslyas the Hanner distribution for small particles. It is continuous anddoes not suffer from the defect that particles below a0 are not al-lowed. It is therefore analytically more tractable. We use this distri-bution in place of the Hanner distribution when we wish to curtailthe effects of the small particles. It can be seen in Fig. 6 that theresulting calculations of Afq for the particle size distributions ofEqs. (2) and (3) are quite close and can be brought into almost exactagreement by a slight adjustment of the parameters.

One might have thought that the 1P/Halley flyby spacecraftGiotto with its many particle detectors would have settled thequestion of the most appropriate particle size distribution for com-etary dust, but this is apparently not the case. In a comprehensivepaper on these results, McDonnell et al. (1987) give a typical massdistribution law of n(m) �m�0.85, but the exponent apparently var-ies at various locations in Halley’s coma. If we understand their pa-per correctly, there seems to be a flattening of the size distributionfor particles K 0:50 lm. There also seemed to be a higher flux oflarger sized particles than implied by the above power law so thata good total dust mass flux from comet Halley could not bedetermined. The mass power law distribution can be convertedto a differential particle size distribution: nðmÞ � m�0:85 ) dn

dm �m�1:85 ) dn

dm � a�5:55, and dnda ¼ dn

dmdmda � a�5:55a2 � a�3:55.

The results from the ‘‘Stardust’’ spacecraft dust measurementsto comet Wild 2 (Green et al., 2007) also did not yield a simplepower-law distribution. They reported a variable distribution dur-ing flyby with an overall mass power law exponent of 0.75(dn

da � a�3:25); but during the second period of high activity smallgrains dominated giving an exponent of 1.13 (dn

da � a�4:4). The small-est grains that they could detect were �3 lm.

More recently, thermal infrared measurements using the Spitzerspacecraft has produced some excellent new data on emission bysolid species in the coma of comets and the ejecta from the DeepImpact spacecraft mission to Tempel 1 by Lisse et al. (2006,2007). The various analyses and papers give somewhat conflictingresults, but it appears that the ejecta plume of 9P/Tempel 1 had aparticle size distribution dn

da � a�3:8 while the quiescent coma fol-lowed ‘‘a less steep power law’’. For Hale-Bopp at 2.8 AU, a distri-bution of dn

da � a�3:6to�4:0 is quoted. It is also noted that the DeepImpact ejecta plume had a significant number of 0.1–1.0 lm parti-cles necessary to produce the sharp mineral emission features seenin the spectra. Hale Bopp displayed less sharp emission features yetstill possessed an excess of small particles, though somewhat few-er in number.

It thus appears that a particle distribution power lawdnda � a�3:5 to �4:5 is most likely. There are also indications from theconsiderations of Hanner (1983a,b), the small particle flattening

of the 1P/Halley Giotto particle size distribution, as well as ourown very preliminary Afq wavelength dependent color simula-tions, that a fall off in the particle number below 1 lm is required.It is also quite possible and probably very likely that different com-ets exhibit different distributions. In the following section on thecalculation of Afq we will treat the power law exponent as a freeparameter to explore its effect on Afq. We also use our particle sizedistribution of Eq. (3) as well as the Hanner distribution both ofwhich have a peak in the distribution at a particular particle size.For both our distribution and that of Hanner we keep the exponentfor large particles, at 4, otherwise the parameter space would be-come too large. In order to keep the total mass loss rate fromdiverging, we limit the largest particle size to 10.0 mm, the largestliftable dust size under the assumption of the DSMC model.

4. Definition of Afq for single particle scattering

The concept of Afq was developed for ground based observa-tions looking through the whole coma from �1 to +1, along a lineof sight z, within a projected distance q from the nucleus. We de-rive Afq for this scenario. We approximate the size distribution bya step function and investigate one individual size bin before mov-ing on to a continuous distribution. Additional assumptions are aspherically symmetric dust outflow model for a comet, having aconstant outflow velocity and having no gain or loss of dust parti-cles. Hydrodynamic (e.g. Finson and Probstein, 1968), and DSMCcalculations show that the dust usually reaches a terminal velocityat several nuclear radii, typically 10–20 km (cf. Fig. 1).

The dust volume density as a function of the radial distance rfrom the comet center is then given by:

nvolðrÞ ¼Q d

4p vd

1r2

Qd (particles/s) is the dust production rate from the nucleus for par-ticles of radius a; r is the distance from the comet center, but cannotbe smaller than the nucleus radius rn.

The corresponding column density can be derived along a lineof sight, of the projected distance q in the sky from the nucleuscenter, where we take the limits z1, z2, going through the wholecoma from �1 to +1.

ncolðqÞ ¼Z z2

z1

nvolðrÞdz ¼ Q d

4pvd

Z z2

z1

dzq2 þ z2 ¼

Q d

4vd� 1

q

Total number of dust particles in a column within a circular observ-ing aperture of radius q is

N ¼Z q

0nðqÞ2pqdq ¼ Q d

4vdxZ q

0

1q

2pqdq ¼ Q d

4vd2pq

We note that with an accurate dust outflow model such asDSMC, the column density along the spacecraft line of sight z1, z2

can be calculated using numerical integration and the restrictionsof constant outflow velocity and spherical symmetry no longer ap-ply. The dust production rate then refers to a specific nucleus sur-face area sampled by the spacecraft line of sight e.g. looking downat the nadir towards the nucleus. As mentioned in the introductionwe do not pursue this scenario here, but have used it for the esti-mation of Rosetta instrument specific expected intensities and sig-nal-to-noise ratios.

To calculate Afq, the specific Intensity or radiance of light IðkÞscattered by a column of particles is required. It is assumed thatthe coma is optically thin and that single scattering applies. Thisis the case for most comets since it is well known that backgroundstars are easily seen through the coma. It is likely though, that veryactive comets, like Hale-Bopp, are optically thick in the inner fewthousand km of the coma. This case is not considered in this paper.

726 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

The derivation below is confined to particles of a single type andsize. Later it will be expanded to a distribution of sizes, by addingthe contributions to the Intensity in the various size bins. It canalso be expanded to include an ensemble of types (e.g. ice anddust) by summing up the contributions of the various particletypes. For an optically thin coma the Intensity of light IðkÞ (pho-tons/s cm2 lm sr) or (W/m2 lm sr) scattered by a column of parti-cles of column density ncol is given by:

IðkÞ ¼ F iðkÞncolðqÞrgeomqscaðkÞpðgÞ4p

F iðkÞ is the incident parallel flux (photons/s cm2 lm) or (W/m2 lm); rgeom = pa2 is the geometric particle cross section (m2), abeing its radius; qscaðkÞ is the scattering efficiency of the dust parti-cle; rscaðkÞ ¼ rgeom � qscaðkÞ, the scattering cross section of the dustparticle is also used some times; p(g) is the phase function normal-ized so that

R4p pðgÞdX ¼ 4p; for isotropic scattering p(g) = 1.

The scattering efficiency of the dust particle qscaðkÞ is a functionof the size and index of refraction of the particle, as well as thewavelength. For spherical particles it can be determined usingMie scattering theory.

For an infinitely deep atmosphere, where the optical depth is ex-pressed in terms of the extinction cross section and not the scatter-ing cross section, the single scattered component Is can be written as

Is ¼-4p

pðgÞ F iðkÞl0

lþ l0

where - ¼ qscaðkÞqext ðkÞ

¼ qscaðkÞqscaðkÞþqabsðkÞ

is the single scattering albedo while l0

and l are the cosines of the angles of incidence and emergence,respectively; cf. Chandrasekhar (1960), Section 47.1, Hapke (1981).

If this equation is not integrated to infinity it can be rewritten inthe optically thin case as:

Is ¼ - rgeom qextðkÞpðgÞ4p

zl

nvolF iðkÞ ¼ -rgeomqextðkÞpðgÞ4p

ncolF iðkÞ;

where zl is the slant path onto the scattering volume, and thus z

l nvol

becomes ncol.The above equation makes it appear that the single scattering

albedo - should be used in the equations for the scattered radi-ance and this probably led to the definition of A, in Afq as an albe-do. However we note that -qextðkÞ ¼ qscaðkÞ, and this equationreverts to our expression above. Thus the single scattering albedoonly enters into the equations for single scattering if the opticaldepth is expressed in terms of the extinction cross section of themedium. Similarly we note that the terms Bond albedo or the geo-metric albedo do not apply to scattering by small particles but bothare used for large solid bodies in the Solar System.

The flux FobsðkÞ (photons/s cm2 lm) or (W/m2 lm) gathered by aninstrument at the telescope can now be calculated. Using a circularaperture of angular radius h centered on the comet nucleus, equiva-lent to a projected radius q at a geocentric distance D in the sky it is:

Fobs ¼Z

IðX; kÞdX

dX ¼ 2pqdqD2 for a comet observation from the Earth centered on the

nucleus; F iðkÞ ¼ FsðkÞr2

hfor a comet at a heliocentric distance rh from

the Sun, and FsðkÞ is the solar flux at 1 AU;

Fobs ¼Z q

0

Fs

r2h

rgeomqscaðkÞpðgÞ4p

ncolðqÞ2pqdq

D2

¼ Fs

r2h

rgeomqscaðkÞpðgÞ

4pD2

Q d

4vd2pq ¼ Fs

r2h

rgeomqscaðkÞpðgÞ

4pD2 N

Following A’Hearn et al. (1984), we define f as the ratio of thesum of the geometric cross sections of the dust particles withinthe observation aperture to the actual observation aperture areaof radius q.

f ¼ Nrgeom

pq2 or Nrgeom ¼ fpq2

Thus

Fobs ¼Fs

r2h

qscaðkÞpðgÞ

4pD2 Nrgeom ¼Fs

r2h

qscaðkÞpðgÞ

4pD2 fpq2

¼ Fs

r2h

q4D2 pðgÞqscaðkÞfq ð4Þ

We now equate qscaðkÞpðgÞfq with Afq, and we get:

Afq � qscaðkÞpðgÞfq ¼Fobsr2

h4D2

Fsqð5Þ

This is the equation that is in standard use for calculating Afq fromobservational data.

Inserting Fobs from Eq. (4) into expression (5) we obtain anexpression relating Afq for particles of a particular type with indexof refraction n and radius a to their production rate and outflowvelocity.

AfqðaÞ ¼ rgeomðaÞqscaða;n; kÞpða;n; k; gÞQ dðaÞ2vdðaÞ

¼ Ddða;n; k; gÞQ dðaÞ2vdðaÞ

For convenience of notation we define a constant Ddust ¼Ddða;n; k; gÞ ¼ rgeomqscaðkÞpðgÞ; which contains the scattering prop-erties of the dust. The second term contains the particle outflowvelocity and the dust production rate which is a function of theparticle size distribution.

The total observed Afq is therefore the sum of the above over allparticle sizes and types:

Afq ¼X

a

Ddða;n; k; gÞQdðaÞ2vdðaÞ

¼ p2

Za2qscaða;n; kÞpða;n; k; gÞ

dQdðaÞda

1vdðaÞ

da ð6Þ

In this paper we calculate Afq using the first term of Eq. (6), adiscrete sum whose particle production rates are given by theDSMC model and various particle size distributions. The secondterm is an integral that could be used if continuous analyticexpressions for the factors within the integral were available.

We compare our formulation to that currently in use in the lit-erature for determining mass loss rates, e.g. Agarwal et al. (2007,2010), Weiler et al. (2003). The Weiler et al. formulation is basedon the derivation by Jorda (1995), which is stated to be valid onlyfor large particles.

Afq ¼ QN2p2ABðkÞDðbÞZ a2

a1

f ðaÞa2

vðaÞ da

In this equation AB is called the ‘‘Bond albedo’’ and a value of 0.3 isassumed for this value.

The quantity f(a) is the differential particle size distribution cor-responding to our dQdðaÞ

da . Rather than using the appropriate phasefunction for each particle size a common phase function, D(b), gi-ven by Divine et al. (1986) is used. The above expression is validfor large particles (x > 10 or so), because the scattering efficiencyand phase function become relatively constant (cf. Figs. 3 and 4).Thus these two quantities can be taken outside the integral forAfq. (The expression differs by 4p from our Eq. (6) because thephase integral is normalized to 1 instead of 4p, as we have chosento do). A similar equation is used by Agarwal et al. (2007, 2010),who use a formulation where AB is called a ‘‘geometric albedo’’for which they use values between 0.022 and 0.044. We couldnot find in the literature a calculation of Afq, or the mass loss rate

Fig. 3. Mie scattering efficiencies for spherical particles versus the x parameter, where x = 2pa/k. Each plotted x value represents an average over a size distribution to smooththe fine structure of the Mie scattering. Plots are shown for nreal = 2.0 and nreal = 1.6 and a variety of imaginary indices of refraction ranging from i = 0.001 which is essentiallytransparent to i = 1.60 which is opaque. The scattering efficiencies peak for x parameters between 1.0 and 10.0 (particles sizes of 0.16–1.6 lm, for a wavelength of 1.0 lm).The efficiencies go to zero for small particles and reach an asymptote of �1.2 for large particles. While the index nreal = 1.6 shifts the maximum efficiencies to slightly larger x,it does not cause major differences in the resulting Afq, as shown in Fig. 5.

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 727

that took the scattering properties and phase function for all parti-cle sizes into account.

4.1. Results of Afq calculations

We are now ready to calculate Afq for various particle size dis-tributions and various scattering geometries, via Eq. (6) derived inSection 4. The particle velocities, and the particle size distributionsor equivalently Qd(a) have been discussed in Sections 2 and 3. Forall of our calculations that follow, the particle size limits that weuse are 0.010 lm to 1.0 cm. We next proceed to the particle scat-tering efficiency qscaða;n; kÞ and the phase function, pða; n; k; gÞwhich must be calculated for each particle size bin using Mie scat-tering. The contributions from all the size bins then have to beadded up. Until this sum has actually been performed it is not atall intuitively obvious how the scattering properties of the particlescombine with the dust size distribution to yield a resultant Afq.The applicability of Mie scattering and its potential short comingsfor our calculations of Afq will be discussed in Section 4.2.

Fig. 3 shows the scattering efficiency for nreal = 2.0 and nimag

ranging from 0.001 (essentially transparent) to 1.60 (very stronglyabsorbing and opaque). The scattering efficiency is plotted versusthe parameter x = 2pa/k. The curves represent averages over arange of sizes. For the averaging we have used the size distributionof Eq. (2.56) Hansen and Travis (1974) with a b value of 0.10. With-out averaging, the scattering efficiencies display numerous strongundulations and wiggles. This fine structure is caused by interfer-ence between the diffracted and the transmitted waves as wellas surface electromagnetic waves which travel around the spheregiving off energy in all directions (Hansen and Travis, 1974). Sinceit is unlikely that a single size predominates in the coma we felt

that averaging is readily justified. The curves all show similarities:for small particles the scattering efficiency approaches Rayleighscattering and goes to zero, while for large particles x > 10 the scat-tering efficiency reaches an asymptote of approximately 1.2. In be-tween is a maximum which depends on the imaginary index ofrefraction. We also give a few examples for nreal = 1.60. Thesecurves are similar to those for nreal = 2.0 but shifted to slightly lar-ger x values.

The composition of coma dust is presently not known with pre-cision but thermal infrared spectra using the ISO and Spitzer satel-lites as well the ejecta from 9P/Temple 1 seen by the Deep Impactspacecraft have given us good insight into the dust emission by so-lid species in the coma of comets (Crovisier et al., 1997; Lisse et al.,2006, 2007). We list in Table 1 a baseline dust coma compositiontogether with optical constants at 1.0 lm. We pick a wavelengthof 1.0 lm which is roughly in the middle of the range of the ‘‘VIR-TIS’’ visible/infrared spectrometer on the Rosetta mission space-craft. The optical constants were taken from (Edoh, 1983; Jaegeret al., 1994; Dorschner et al., 1995). We give both the complex in-dex of refraction and the equivalent absorption coefficient in cm�1.For the majority of our calculations we chose a complex index ofrefraction n = 2.0 + 0.10i, which results in a slight weighting to-wards carbon which might dominate the low surface and comadust albedo of comets. However, we also carry out calculationsfor nreal = 1.80 and nreal = 1.60.

In Fig. 4 we present the phase functions for various particlesizes and indices of refraction. The phase function gives the scatter-ing efficiency versus scattering angle. Our integral over all scatter-ing angles is normalized to 4p. We note that the phase angle is180� – the scattering angle. Just as with the Mie scattering efficien-cies, the values presented are averages over a range of phase an-

Fig. 4. Mie phase curves versus scattering angle for various average x values. (Note that the phase angle is 180� minus the scattering angle). All curves are normalized so thatthe integral is 4p. As in Fig. 3, each curve represents an average over a range of particle sizes. The curve for the smallest x value is essentially that for Rayleigh scattering. Forlarge particles there is a strong forward scattering component. The figure demonstrates that particles with an imaginary index of 0.10 exhibit very little back-scattering, andthe imaginary index must be reduced to produce significant back-scattering.

Table 1Baseline dust composition and optical constantsa used in our work.

Refractive index Absorption coefficient

33% Olivine n = 1.80 + 0.10i kv = 12,000 cm�1

33% Pyroxene n = 1.70 + 0.04i kv = 5000 cm�1

33% Carbon n = 2.00 + 0.50i kv = 63,000 cm�1

a The optical constants for carbon are from Edoh (1983). For pyroxene and olivinethey are from Dorschner et al. (1995) and represent rough averages of intermediateFe content for wavelengths between 0.5 and 1.0 lm.

728 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

gles. The curve for xav = 0.43 closely resembles the Rayleigh scat-tering phase curve. For large xav values of 62.83 and 198.67 thereis a strong forward scattering component. For typical observationalphase angles of �40� (scattering angle of 140�), the values for largeparticles (x P 6) are about a factor of 10 below those of small par-ticles (x � 0.5). Thus assuming a constant phase function for allparticle sizes or assuming isotropic scattering can result in sizeableerrors for the Afq calculation.

Table 2Dust mass loss rates, Afq values and dust/gas ratios for various particle size distributions.

Particle size distribution function Dust mass loss ratea (kg/s) Afq at 40� (m

n = 2.00 + 0.10iQ � a�2 dQ/da � a�3 (396) (1.67)Q � a�2.5 dQ/da � a�3.5 22.8 0.598Q � a�2.75 dQ/da � a�3.75 7.54 1.01Q � a�3 dQ/da � a�4 4.66 2.29Q � a�4 dQ/da � a�5 (396) (122)UF (Eq. (3)) ap = 0.095 lm 3.21 0.633UF (Eq. (3)) ap = 0.50 lm 3.11 0.151

n = 1.80 + 0.04iQ � a�3 dQ/da � a�4 4.66 2.27UF (Eq. (3)) ap = 0.095 lm 3.21 0.740UF (Eq. (3)) ap = 0.50 lm 3.11 0.143

a The dust mass loss rate is calculated from 0.010 lm to the maximum liftable dust sb Assuming an Afq of 2.0 m and a gas production of 153 kg/s (5 � 1027 molecules/s).

We note that for the imaginary index of 0.10 very little back-scattering occurs which is in contradiction to the observed behav-ior. Exploratory Mie scattering calculations showed thatbackscattering is produced mainly by intermediate to large sizeparticles. However, the absorption using an imaginary index of0.10 is large enough that light within such particles is absorbedthus suppressing most of the back scattering component. To en-hance the effect of back scattering we examined the effect ofdecreasing the imaginary part of the refractive index. As an exam-ple, we plot in Fig. 4 curves for the size parameter of 9.22 (corre-sponding to an a of 1.47 lm at a wavelength of 1.0 lm) andvarious imaginary indices. It can be seen that back scattering canbe increased considerably by decreasing the imaginary index ofrefraction from 0.10 to 0.04, 0.01 and 0.001. For the latter two val-ues, back scattering is increased by well over an order of magni-tude. This will be discussed in more detail in Section 4.3.

The resulting Afq calculations for a variety of potential simpleexponential particle size distributions are displayed in Fig. 5. TheAfq values presented were calculated for a phase angle of 40�

) Within 0.1–1.0 lm (%) _m=Afq (kg/(s m)) CG estimateb (dust/gas)

(3.50) (237) (3.09)54.2 38.1 0.49879.0 7.47 0.09887.5 2.03 0.027(82.3) (3.25) (0.042)75.2 5.07 0.06615.2 20.6 0.269

89.9 2.05 0.02782.0 4.34 0.05728.0 21.8 0.284

ize radius of 1.0 cm. The grain density used is 1 g/cm3.

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 729

which is typical for Earth-based cometary observations. The result-ing Afq sums together with their mass loss rates are listed in Ta-ble 2. For illustrative purposes and to examine the trend of Afqwith particle size distribution, we have included in Fig. 5 the some-what extreme distributions of dQ/da � a�5 and dQ/da � a�3. Theparticle size distribution dQ/da � a�5 (Qd(a) � a�4) results inunreasonably large Afq values, partially due to our normalizingto the other distributions at 10 lm. With this size distribution,the major contribution to Afq is given by small particles (99.6%of the contribution is for particles between 0.01 and 1.0 lm), whilethe large particles contribute essentially nothing. On the otherhand, for the distribution dQ/da � a�3 (Qd(a) � a�2), the large par-ticles contribute most to Afq, and this distribution is thus quitesensitive to the large particle cut-off.

It is noteworthy that for the likely particle size distributions,discussed in Section 3, dQ/da � a�3.5 to a�4, there is a natural cutoff in Afq for both small and large particles. The contribution ofthe small particles falls off quite fast because their scattering effi-ciency times the geometric cross section decreases as a�6, eventhough their number density goes up. The contribution of the largeparticles decreases because their number density falls off rapidlywhile the scattering efficiency times the geometric cross sectiononly increases as �a2. The largest contribution to Afq (roughly50–90%; see Table 2) is produced by particles in the size range0.10–1.0 lm. Thus calculations of Afq or its application to massloss rates using a large particle approximation, as has been donein the past, cannot represent a physically realistic model.

To check the sensitivity of the resulting Afq calculations tochanges in the index of refraction, we also plot on this figure two

Fig. 5. Results of Afq calculations versus particle size for a variety of particle size distribuat a particle size of 10 lm. The small and large particle cut-offs are respectively at 0.010 langle of 140�). The total Afq is the sum of the individual Afq’s over all particle size bidemonstrate the effects of more extreme changes in the particle size distribution. The formparticles dominate making this distribution very sensitive to the large particle cut-off. Inexhibit natural contribution limits to Afq by both small and large particles. Particles betwfor the index nreal = 1.60 are also presented and show that the effect on the resulting Afqfigure use an imaginary index of refraction of 0.10.

calculations using the index n = 1.60 + 0.10i. While the resultingAfq values are somewhat lower, they do not differ substantiallyfrom those with nreal = 2.0. Additional calculations using an indexof 1.80 + 0.04i are listed in Table 2 and are discussed in conjunctionwith the dust phase curve in Section 4.3. The resulting Afq valuesare relatively insensitive to the exact choice of the refractive indexand any such variations are overwhelmed by those incurred usingvarious particle size distributions.

In Fig. 6 we present our calculation of Afq values using smallparticle cut-offs. We plot two of our distributions (Eq. (3)) andtwo Hanner dust size distribution (Eq. (2)). The first two use a peakparticle radius of 0.095 lm, while the second two use a peak par-ticle radius of �0.50 lm. For the large particles, all use a differen-tial particle size law of dQ/da � a�4. For comparison we also re-plotthe distribution of dQ/da � a�4 from Fig. 5. We first note that ourdistributions and those of Hanner agree quite closely and slightlytweaking the parameters, which would fall well outside of anyobservational verification, could yield identical total Afq values.Compared to a distribution dQ/da � a�4, using a peak particle ra-dius of 0.095 lm diminishes the effect of the small particles some-what and lowers the Afq sum, but not excessively. On the otherhand, using a peak particle radius of 0.50 lm severely curtailsthe contribution of small particles and lowers the resulting Afq va-lue considerably.

To put the resulting Afq values into perspective they must becompared to their dust mass loss rate for the same particle size dis-tribution. Since we normalize the particle number density in the10.0–14.7 lm bin, it follows that both Afq and the mass loss rateare similarly normalized. We plot the dust mass loss rate for the

tions. The particle number density for all particle distributions has been normalizedm and 1.0 cm. The calculations were performed for a phase angle of 40� (scattering

ns. The particle size distributions of dQ/da � a�5 and dQ/da � a�3 are included toer yields a very high Afq dominated by small particles, while for the latter the large

terestingly, the most probable distributions between dQ/da � a�3.5 and dQ/da � a�4

een 0.10 and 1.0 lm contribute the major fraction to the resulting Afq. Calculationsis moderate compared to changing the particle size distribution. All curves in the

Fig. 6. Calculations of Afq for small particle cut-off distributions. We compare our (UF) distributions for ap = 0.095 and 0.50 lm (Eq. (3) in the text) to Hanner sizedistributions with ap = 0.095 and 0.475 lm (Eq. (2) in the text) respectively. As expected the resulting Afq values are quite similar. For comparison the calculation for dQ/da � a�4 from Fig. 5 is repeated. By curtailing the contributions by particles below 1 lm, the resulting Afq is clearly reduced considerably, especially for the distribution withap = 0.50 lm. The small and large particle cut-offs are respectively at 0.010 lm and 1.0 cm.

730 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

various distributions in Fig. 7, and provide tabular values in Table 2.The table also lists the _m=Afq ratio which is independent of our nor-malization at 10 lm. The particle distribution of dQ/da � a�3 clearlyweights the large particles very strongly, is very much dependent onthe large particle cut-off and gives rather large values for the massloss rate. By contrast the distribution dQ/da � a�5 over weightsthe small particles but it is not affected strongly by the large parti-cles. The particle distribution used by the DSMC model (dQ/da � a�4) gives a constant mass for every particle size bin and thusthe total mass loss is dependent on the selected small and large par-ticle size cut offs. As mentioned earlier, we limit the large particlesize to 610 mm, the largest liftable size under the assumption ofthe DSMC model. This large particle size cut-off is not quite asrestrictive as it seems. For example, for dQ/da � a�4, if we extendthe large particle size cut-off to 1 m, we would only add 6 bins(see discussion in Section 4.4) or about 20% to the total mass sum.

4.2. On the applicability of Mie scattering for non-spherical particles

Mie scattering, strictly speaking, only applies to spherical parti-cles. On the other hand larger cometary particles are presumed tobe fluffy fractal aggregates of smaller particles. Pollack and Cuzzi(1980) have shown that for randomly oriented non-spherical par-ticles, Mie theory is still quite rigorous in the small particle regimewith x ¼ 2pa

k 6 5. (We note that this is only correct for calculatingthe scattering efficiencies and it does not apply if polarization isto be considered.) Inspection of Fig. 3 shows that this regime in-cludes the maxima in the scattering cross section and thus the ma-jor contribution to our Afq calculations.

For non-spherical particles with x P 5 the situation becomesmore complex. Pollack and Cuzzi (1980) show that the applicabil-ity of Mie scattering can be extended to a size regime of aboutx = 15–20, as long as the particles are compact and do not deviatefrom a roughly spherical shape, e.g. octahedral particles or roughspheres. We ourselves have carried out scattering calculations

using rotationally symmetric ellipsoids, cylinders and Chebyshevparticles, according to the work by Mishchenko (1991), havingequivalent 2 lm spherical grain (x = 12.56 at a wavelength of1 lm) geometric cross sections or volumes. We found the resultingphase functions and scattering cross sections to agree with spher-ical Mie particles to within a few percent. Thus if cometary parti-cles in that size regime are of such a shape or similar shape ourcalculations are still quite rigorous.

For particles that deviate substantially from spheres such ascubes, platelets or flakes, the phase function can differ consider-ably from Mie calculations for spheres. Pollack and Cuzzi (1980)propose a semi-empirical approximation to fit the scattering prop-erties and phase functions of a variety of randomly oriented suchparticles to experimental measurements. In addition to the com-plex index of refraction, these calculations require several addi-tional free parameters such as the particle shape, the particle sizedistribution of particles with this shape, as well as two moreparameters. For their laboratory fits, the particle shape is knownfrom the experimental data. However we do not presently havethat information for cometary coma particles thus making thistype of calculation moot.

On the other hand, particles may not be compact but ratheropen and fluffy. For this a better future approach is to utilizeimprovements in high-speed computers which allow calculationof the scattering properties of aggregate fractal particles using Dis-crete Dipole Approximations. Examples of such calculations for theinterpretation of cometary polarization are given in Kolokolovaand Mackowski (2012) for aggregates of 1024 monomers each0.10 lm in radius. The authors state that presently it is difficultto extend the number of particles because of the large amount ofsuper-computer time necessary. It is not clear how to define thex parameter for such open particles. Arranged in a cube, 1024 par-ticles would have a porosity of 80% and an equivalent radius of1.7 lm which might be construed as an x parameter of �11 at awavelength of 1 lm. Thus the present capability limit of Direct

Fig. 7. Mass loss rate in kg/s for the particle size distributions used in Figs. 4 and 5. The density used for the particles is 1.0 g/cm3 and the small and large particle cut-offs arerespectively at 0.010 lm and 1.0 cm. The more extreme particle size distributions of dQ/da � a�5 and dQ/da � a�3 give large mass loss rates, the former via the small particlesand the latter through the large particles (they both yield the same values because of our normalization at 10 lm, in the middle of our range). The distribution dQ/da � a�4

gives a constant mass loss rate for the various particle size bins whose sum thus depends only on the large and small particle cut off. The distributions cutting out thecontributions from the small particles concomitantly yield somewhat lower mass loss rates.

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 731

Dipole Approximation calculations is a relatively modest x param-eter for which Mie scattering might give a reasonable approxima-tion for the scattering cross section; a conclusion which wouldhave to be confirmed by actually performing such calculations.

In view of the present uncertainty in the shape of cometary par-ticles, our use of Mie scattering can give a first look at the calcula-tion of Afq. It has the advantage that the effects of varying indicesof refraction and particle size distributions can quickly be exploredand tested. In order to carry out meaningful calculations for thescattering efficiencies of aggregate particles larger than x � 5, theshape of the particles would have to be known, as well as their sizedistributions before break-up. (Because of the high impact veloci-ties, the original fluffy particles are not well characterized by thepresent particle detector experiments). Advances in super-com-puter technology also would be necessary. We have alreadypointed out that it is the particle size distribution which results inthe greatest uncertainty in the ratio dust/gas mass production(cf. Table 2), and not the scattering cross sections. Figs. 5 and 6show that for most particle size distributions the largest contribu-tion to Afq is by particles below 1 lm in size (x < 6.28) for whichMie calculations of randomly oriented non-spherical particles stillgive quite reliable results. If the particles are open and cross sec-tions and phase functions become available, they can be substi-tuted into our equations and result in more accurate calculationsof the dust Afq in comets.

4.3. The dust phase curve

Our calculations so far concentrated on a phase angle of 40� sothat they could readily be compared to observations made close tothis phase angle. However, our calculations can be expanded to cal-culate Afq at a variety of phase angles thus producing a phase curvethat can be compared to observational data. The results of this areshown in Fig. 8.

We are aware of two observational phase functions. The firstone was assembled by Divine et al. (1986) in preparation for the

1P/Halley observations and is plotted in Fig. 8. This phase functionseems to be reasonably reliable except for the region below about30�, where it was simply approximated by a straight line reachingan unrealistically low value at 0� scattering angle. A second phasefunction has been assembled by Schleicher (private communica-tion, 2012), and is also plotted in Fig. 8. (A discussion of this phasecurve and numerical values can be found at: http://asteroid.low-ell.edu/comet/dustphase.html.) This phase curve is made up oftwo parts. The data-set for the smaller phase angles (scattering an-gles P 140�), is taken from the Schleicher et al. (1998) work on 1P/Halley, while the data-set for small scattering angles is taken fromMarcus (2007). The data for the small scattering angles come fromfive comets all observed at heliocentric distances of �0.10–0.20AU. At such low heliocentric distance a considerable number oflarge particles can be lifted from the surface. The forward scatter-ing component of a coma dust distribution comes largely fromsuch large particles (cf. Fig. 4). The resulting phase function willthus exhibit quite strong forward scattering and is thus likely notrepresentative for the calculations we carried out for heliocentricdistances of 1.0 AU or larger. To lessen this bias, we have chosento plot in Fig. 8 the data for the comet with the largest heliocentricdistance of 0.260 AU (C/1980 Y1, Bradfield).

Also plotted in Fig. 8 are four representative phase curves fromour calculations. Our curve using an index of refraction ofn = 2.0 + 0.10i only exhibits week back scattering. We tried a vari-ety of particle size distributions with this index of refraction butnone yielded any significant back scattering. Small particles donot exhibit back scattering and the absorption with an imaginaryindex of 0.10 is large enough that the light within the large parti-cles is absorbed and thus these particles also do not contribute toback scattering (cf. Fig. 4).

We then tried reducing the imaginary index of refraction, calcu-lating phase curves for n = 1.80 + 0.001i, n = 1.80 + 0.01i, andn = 1.80 + 0.04i. We chose the real part of the index to be 1.80 in-stead of 2.0 because we felt that particles with less absorption mostlikely are silicate particles such as pyroxene or olivine with a lower

Fig. 8. Comparison of observed phase functions with our calculations. All curves are normalized to 10.0 at 90�. The observed phase functions are from Schleicher et al. (1998)for scattering angles > 90� combined with data from Marcus (2007) for scattering angles <90�. We also plot the earlier phase curve by Divine (1986). The Marcus data-setderives from comets at heliocentric distances between 0.10 and 0.20 AU and is thus very likely biased towards strong forward scattering by large particles. The plotted curverepresents data for Comet C/1980 Y1, Bradfield at a heliocentric distance of 0.266 AU as probably the least biased. The calculated curves show that an imaginary index of 0.10cannot reproduce the observed backscattering near 180�. By decreasing the imaginary index, a mixture of dark material (nimag P 0.10) and brighter material (nimag 6 0.04) canfit the observed backscattering quite well.

732 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

real part of the index of refraction (cf. Table 1). All these casesyielded a considerable amount of backscatter, much too high forn = 1.80 + 0.001i, still somewhat too high for n = 1.80 + 0.01i (shownin Fig. 8), and fairly closely matching our data for n = 1.80 + 0.04i.Choosing an index of n = 1.80 + 0.04i, affected the resulting Afq at40� phase angle in only a minor way, as shown in Table 2.

The observed backscatter can be matched quite closely by usingan index of n = 1.80 + 0.04i, and a small particle cut off ofap = 0.095 lm. Unfortunately this is not a diagnostic definitive re-sult. The observed phase curve can also be matched by a mixtureof �60% dark organic CHON particles (made up mostly of the or-ganic elements Carbon, Hydrogen, Oxygen, and Nitrogen), with lit-tle backscattering and �40% brighter silicate particles with anindex of n = 1.80 + 0.01i. Various other mixtures can also result inmatching the observed back scattering and thus a unique solutionwithout additional other data cannot be established. However, wecan reach the reasonable firm conclusion that dark CHON particleby themselves with an index of n = 2.0 + 0.10i, cannot match thedata. If comets are observed with large backscattering, they musthave a substantial amount of lighter silicate dust component, whilecomets exhibiting almost no backscattering have very little of this

Table 3Afq and gas production rates for 67P/Churyumov-Gerasimenko.

Date rh (AU) Phase angle (�) A

1982 October–1983 January (Av) �1.36 �20–38 21996 January–February (Av) �1.32 �44–48 11996 February 17 1.35 44.9 2DSMC model 1.29 40 0.

1. Schleicher (2006).2. Fink (2009).3. Gas production rates from Tenishev et al. (2008).4. Afq results from present paper.

material. This could provide a test for the gross variation in dustcomposition released by various comets. Reliable phase curvesfor comets are however difficult to establish.

4.4. The scalability of the _m=Afq ratio

We briefly address a subject, which has arisen, regarding thescalability of our results for different outflow velocities and extrap-olating to larger liftable dust sizes. There are various reasons to re-scale dust velocities including a variation in gas production rate, orthe reader might be interested in adopting dust terminal velocitiesfrom other works, e.g. by Sekanina (1981). For each bin, our pre-sented _m=Afq can be rescaled to a different dust terminal velocity,as desired:

AfqðaÞ � Q dv a2. The mass loss rate is _mdðaÞ � Q da3. Thus its ratio

is _mdAfq ðaÞ � va:

For example, if the velocities in all bins are halved from our val-ues in Fig. 1, the corresponding _m=Afq is also halved.

The presented _m=Afq can also be extrapolated to a different lift-able dust size. As illustrated in Fig. 1, the velocity of grains larger

fq (m) QH2O (s�1) QCO (s�1) Qgas (kg/s) Ref.

.33 6.1 � 1027 182 1

.27 4.6 � 1027 137 1

.83 1.0 � 1028 299 215–2.3 4.76 � 1027 2.4 � 1026 153 3, 4

U. Fink, M. Rubin / Icarus 221 (2012) 721–734 733

than 1 lm goes as v � a�0.5 and thus _mdAfq ðaÞ � a0:5. A detailed model

of maximum liftable dust sizes in jets as well as an analytical for-mulation to calculate these in a simple spherically symmetric casecan for instance be found in Combi et al. (2012). This formula in-cludes the effect of the size of the nucleus where, for a particulargas production rate, a smaller radius will yield a higher gas densityand thus a larger liftable particle size.

To carry out this extrapolation it should be noted that in our cal-culations, each decade in radius is divided into 6 logarithmicallyspaced bins, namely 1.00–1.47, to 6.81–10.00. The points are al-ways placed at the lower limit and give the integral over the wholebin, so e.g. the Afq at 215 lm in Fig. 5 indicates the total contribu-tion to Afq from grains with radii from 215 to 316 lm. The totalAfq or _m is listed in the figure legends over the whole size rangefrom 10�8 to 10�2 m and the user can then readily add or subtractvalues for desired whole bins.

5. Discussion and conclusions

Our results are summarized in Table 2. In this table we list thedust mass loss rate, _m, in kg/s, Afq values in m, and the dust-to-gas ratio ( _m=Afq) for various particle size distributions. We notethat while the Afq values and _m values in Table 2 can be signif-icantly affected by our normalization to equal particle numberdensity at 10 lm, the dust-to-gas ratio is independent of this.We also give the fraction of Afq which is due to particles in theradius range 0.10–1.0 lm. In addition to Afq values using a refrac-tive index of 2.0 + 0.10i, we list values for the index of 1.80 + 0.04iwhich were necessary to obtain a good fit to the observed back-scattering phase curves. The resulting Afq values at a phase angleof 40� changed only in a minor way. The somewhat extreme par-ticle size distributions of dQ/da � a�5 and a�3 which were calcu-lated mostly for illustrative purposes are put in parentheses.Ignoring these, the Afq values range from a low of 0.14 m forthe particle size distribution that severely curtails the effects ofthe small particles to a value of 2.29 m for the distribution dQ/da � a�4. For most of the distributions, except that withap = 0.50 lm, the contribution to Afq by particles between 0.10and 1.0 lm ranges between 50% and 90%. As a caveat we notethat the particle size distribution and the dust/gas ratio in thecoma of CG is not necessarily representative of the particle sizedistribution on the surface or the bulk dust/ice ratio in the nu-cleus of the comet.

Observed Afq values and gas production rates for 67P/Churyu-mov-Gerasimenko are listed in Table 3. Also listed are the valuesused for the DSMC model. For the latter, the Afq range given is thatfor the most plausible particle size distributions explained above.Comet CG exhibited a fair amount of activity so that both the ob-served gas production rate and Afq show considerable scatter.The post-perihelion values appear to be about a factor of 2.5 higherthan the pre-perihelion values. The values in Table 3, taken fromSchleicher (2006), are therefore averages, with some preference gi-ven to the higher post-perihelion values. In general we can say thatat a heliocentric distance of 1.35 AU the water production rates areof the order of 5–10 � 1027 molecules/s, in close accord with thevalues used in the DSMC model, while observed Afq values are ofthe order of 2 m with a considerable scatter pre/post-perihelionand with changing cometary activity.

We also calculate phase curves for various particle size distribu-tions and indices of refraction and compare them to observedphase curves. We find that using an imaginary index of refractionof i = 0.10 does not produce sufficient backscattering to matchthe observational data. However a mixture of dark particles withi P 0.10 and brighter silicate particles with i 6 0.04 can matchthe observed phase curves quite well.

From the _m=Afq ratio tabulated we can estimate dust/gas massproduction ratios for the coma of 67P/Churyumov-Gerasimenko,using the Afq of 2 m from above. These are listed in the last columnof Table 2. We find the dust/gas mass ratio at 1.3 AU is confined toa range of 0.03–0.3 with possibly a most likely value around 0.10.Thus it appears that the dust mass loss rate is considerably less,probably about a factor of 10 smaller, than the gas outflow. Thisprovides a useful input to the Rosetta mission dust environmentmodeling. It should be possible to refine this number once gooddust shapes and size distributions for CG become available. It willbe interesting to see the many measurements that the Rosetta mis-sion can bring to bear on this issue.

In addition we conclude:

– The original definition of A in Afq is problematical and Afqshould be: qscaðn; kÞ � pðgÞ � f � q.

– The ratio between Afq and the dust mass loss rate is dominatedby the particle size distribution.

– For most particle size distributions presently in use small parti-cles in the range from 0.10 to 1.0 lm contribute a large fractionto Afq.

– For the commonly used particle size distribution, dn/da � a�3.5

to a�4, there is a natural cut off in Afq for both small and largeparticles.

– Simplifying the calculation of Afq by considering only large par-ticles and approximating qsca does not represent a realistic model.Mie scattering theory or, when they become available, more com-plex scattering calculations for fluffy aggregates must be used.

Acknowledgments

The work of Uwe Fink was supported by a NASA Rosetta GrantJPL-1270067. The work of Martin Rubin was supported by JPL Sub-contract 1266313 under NASA Grant NMO710889. We extend ourspecial thanks to Martin Tomasko and Lyn Doose for many insight-ful discussions and above all for making their IDL version for Miescattering available to us. We also thank Melissa Dykhuis for re-coding this program for easier user interface and averaging of scat-tering efficiencies.

References

Agarwal, J., Müller, M., Grün, E., 2007. Dust environment modelling of Comet 67P/Churyumov-Gerasimenko. Space Sci. Rev. 128, 79–131.

Agarwal, J., Müller, M., Reach, W.T., Sykes, M.V., Boehnhardt, H., Grün, E., 2010. Thedust trail of Comet 67P/Churyumov-Gerasimenko between 2004 and 2006.Icarus 207, 992–1012.

A’Hearn, M.F., Schleicher, D.G., Feldman, P.D., Millis, R.L., Thompson, D.T., 1984.Comet Bowell 1980b. Astron. J. 89, 579–591.

Baum, W.A., Kreidl, T.J., Schleicher, D.G., 1992. Cometary grains. Astron. J. 104,1216–1225.

Chandrasekhar, S., 1960. Radiative Transfer. Dover, New York.Combi, M.R., Tenishev, V.M., Rubin, M., Fougere, N., Gombosi, T.I., 2012. Narrow dust

jets in a diffuse gas coma: A natural product of small active regions on comets.Astrophys. J. 749 (29), 13p.

Crovisier, J. et al., 1997. The spectrum of Comet Hale-Bopp (C/1995 01) observedwith the Infrared Space Observatory at 2.9 AU from the Sun. Science 275, 1904–1907.

Divine, N. et al., 1986. The comet Halley dust and gas environment. Space Sci. Rev.43, 1–104.

Dorschner, J., Begemann, B., Henning, T., Jaeger, C., Mutschke, H., 1995. Steps towardinterstellar silicate mineralogy. II. Study of Mg–Fe-silicate glasses of variablecomposition. Astron. Astrophys. 300, 503–520.

Edoh, O., 1983. Ph.D. Thesis, Optical Properties of Carbon from the Far Infrared tothe Far Ultraviolet, University of Arizona, Tucson, AZ.

Fink, U., 2009. A taxonomic survey of comet composition 1985–2004 using CCDspectroscopy. Icarus 201, 311–334.

Finson, M.J., Probstein, R.F., 1968. A theory of dust comets. I. Model and equations.Astrophys. J. 154, 327–376.

Green, S.F. et al., 2007. Stardust wild 2 dust measurements. In: Proc. ‘Dust inPlanetary Systems’, ESA SP-643, pp. 35–44.

734 U. Fink, M. Rubin / Icarus 221 (2012) 721–734

Hanner, M.S., 1983a. The nature of cometary dust from remote sensing. In: Gombosi,T.I. (Ed.), Cometary Exploration, vol. 2. Akademiai Kiado, Budapest, pp. 1–22.

Hanner, M.S., 1983b. Cometary dust studied with infrared photometry.Naturwissenschaften 70, 581–585.

Hansen, J.E., Travis, L.D., 1974. Light scattering in planetary atmospheres. Space Sci.Rev. 16, 527–610.

Hapke, B., 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res.86, 3039–3054.

Jaeger, C., Mutschke, H., Begemann, B., Dorschner, J., Henning, T., 1994. Steps towardinterstellar silicate mineralogy. 1: Laboratory results of a silicate glass of meancosmic composition. Astron. Astrophys. 292, 641–655.

Johnson, J.R., Fink, U., Larson, S.M., 1984. Charge Coupled Device (CCD) spectroscopyof comets: Tuttle, Stephan-Oterma, Brooks 2 and Bowell. Icarus 60, 351–372.

Jorda, L., 1995. Ph.D. Thesis, Observatoire de Paris-Meudon.Kolokolova, L., Mackowski, D., 2012. Polarization of light scattered by large

aggregates. J. Quant. Spectrosc. Rad. Trans. http://dx.doi.org/10.1016/j.jqsrt.2012.02.002.

Lisse, C.M. et al., 2006. Spitzer spectral observations of the deep impact ejecta.Science 313, 635–640.

Lisse, C.M., Kraemer, K.E., Nuth, J.A., Li, A., Joswiak, D., 2007. Comparison of thecomposition of the Tempel 1 ejecta to the dust in Comet C/Hale Bopp 1995 O1and YSO HD 100546. Icarus 186, 69–86.

Marcus, J.N., 2007. Forward-scattering enhancement of comet brightness. I.Background and model. Int. Comet Quart. 29, 39–66.

McDonnell, J.A.M. et al., 1987. The dust distribution within the inner coma of CometP/Halley 1982i – Encounter by Giotto’s impact detectors. Astron. Astrophys.187, 719–741.

Mishchenko, M.I., 1991. Light scattering by randomly oriented axially symmetricparticles. J. Opt. Soc. Am. A 8, 871–882.

Newburn, R.L., Spinrad, H., 1985. Spectrophotometry of seventeen comets. II – Thecontinuum. Astron. J. 90, 2591–2608.

Pollack, J.B., Cuzzi, J.N., 1980. Scattering by non-spherical particles of sizecomparable to wavelength – A new semi-empirical theory and its applicationto tropospheric aerosols. J. Atom. Sci. 37, 868–881.

Schleicher, D.G., 2006. Compositional and physical results for Rosetta’s new targetComet 67P/Churyumov Gerasimenko from narrowband photometry andimaging. Icarus 181, 442–457.

Schleicher, D.G., Millis, R.L., Birch, P.V., 1998. Narrowband photometry of Comet P/Halley: Variation with heliocentric distance, season, and solar phase angle.Icarus 132, 397–417.

Sekanina, Z., 1979. Expected characteristics of large dust particles in periodic CometHalley. ESASP 153, 25–34.

Sekanina, Z., 1981. Distribution and activity of discrete emission areas on thenucleus of periodic Comet Swift-Tuttle. Astron. J. 86, 1741–1773.

Tenishev, V.M., Combi, M.R., Davidsson, B.J.R., 2008. A global kinetic modelfor cometary comae: The evolution of the coma of the Rosetta targetcomet Churyumov-Gerasimenko throughout the mission. Astrophys. J. 685,659–677.

Tenishev, V.M., Combi, M.R., Rubin, M., 2011. Numerical simulation of dust in acometary coma: Application to Comet 67P/Churyumov-Gerasimenko.Astrophys. J. 732, 104–120.

Weiler, M., Rauer, H., Knollenberg, J., Jorda, L., Helbert, J., 2003. The dust activity ofComet C/1995 O1 (HaleBopp) between 3 AU and 13 AU from the Sun. Astron.Astrophys. 403, 313–322.