Wirtanen dust parameters provided by in-situ and ground-based observations

\ PERGAMON

Planetary and Space Science 36 "0888# 716Ð726

Constraints on comet 35P:Wirtanen dust parameters provided by in!situ and ground!based observations M[ Fulle Osservatorio Astronomico di Trieste\ Via Tiepolo 00\ I!23020 Trieste\ Italy Received 14 March 0887^ received in revised form 03 August 0887^ accepted 5 September 0887

Abstract A dust environment working group was encouraged by ESA to provide coma dust environment models useful to plan the ROSETTA operations around the nucleus of short period comet 35P:Wirtanen[ Among the many parameters describing the dust released from the nucleus surface\ special care was devoted to the dust size distribution[ Its present uncertainty makes all environment models sensitive\ mainly\ to which actual size distribution is adopted[ In fact\ it must be stressed that no other cometary dust parameter can be derived\ such as dust loss rate or dust to gas ratio\ if the size distribution remains undetermined[ This paper will focus\ therefore\ on the available information on cometary dust size distributions\ starting from the in situ measurement cornerstone provided by the GIOTTO!DIDSY results[ Available ground!based observations are then reviewed\ in order to disentangle the real sensitivity of them to this quantity^ the size distribution is always embedded together with other dust parameters\ and its in~uence on the published results is often forgotten[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ Introduction In 0886\ ESA encouraged the formation of a working group\ whose aim is the de_nition of a common frame! work useful to de_ne the dust environment inside which the Rosetta probe will operate[ The _rst output of this   group "Muller and Grun\ 0886# points out that the dust size distribution is the parameter whose uncertainties introduce the largest ~uctuations in the expected dust ~uxes[ In fact\ it is impossible to evaluate any other dust parameter if the size distribution remains unknown[ Therefore\ this paper will be devoted to reviewing all available information about this elusive cometary quan! tity\ starting from the cornerstone of the GIOTTO! DIDSY in situ measurements[ Later\ all available ground!based dust observations will be reviewed\ in order to check if there is data or rationale which may provide further constraints to "or being in clear disagreement with# the in situ results[ As an example of the importance of the size distri! bution\ let us take into account the well known Afr parameter\ which was introduced by A|Hearn and Sch! leicher "0873# to measure the dust activity of comets[ This parameter is simply provided by the coma and solar brightness ratio multiplied by the size r of the observation

 E!mail] fulleÝts[astro[it

diaphragm times the unknown dust albedo A[ In the hypothesis that the dust coma brightness is inversely pro! portional to the distance between the coma point and the nucleus\ the multiplication by r compensates the measure dependence on the adopted r value[ It follows that Afr has linear dimensions\ so that it can be interpreted as the size of the equivalent dust disk providing the observed coma brightness[ Although the quantity Afr is simple to measure and\ although it is always interpreted in terms of dust comet activity\ it is hard to extract from this quantity alone any useful information on the real dust activity of any comet[ In fact\ we can model it only assuming that the dust is ejected isotropically from the nucleus and that the dust crosses the observation dia! phragm with constant velocity[ It is clear that both these hypotheses are in contrast with any realistic dustÐgas drag model of the inner coma[ But\ even assuming these simplifying hypotheses\ the quantity Afr remains proportional to the dust loss rate times the dust cross section divided by the dust velocity[ The dust cross section must be averaged over all the dust sizes\ so that it heavily depends on the size distribution[ Since the size distribution and the dust velocity may show time variations even larger than those of the dust loss rate\ it is clearly dangerous to use Afr alone to derive any time dependence of the dust loss rate of any comet[ In other words\ Afr can be used only to constrain dust environment models\ which must be consistent with "i[e[

9921!9522:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved[ PII] S 9 9 2 1 ! 9 5 2 2 " 8 7 # 9 9 0 9 2 ! 1

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M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726

must provide# the observed Afr[ It cannot be used in any way to derive\ from pure observations\ any time evolution of the cometary dust physics[ In order to reach this goal\ it is dangerous even to adopt the mean size distribution provided by dust environment models[ In fact\ the size distribution which should be adopted to derive from Afr the dust loss rate at the observation time is the time! dependent size distribution] the di}erences in the loss rate derived by adopting the proper time!dependent size distribution or the time!averaged one may be of orders of magnitude[   For dust masses m × 09−01 kg\ Muller and Grun "0886# adopted a power law of the mass "or of the size#[ Special care will be devoted to check if such an assumption is consistent with the available data[ Moreover\ the DIDSY results pointed out that most of the mass was released in the form of large grains[ It turns out that the size distribution power index is mostly constrained by the actual behaviour of the size distribution at large sizes[ Therefore\ the _rst e}ort will be to derive the power index range of the distribution of large grain consistent with the DIDSY results[ This range should be interpreted as the lower limit of the range in general assumed by the power index of the time!averaged size distribution[ In fact\ although the DIDSY measurements covered a few hours time interval\ recent DIDSY data _ts point out that they depend on the dust ejection occurred over periods of months preceding the ~yby "Fulle et al[\ 0884#[ Then\ it will be checked if ground!based observations provide further constraints to the time!averaged size distribution[ Special care will be devoted to dust tail models\ which in the pre!Halley era provided the only available infor! mation about the dust size distribution[ Deep analysis will be devoted to all models applied so far\ with the result of showing that most of the inconsistencies among these results were mainly due to real errors in the model applications[ It is necessary to point out clearly these errors\ to avoid the confusion that in the past a}ected this branch of cometary science[

1[ The in situ experiments The GIOTTO!DIDSY data required many years to be properly re!calibrated\ and the _nal DIDSY distribution regarding the Halley ~yby was presented by McDonnell et al[ "0880#] the main di}erence with respect to previous papers regards the distribution at the largest masses^ the core of our problem[ Therefore\ results based on pre! liminary DIDSY distributions should be considered wrong independent of the correctness of the adopted model\ because they _t substantially wrong data[ One example of these wrong results is provided by the _t performed by Divine and Newburn "0876#\ which com! pletely misses the well known large grain excess charac! teristic of the DIDSY _nal size distribution[

The proper output of the DIDSY experiment was not the size distribution\ but the ~uence\ i[e[ the time integral of the dust ~ux expressed versus the dust mass[ In order to derive from the ~uence the size distribution of the dust released from the nucleus surface\ it is necessary to adopt models of the dust dynamics between the surface and the probe[ The simplest approach is to adopt isotropic dust ejection with time constant dust velocity\ which depends on the dust size as a power law with index k[ Hydro! dynamic models of the dust gas interaction in the inner coma provide k  −0:1 for large grains "Crifo\ 0880#[ Moreover\ it is usual to assume implicitly that for each dust mass\ a single velocity is possible[ Then\ if the cumu! lative ~uence has a power index b\ the cumulative mass distribution at the nucleus has a power index g  b−0:5\ and the corresponding di}erential size distribution has a power index a  2g−0[ It turns out that\ for the dust mass range 09−01 ³ m ³ 09−7 kg\ a  −3[1[ For larger masses\ there is a strong excess of grains up to m  09−5 kg\ and then the ~uence slope is ill de_ned\ due to poor experiment statistics[ A constraint is given by the single GRE event "Edenhofer et al[\ 0875#\ which constrains the index to a  −3[1 even for 09−5 ³ m ³ 09−2 kg[ However\ it is impossible to _t properly the ~uence with single event statistics[ In any case\ over the whole mass range 09−01 ³ m ³ 09−2 kg\ it is impossible to _t the size distribution with a constant power index[ When we _x the ~uence of m  09−01 kg\ if we adopt a  −2[2\ we obtain a ~uence which is not lower than the observed one over all the mass range\ and with a  −3[1 a ~uence not larger[ At m  09−7 kg\ a  −3[1 well _ts the observations\ whereas a  −2[2 overestimates them by a factor 09[ At m  09−5 kg\ a  −3[1 underestimates the observations by a factor of 09\ whereas a  −2[2 perfectly _ts the observations[ For the single GRE event\ both indices make an error of a factor of 09\ above or below the observations[ Therefore\ we can conclude that the isotropic approach _ts the DIDSY results with −3[1 ³ a ³ −2[2[ We point out that there are observational constraints against k  −0:1 "Fulle and Sedmak\ 0877^ Cremonese and Fulle\ 0878#] the real range of possible k values is −0:1 ³ k ³ 9\ so that the isotropic approach would really provide −3[1 ³ a ³ −1[7[ The obtained a range cannot however be considered satisfactory\ because it is provided by a surely unrealistic model[ Fulle et al[ "0884# developed an alternative approach which is able to avoid the most unrealistic hypotheses of the isotropic approach[ First of all\ the hypothesis of a single possible dust ejection velocity for each dust mass was rejected[ It should be pointed out that this hypothesis is very dangerous in the case of analyses of dust ~uxes collected by a space probe[ The space volume covered by a probe is an extremely low fraction of the whole coma volume\ so that small errors in the dust ejection velocity imply no impact with the spacecraft[ The

M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726

inverse approach is much more well de_ned\ and in fact it gave signi_cant improvements in the DIDSY data _t[ In particular\ it naturally predicts impacts with grains of any size even at large distances from the ~yby\ a mystery for classical fountain models[ Fulle et al[ "0884# computed the ejection velocity which the dust should have to impact the probe at each position of its path\ and named this quantity the impact velocity[ The impact velocity has nothing to do with the dust physics and dustÐgas inter! action in the inner coma[ It is simply the Keplerian vel! ocity which connects at any time the nucleus surface to the probe[ Therefore\ it is completely parameter free\ depending only on the ratio between the solar radiation pressure and gravity forces\ b  0−m  CQ"rdd#−0\ where C  0[08 09−2 kg m−1\ rd is the dust bulk density\ and d is the dust diameter[ Q is the so called scattering e.ciency\ Q  0 for absorbing grains larger than 9[0 mm "Burns et al[\ 0868#[ Then\ the ~uence model must de_ne the probability to have e}ectively this impact velocity\ and the dust physics and dustÐgas interaction enters here[ Moreover\ since the impact velocity is a vector\ it is necessary not only to de_ne a probability function of the ejection velocity absolute value\ but also of its direction\ i[e[ it is necessary to de_ne the anisotropy of the dust ejection[ In this way\ also\ the second unrealistic hypothesis of the isotropic approach is improved\ facing more realistic anisotropic comae[ For both the probability distributions\ Gaussian functions were adopted[ The ejection velocity absolute value distribution was centered on the classical values of the dust ejection velocity "which depends on the size according to a power law with index k  −0:1#\ while the ejection velocity direction distribution was centered along the sun direction[ We point out that the time range of dust ejection relevant for the DIDSY experiment was of the order of months\ so that this anisotropy dis! tribution should be considered as a time averaged ani! sotropy function[ Then\ the two ejection velocity distributions are characterized by dispersion values\ ns for the absolute velocity one\ and zs for the anisotropy one\ where z denotes the ejection zenithal angle[ Fulle et al[ "0884# showed that the isotropic results are perfectly reproduced when zs  p\ i[e[ for an isotropic dust ejection[ This proves that the model is consistent with all ~uence de_nitions\ and that the introduction of a wide velocity distribution does not introduce spurious results in the isotropic case\ as it must be[ Everything changes when zs ð p] an input size distribution with con! stant power index a over the whole mass range of the DIDSY experiment perfectly _ts the excess of large grains[ This proves that the DIDSY data do not contain enough information to constrain the source dust size distribution better than a power law with constant index for the mass range 09−01 ³ m ³ 09−2 kg[ This fact also suggests that the large grain excess is purely due to dust dynamical e}ects[ Moreover\ not only the total DIDSY

718

~uence was _tted^ for the _rst time\ also the pre! and post! ~yby ~uences were well explained in terms of ejection anisotropy[ A striking feature of the pre! and post!~yby ~uences is that they are identical at m  09−00 kg\ whereas at m  09−4 kg the pre!~yby ~uence is ten times larger than the post!~yby one[ This cannot be explained by time changes of the loss rate[ The adopted ejection anisotropy\ combined with the GIOTTO path orientation\ allows us to _t this di}erence between large and small dust masses[ Fulle et al[ "0884# _tted most DIDSY features by adopt! ing a constant power index −2[6 ³ a ³ −2[2[ In this _rst anisotropic approach\ the goal was to _t the ~uences only\ with no attention to the DIDSY ~ux[ To reach this goal\ a dust size independent zs was su.cient[ Levasseur!Regourd et al[ "0886# pointed out that the assumption of a size dependent zs allows us to _t both the total\ pre! and post!~yby ~uences\ and the DID3 dust ~ux\ and the OPE data "Levasseur!Regourd et al[\ 0875#\ assuming a constant geometric albedo over the dust mass range 09−01 ³ m ³ 09−2 kg[ In this approach\ special care was devoted also to other dust parameters\ such as the dust bulk density rd[ It was found that it is possible to _t most DIDSY and OPE data when constant values of the following parameters are adopted] 091 ³ rd ³ 2 092 kg m−2 and −2[9 ³ a ³ −1[4 "Table 0#[ From all these applications\ we can conclude that the DIDSY experiment was heavily in~uenced by the dust dynamics\ to which most of its features may be due\ with poor constraints\ to the size distribution[ In other words\ it is impossible to extract information on the dust size distribution from in situ experiments\ if simultaneous dust velocity measurements are not performed[ Taking into account all available DIDSY data _ts\ isotropic or anisotropic\ we must conclude that a power law with constant index is su.cient to describe the Halley dust size distribution\ and that the power index is actually in

Table 0 Parameters best _tting the DIDSY and OPE data for 09−01 ³ m ³ 09−2 kg "Levasseur!Regourd et al[\ 0886# rd kg m−2

a

Ap

M g

m

ns m s−0

zs deg

091 091 4×091 4×091 4×091 2×092 2×092

−1[4 −2[9 −1[4 −1[6 −2[9 −1[4 −2[9

9[994 9[90 9[92 9[92 9[93 9[02 9[05

9[8 0[1 0[1 0[9 0[1 0[3 0[1

29[9 2[9 39[9 04[9 2[4 24[9 3[9

89 89 79 79 64 34 34

4[4 8[9 8[9 09[9 00[9 08[9 19[9

For each value of the dust bulk density rd and of the source dust size distribution power index a\ all the other parameters were obtained by _tting the DIDSY and OPE data[ Ap is the geometric dust albedo for the phase angle of 62>\ M is the total collected dust mass\ m is the dust to gas mass ratio\ ns and zs are the dispersion values of the ejection velocity vector "the latter for the largest grains#[

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729

the range −3[1 ³ a ³ −1[4^ when we take into account only realistic anisotropic models\ −2[6 ³ a ³ −1[4[ It must be pointed out that these results deal with Halley only\ and with the DIDSY experiment alone[ The GIOTTO Extended Mission "GEM# allowed us to collect other DIDSY data for a short period comet\ 15P:GriggÐ Skjellerup "McDonnell et al[\ 0882#] the obtained ~uence is perfectly consistent with the Halley one\ although the statistics are so poor as not to allow realistic models to be built on it[ However\ this points out that the large grain excess was not an artifact of the Halley ~yby] it is a common feature which can be due to dust dynamics in both cases[ No deep analysis was performed so far on the VEGA data "Mazets et al[\ 0876#[ However\ we point out that their upper mass limit is 09−6 kg only\ so that anyway they will poorly constrain the dust size distribution at the large masses\ where most information is required[ It is planned in the near future to perform also the same probabilistic anisotropic analysis as the VEGA data\ which was so successfully applied to the DIDSY data[

2[ The FinsonÐProbstein model of dust tails Dust tail observations are the only ground!based ones allowing us to infer\ by means of proper models\ the dust size distribution[ In fact\ the solar radiation pressure acts as a mass spectrometer\ e}ectively placing grains of di}erent b  0−m parameter\ i[e[ of di}erent mass\ in di}erent positions[ In the case of zero dust ejection velocity\ a dust tail is well represented by the well known synchroneÐsyndyne network[ For the realistic case of non zero dust ejection velocity\ deconvolution models are necessary in order to disentangle the brightness con! tribution due to the size distribution from that due to the size and time dependence of the dust velocity[ Therefore\ dust tail models o}er the only available tool to infer properly the time dependence of both the dust size dis! tribution and the dust ejection velocity over long time intervals\ thus allowing us to compute properly the dust mass loss rate and then the dust to gas ratio[ Moreover\ they o}er a big advantage which is peculiar to them and often forgotten] the dust mass loss rate estimates provided by these models are completely independent of the poorly known dust bulk density rd[ In fact\ the dust tail brightness I"x\ y# is I"x\ y#  BD"x\ y# þ  BÐÐd"x\ y\ t\ 0−m#N"t#s"t\ 0−m# dt d"0−m#

"0#

where x\ y are the sky coordinates\ B is the brightness "thermal or optical# of a grain of unit cross section\ d is the grainþnumber per unit sky surface coming from tail models\ N is the dust number loss rate and s is the mean cross section of all the grains\ depending on the size distribution[ Tail models d depend on the quantity

b  0−m\ i[e[ on the product rdd[ Then\ we must express all quantities as it follows p ð"rdd#1Ł p s  ðd 1Ł  3 3 r1d

"1#

þ þ p ð"rdd#2Ł M  N rd 5 r2d

"2#

þ so that the relation between the dust mass loss rate M and tail brightness I"x\ y# is independent of rd þ ð"rdd#1Ł 2 dt d"0−m#[ I"x\ y#  BÐÐd"x\ y\ t\ 0−m#M"t# 1 ð"rdd#2Ł "3# The _rst quantitative model able to _t dust tails was the well known FinsonÐProbstein model "Finson and Probstein\ 0857#[ They computed in an analytical way the dimensionless brightness of a single synchronic tube þ N"t# f "t\ 0−m# dt "4# dD"x\ y\ t#  dr 1tn"t\ 0−m# d"0−m# so that the brightness of the whole dust tail is simply proportional to the numerical time integral of eqn "4#[ Since eqn "4# comes from an analytical integral\ it must satisfy the following condition\ otherwise eqn "4# is wrong] dr n"t\ 0−m#t Ł d"0−m# 0−m

"5#

which was named the hypersonic condition[ In eqns "4# and "5#\ r is the parametric coordinate along the synch! rone\ t is the time interval between dust ejection and tail observation\ n"t\ 0−m# is the dust ejection velocity and f "t\ 0−m# is the b distribution with power index −a−3[ In fact\ f "t\ 0−m# takes into account the brightness depen! dence on the dust cross section[ Moreover\ there is ano! ther strong approximation in the FinsonÐProbstein model\ often forgotten] the dust shells are assumed to expand linearly in time[ Kimura and Liu "0866# pointed out that this approximation holds only if the time t cor! responds to anomaly intervals much smaller than p] if this anomaly interval is equal to p\ then the synchronic tube is no more built up by the envelope of dust spherical shells^ on the contrary\ it becomes a 1D structure on the comet orbital plane[ This fundamental e}ect is necessarily due to the heliocentric Keplerian dynamics of dust\ which reaches its second orbital node at the observation time[ The spherical shell approximation and the hypersonic one have a fundamental consequence on the FinsonÐ Probstein model] when we adopt it\ it is impossible to infer any information about the size distribution of large grains\ because for large grains the two FinsonÐProbstein conditions are never satis_ed[

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3[ Perspective anti!tail observations

D"r\ u# 

Cometary anti!tails are extremely useful to infer the required information on the dust size distribution\ because they are always composed of very large grains\ in a mass range exactly corresponding to that of the DIDSY experiment[ Despite the fact that for anti!tails the two FinsonÐProbstein conditions are never satis_ed\ so that the FinsonÐProbstein model cannot provide any useful information on the size distribution of large grains\ it was systematically applied to provide the a value used to model the GIOTTO ~yby to Halley[ It is useful to enter into the details of these anti!tail models\ because they allow us to understand clearly why the adopted a value for the Halley models was wrong[ Not only were the hypotheses unrealistic\ but their application also was wrong[ In fact\ had the FinsonÐProbstein model been applied correctly\ it would have allowed us to predict correctly the DIDSY ~uences\ although these predictions would have been based on a wrong model[ Sekanina and Schuster "0867# analysed the perspective anti!tails of two short period comets\ 1P:Encke and 5P:D|Arrest[ They measured the power law of the anti!tail brightness with respect to the nucleus distance[ In order to avoid the hypersonic condition\ they made the absurd hypothesis "they admit this in the same paper# that the dust was released exactly at zero velocity[ Then\ the dimensionless tail brightness is simply given by þ Nf "t\ 0−m# dt d"0−m# "6# D"r\ u#  r dr du where r\ u substitute x\ y[ For anti!tails\ we can express the synchrone by means of the following parametric equations 0−m 

r \ t  h"u#to ro

"7#

where ro and to are parameters independent of 0−m[ Then\ since f "t\ 0−m#  "0−m#−a−3\ eqn "6# becomes þ to r D"r\ u#  N 1 r o ro

−a−4

01

[

"8#

Sekanina and Schuster "0867# measured an averaged rad! ial brightness power index of −9[74 so that they deduced a  −3[04[ This is the source of the misleading security that all the dust mass from comets was in small grains\ with fatal consequences for the planning of the GIOTTO mission[ Sekanina and Schuster "0867# admit that it is absolutely improbable that the dust ejection velocity was lower than 0 m s−0 for the observed anti!tail] this implies that these anti!tails were thick\ so that their brightness can be computed by means of the time integration of the FinsonÐProbstein synchronic tubes

g

þ Nf "t\ 0−m#

u1

u0

720

1h"u#n"t\ 0−m#

dr d"0−m#

dh"u# du du

"09#

where n"t\ 0−m#  no"t#"0−m#−k\ so that eqn "09# becomes\ in the approximation of a dust shell size com! parable to the nucleus distance and of quantities smoothly time dependent þ N r D"r\ u#  1noro ro

k−a−3

01

"00#

so that the observed radial brightness index −9[74 pro! vides a  −2[54 if k  −0:1\ or even a  −2[04 if k  9[ Sekanina and Schuster "0867# had all the tools to predict properly the DIDSY results[ However\ this would have been completely fortuitous\ because neither the hyper! sonic condition\ nor the dust spherical shell condition are satis_ed in eqn "00#[ The proper way to analyse anti!tail observations is by means of neck!line structures or tail numerical models[ However\ we must point out that a ³ −3 is not supported by any ground!based anti!tail observation[

4[ Neck!lines structures Kimura and Liu "0866# called the 1D structure orig! inated by the collapse on the comet orbital plane of the 2D synchronic tube at its second orbital node a neck! line[ When the Earth crosses the comet orbital plane\ the neck!line appears as a bright spike in the dust tail[ According to the observation perspective condition\ it may appear as a Sunward spike "the best example is the well known spike of comet ArendÐRoland#\ or over! lapped to the main dust tail "the best example was shown by comet Halley in May 0875#[ As we have seen\ we cannot model it by means of the FinsonÐProbstein method[ Fulle and Sedmak "0877# developed a quan! titative model which necessarily takes into account the wide velocity probability distribution\ and which is able to _t the photometry of neck!lines[ It was applied to comet Bennett 0869II "Fulle and Sedmak\ 0877#\ to comet Halley "Cremonese and Fulle\ 0878# and to comet Austin 0889V "Fulle et al[\ 0882#[ Since neck!line photometry was performed before the publication of the _nal papers devoted to DIDSY data\ the neck!line _ts of comets Bennett and Halley provide true predictions of the size distribution of large cometary grains[ Fulle and Sedmak "0877# showed that the neck!line surface density is

g



D"x\ y#  N

9

f "0−m#h"Vo#z exp"−z# d"0−m# ðx−"0−m#s cosVmŁ1¦ðy−"0−m#s sinVmŁ1 "01#

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721

Vo  arctan

z

y−"0−m#s sinVm x−"0−m#s cosVm

"02#

a1ðx−"0−m#s cosVmŁ1¦b1ðy−"0−m#s sinVmŁ1 n1"0−m#

"03#

where N is the number of grains composing the neck! line\ a\ b\ s and Vm are neck!line parameters depending only on Keplerian dynamics "and therefore precisely known#\ and h is the anisotropy distribution[ If the dust ejection is isotropic "h  const# and the hypersonic condition n"0−m# ð "0−m#sa is satis_ed\ then eqn "01# becomes D"x\ y# 

Nbf "0−m# 3zpsn"0−m#

$

× 0¦erf

ax b 1 y1 exp − 1 n"0−m# n "0−m#

% $

%

"04#

where now the x!axis is the neck!line axis[ It is clear that the width of the neck!line provides directly the size dependence of the dust velocity[ These are the only ground!based observations directly able to infer this extremely poorly known dependence\ and in general pro! vide k  −9[0429[94[ When n"0−m# is known\ then the neck!line brightness along its axis depends directly on the dust size distribution\ which can be derived into all its details[ For comet Bennett 0869II\ the model application provided information for 9[94 ³ rdd ³ 2 kg m−1 cor! responding to the mass range 09−00 ³ m ³ 09−5 kg "if rd  092 kg m−2#\ within the DIDSY experiment mass range[ The dust size distribution showed a strong excess of large grains\ extremely similar to that provided by the DIDSY experiment\ for 09−09 ³ m ³ 09−7 kg^ had the proper tail models been correctly applied to proper obser! vations\ the DIDSY results would have been predicted in all their details[ For comet Halley\ the neck!line pho! tometry provided information on the dust distribution between 09−09 ³ m ³ 09−4 kg\ with a strong excess of large grains for 09−6 ³ m ³ 09−4 kg[ These results refer to time intervals "a few hours# much shorter than those relevant for the DIDSY experiment "months#\ so that they point out that the time!dependent size distribution cannot be properly described by a simple power law[ The large grain excesses are common features for most comets\ and strongly support the conclusion that a × −3[

5[ Numerical inverse dust tail models A shortcoming of neck!lines is their rarity^ only bright comets show them\ they can be observed in particular perspective conditions only and hence provide the size distribution related to a very short time interval[ If the

FinsonÐProbstein method were to avoid its limitations\ which do not allow us to infer any information regarding large grains\ it would be much more powerful than neck! line photometry\ being applicable to every dust tail[ Such model improvements have been now available for ten years\ and are provided by the inverse numerical tail model "Fulle\ 0878#[ It avoids all the limits of the FinsonÐ Probstein approach\ namely] "i# It computes the rigorous heliocentric Keplerian orbits of millions of sampling dust grains\ so that the spherical shell approximation is avoided[ In this way\ it naturally builds up neck!lines\ when they can occur\ thus o}ering a check of the neck!line models^ "ii# It performs both the size and time integral by means of numerical methods\ so that the hypersonic approximation is avoided[ In this way\ it can _t not only the external tail\ but also the inner one\ as close as possible to the dust coma\ where the largest grains usually remain^ "iii# It takes into account anisotropic dust ejections^ "iv# It takes into account wide dust velocity distributions "Fulle\ 0881#^ "v# Regarding the dust size distribution\ it avoids the trial and error procedure typical of the original Fin! sonÐProbstein method\ by means of inverse ill!posed mathematics[ In this way\ the uniqueness of the results "impossible to be established in the original FinsonÐProbstein approach# is recovered in the least square _t sense[ This model was applied to about 04 comets "Table 1#\ thus allowing good statistics of the cometary dust size distribution[ It is among the very few models able to infer\ from the observations\ the time and size dependence of the dust velocity^ this\ as we have seen\ is fundamental to extract properly the dust physics from the available data[ The model outputs are insensitive to the ejection velocity probability distribution\ i[e[ it takes into account that many velocities are possible for every dust mass and ejection time "Fulle\ 0881#[ Moreover\ it is able to provide the time evolution of the size distribution\ so that the mass loss rate consistent with the actual size distribution can be evaluated[ In general\ the time dependent size distribution shows large variations[ For comet Austin 0889V\ the comparison between the time dependent size distribution provided by the inverse numerical model and the neck!line one was performed\ obtaining a good agreement[ From this time dependent size distribution\ the time averaged one was computed\ taking into account the weight of the dust loss rate[ The result for long period comets is −2[5 ³ a ³ −2[9\ perfectly consistent with the DIDSY results[ The only comet showing a very di}erent a power index was SekiÐLines\ for which however heavy dust sublimation processes are very probable\ being a Sun grazing comet "its dust size distribution should be therefore peculiar#[

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722

Table 1 The size distribution power index a provided by the inverse numerical dust tail model Comet

Time!dependent distribution

Time averaged a

Reference

Hyakutake 0885B1 09P:Tempel 1 P:SwiftÐTuttle 1959 Chiron Levy 0889XX 15P:GriggÐSkjellerup Austin 0889V 18P:SW0 Liller 0877V 1P:Encke 5P:D|Arrest SekiÐLines 0851III Kohoutek 0862XII Wilson 0876VII Brad_eld 0876XXIX

−3[6 ³ a ³ −2[9 −3[7 ³ a ³ −2[3 −4[9 ³ a ³ −2[9 −3[4 ³ a ³ −2[9 −4[9 ³ a ³ −2[9 −3[9 ³ a ³ −2[9 −3[4 ³ a ³ −1[7 −3[9 ³ a ³ −2[9 −3[2 ³ a ³ −2[2 −3[7 ³ a ³ −1[7 −4[9 ³ a ³ −2[4 −4[9 ³ a ³ −3[9 −4[9 ³ a ³ −2[9 −2[7 ³ a ³ −1[7 −3[1 ³ a ³ −2[9

−2[529[1

A+A 213\ 0086 A+A 200\ 222 A+A 181\ 293 A+A 171\ 879 PSS 31\ 152 A+A 165\ 47 A+A 161\ 523 Nat[ 248\ 31 A+A 142\ 504 A+A 129\ 119 A+A 129\ 119 A+A 106\ 172 A+A 106\ 172 AJ 099\ 0174 Dusty Ob[ 062

Regarding short period comets "the most interesting cases for the ROSETTA mission#\ the small dust tail of 15P:GriggÐSkjellerup maintained memory of the dust loss over so short a time interval\ as to make the com! putation of the time averaged size distribution mean! ingless[ For 09P:Tempel 1\ the dust size distribution showed such large time variations\ as to give poor mean! ing to its average[ For comets 1P:Encke and 5P:D|Arrest\ the averaged a is about −2[6\ in any case not far from the a discussed in Section 3\ devoted to anti!tails] this further con_rms that wrong models can sometimes pro! vide right results[ Available information seems to suggest that the a index of short period comets is somewhat lower than for long period ones\ although de_nitely larger than −3[ In all inverse tail model applications\ special care was devoted to establish the _t sensitivity to the velocity power index k[ In perfect agreement with the neck!line photometry\ the best tail _ts were in general obtained for k  −9[129[94[ No general consensus was reached about the reliability of the inverse numerical tail model\ mainly due to its complexity\ which makes its outputs not easily under! standable[ In any case\ at least the mathematical approach to the problem is rigorous\ being the ill!posed inverse problem mathematics a well studied science[ It was often stated that the results of these tail _ts should be further constrained by other observations\ e[g[ the observed light curves[ The problem is that nobody knows the real meaning of a light curve overall\ with respect to the dust loss rate[ Moreover\ every model can infer information over a limited dust size range\ so that incon! sistencies of results from di}erent models may simply point out that di}erent size ranges were taken into account[ There is no hope of transforming the complex problem of deriving the size distribution from ground! based observations into a simple one[ Dust tail bright!

−2[229[1 −2[129[0 −2[129[0 −2[929[1 −2[229[2 −2[429[1 −2[529[2 −2[729[0 −3[029[5 −2[229[3 −2[929[0 −2[129[1

nesses always will depend on the time!dependent size distribution\ which must be described by hundreds of free parameters^ there is no hope of avoiding any inverse approach[ Tail _ts with time!independent size dis! tribution would be simply unrealistic ones\ giving too much weight to other parameters\ which must be tuned to reach a satisfactory tail _t[ The best way to test the reliability of inverse tail models is to compare the results of di}erent models developed by independent groups[ A comparison between the original FinsonÐProbstein model and the rigorous numerical one\ taking into account the heliocentric Keplerian dust dynamics\ is o}ered by the _t of the dust tail of comet ArendÐRoland performed by Kimura and Liu "0866#[ Finson and Probstein "0857# claimed that the best _t of the tail features is most sensitive to small grains[ On the contrary\ Kimura and Liu "0866# showed that this choice was completely arbitrary\ since a × −3 allowed them to _t all the dust tail features[ Moreover\ for the _rst time\ they obtained k  −0:3\ in contrast with isotropic dustÐ gas drag models[ However\ since they adopted the usual direct approach\ it was impossible to test the real k and a ranges consistent with the observations[ Richter and Keller "0877# used a rigorous heliocentric Keplerian numerical model to _t the perspective dust anti!tail of comet Kohoutek[ However\ they used the usual direct approach\ so that it is impossible to be sure of the unique! ness of the k and a parameters they adopt[ Moreover\ as Kimura and Liu "0866#\ they adopted a time constant size distribution[ Nevertheless\ they _tted the Kohoutek anti!tail with −2[6 ³ a ³ −2[0\ in perfect agreement with the results in Table 1[ Recently\ Waniak "0881\ 0883# developed an inverse tail model independent of that providing the results listed in Table 1[ Almost all the results of both models were in close agreement[ In particular\ for comet Halley\ the dust

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M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726

velocity power index k versus the dust size was sig! ni_cantly larger than the value k  −0:1 provided by isotropic dustÐgas interaction models of inner coma] k  −9[1429[0 "Waniak\ 0881#\ in perfect agreement with the outputs provided by both neck!line and inverse numerical tail models[ The most interesting result for this paper is the time averaged dust size distribution power index a  −2[629[1 "Waniak\ 0881#[ These results were further con_rmed in the analysis of comet Wilson 0876VII] k  −9[1429[96 and a  −2 in average "Wan! iak\ 0883#[ However\ it must be remembered that in these models the dust size distribution is assumed constant in time\ so that the brightness tail details were _tted by means of special dust ejection patterns[ It is unclear whether the dust tail details are most sensitive to the dust ejection patterns or to the time changes of the dust size distribution\ which\ in any case\ should be correlated[ Nevertheless\ Waniak|s approach is one of the very few covering all the possible a and k values\ thus showing that\ when these parameters are left free of converging towards their most probable values\ numerical tail mod! els provide consistent results[

6[ Cometary trail observations Dust trails were discovered by the infrared satellite IRAS "Sykes et al[\ 0875#\ and provided further evidence that the population of large grains released from short period comets was much more signi_cant than that sug! gested by wrong applications of tail models to perspective anti!tails[ However\ no quantitative model of the observed trails was performed so far in order to extract from the trail photometry the most probable dust size distribution "Sykes et al[\ 0889#[ Dynamical models poin! ted out that their general photometric behaviour required the presence of cm!sized grains or even larger[ Qualitative trail width _ts allowed extraction of ejection velocities\ but it is unclear to which dust size they relate[ Until synthetic quantitative models are applied to available trail data\ it will not be possible to consider these results as conclusive[ We point out that the problem is similar to the dust tail modelling\ with a further di.culty] trails depend on a further parameter with respect to tails\ the number of comet revolutions which contribute to the trail formation[ For this reason\ priority should be given to the trails associated to short period comets which in a near past changed their orbit\ thus providing a sure upper limit to the trail age "e[g[ 54P:Gunn#[  Grun "0886# pointed out that the trail brightness gives a very useful constraint to the dust size distribution power index a[ In fact\ a result of this review is that the a lower limit is quite well de_ned "ranging between −2[8 and −2[6#\ whereas no clear a upper limit seems established\ at least by the DIDSY data analysis "ranging between −2[2 and −1[4#[ Since the trail brightness should

increase when a increases\ the trail brightness\ or even its non detection\ would give a sure a upper limit[ In particular\ he suggested that the nondetection of trails in the visible wavelengths already provides an available upper limit for the a index[ These computations were already performed[ Fulle "0889a# considered the outputs of the inverse numerical tail model applied to short period comet 1P:Encke "Table 1# as input to dynamical models able to compute both the trail morphology\ and its bright! ness in visible wavelengths[ It resulted that\ if the trail was built up within the past 21 comet revolutions\ the trail brightness would be R × 17 mag arcsec−1\ which is not reachable by any ground!based observations[ In the near future\ further computations will be performed tak! ing a as a free parameter\ to check which a range would lead to trails detectable by means of optical observations[ 7[ Meteor stream observations Meteor observations provide extremely good statistics of counts versus magnitude bins[ Within these data\ there is hidden much information regarding the cometary dust size distribution^ the problem is to extract it by means of proper meteor models[ Moreover\ the selective processes which act on the dust size distribution at its source remain unknown during the evolution from the parent dust tail to the dust trail and then to evolved meteoric streams[ Therefore\ an agreement between the dust size distri! bution index a provided by direct comet observations and that provided by meteor observations can be considered meaningful only in the implicit hypothesis that such selec! tive processes do not change the dust size distribution[ This hypothesis has to be veri_ed[ Nevertheless\ such a comparison may be interesting\ at least to check if the main result obtained so far "a × −3\ i[e[ all the dust mass is released in form of large grains# is con_rmed by meteor observations[ The conversion from meteor magnitudes to meteor masses depends on many parameters[ There is general consensus that most observations regard the mass range 09−6 ³ m ³ 09−3 kg "Jenniskens\ 0883#\ i[e[ exactly the mass range required to check the DIDSY results[ Inside this mass range\ meteor observations provide the mag! nitude distribution power index x[ The conversion of this index to a requires models of the conversion from meteoric mass to meteor brightness[ Current models point out that the critical limit a ³ −3 corresponds to x × 3[1 "Jenniskens\ 0883#[ The observations of tens of annual meteor streams provide the limits 1[9 ³ x ³ 2[5 "Jenniskens\ 0883#\ corresponding to −2[7 ³ a ³ −2[4[ Regarding the speci_c case of Halley\ the two associated meteor streams "a Orionids and h Aquarids# provide x  2[0 and x  1[6\ corresponding to a  −2[64 and a  −2[55\ respectively[ This di}erence\ negligible enough for our purposes\ remains however of unknown origin[

M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726

8[ Coma observations in visible wavelengths As stated in the Introduction\ no useful direct con! straint to the dust size distribution can be provided by coma observations in visible wavelengths[ The Afr quan! tity depends on so many parameters\ that every dust size distribution can _t the observed Afr[ It is su.cient to tune other free parameters within physically reasonable limits[ On the contrary\ Afr provides very useful con! straints to dust environment models\ which must be con! sistent with the observed Afr[ We point out that the most critical a range for these observations is given exactly by the most probable one\ −3 ³ a ³ −2] in this case\ if k  9\ most of the received light comes from small grains\ whereas most of the mass is ejected in form of large grains[ Therefore\ the received light can well be com! pletely independent of the released mass[ The obvious fact was often forgotten\ and explains most of the apparent paradoxes in the derived dust loss rate trends\ cometary activity and comet taxonomy "dusty comets and so on#[ Moreover\ the erroneous derivation of comet dust activity from the Afr quantity provided sometimes in the past results in complete disagreement with the outputs of tail models\ the only ones able to provide self!consistent dust loss rates^ since coma observations were much more abundant\ these were often preferred\ but the impli! cations were erroneous[ The observations of so called jets in inner comae pro! vided direct measurements of the expansion velocity of jets and shells\ always ranging in the 49Ð499 m s−0 interval "Jorda et al[\ 0883#[ From these values only\ it was deduced that most of the observed light came from small grains[ If this were true\ it would provide at least a sure upper limit a ³ −2[ However\ the DIDSY data _t poin! ted out that any velocity is possible for any size[ In par! ticular\ neck!line photometry pointed out that cm!sized grains can well have ejection velocities larger than 099 m s−0 at 0 AU[ Recent 2D hydrodynamical models of the inner coma seem to con_rm this conclusion] it is extremely di.cult to de_ne any velocityÐmass relation[ Even large grains can well reach the expansion velocities observed in the coma shells "Crifo\ 0886#[ Therefore\ the measured expansion velocities of jets and shells provide the upper limit a ³ −2 if and only if some velocity!mass relation exists\ and only if the common opinion that small grains have velocities much larger than large ones will be con_rmed[ We point out that\ all the times the velocity size dependence was tested\ k × −0:1 was found\ imply! ing that the common opinion may be false] the dust velocity may well be mass independent[

09[ Coma observations in IR wavelengths The only advantage of thermal IR observations with respect to those performed in visible wavelengths is that

724

the models necessary to extract the dust physics from the observations are independent of the unknown dust albedo "Hanner and Newburn\ 0878#[ However\ with respect to the core of our problems\ i[e[ the dust size distribution\ all the limitations discussed in the previous section apply here[ The fact that IR coma observations are unable to provide any direct information on the dust size distribution was proved in the past\ although this was often ignored[ Crifo "0876# showed that a dust to gas ratio between 9[0 and 29 is perfectly consistent with the IR ~ux and spectra received from Halley[ From Table 0\ we obtain that this means a ³ −1[4\ an upper limit not better than that already obtained by the DIDSY data _t[ In other words\ IR observations are unable to provide any constraint on the dust size distribution[ 00[ Coma observations in mm wavelengths These observations provide _rst quality data for our problem[ In fact\ they are mainly sensitive to the actual population of dust having exactly the sizes relevant to determine a value[ However\ since these are dust coma observations\ they cannot provide any direct information on the a index[ They allow measurement of the dust mass inside the view _eld[ In order to obtain the mass loss rate\ hypotheses must be assumed on the time the dust needs to cross the view _eld[ Nevertheless\ if a × −3\ they provide the most accurate values of the dust mass loss rate starting from coma observations\ because for a × −3 all the dust mass is ejected exactly in the grains of sizes to which the mm observations are most sensitive[ The comparison between the dust mass loss rates provided by mm observations and tail models allows us to check the tail model outputs\ and then the a values provided by the tail models too[ For P:SwiftÐTuttle\ mm observations "Jewitt\ 0885# provide a mass loss rate seven times larger than tail models "Fulle et al[\ 0883\ Table 1#[ For C:Hyakutake\ mm observations "Jewitt and Matthews\ 0886# provide a mass loss rate 09 times larger than tail models "Fulle et al[\ 0886#] these results show that the a values in Table 1 are lower limits for these comets[ In general\ these observations provide among the best checks to the fact that inverse tail models provide reliable lower limits of the dust mass loss rate and a index[ 01[ Radar observations Radar observations operating at cm wavelengths o}er even better tools to infer the actual mass loss rate of large grains[ However\ this powerful method can be applied only to bright comets during rare close Earth passages[ So far\ the method was applied to comets IrasÐArakiÐ Alcock "Harmon et al[\ 0878#\ Halley "Campbell et al[\ 0878# and Hyakutake[ Moreover\ the dust physics can be extracted from the radar data by means of models

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M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726

adopting many parameters[ Among these free parameters\ there is the dust size distribution[ The Halley radar data were found to be consistent with a  −2[4\ but no other value was tested to check which real a range is really consistent with the radar data[ It must be pointed out that the adopted model to interpret the radar data assumes k  −0:1\ so that\ for k  9\ radar observations would be consistent with a  −2[9[ Moreover\ for such large sizes the brightness contribution of a cloud of boul! ders in bound orbits around the comet nucleus cannot be excluded\ although preliminary models of the injection of such boulders in these bound clouds suggest that no more than 9[0) of the total ejected dust mass can reach such a _nal state "Richter and Keller\ 0884^ Fulle\ 0886#[ Nevertheless\ until more realistic 2D models will not de_ne precisely this percentage\ it is not sure what radar observations really observe[

02[ Conclusions The planning of the ROSETTA mission will strongly depend on the time evolution of the dust size distribution[ The only available tool to derive this quantity is given by the inverse tail model[ However\ it cannot provide useful data for the ROSETTA mission\ because "i# the time evolution of the size distribution is probably a chaotic phenomenon\ so that "ii# inverse tail models are unable to foresee the time evolution of the dust size distribution which "iii# can be evaluated a posteriori only[ Further\ "iv# the inferred time evolution of the size distribution is usually a}ected by strong noise\ making these results of limited use "and\ moreover\ they cannot be tested by any other model#[ It follows that our only hope is to model properly the time averaged size distribution[ Its uncer! tainties may be related both to model approximations\ and to the fact that it is the average of the time dependent size distribution\ so that in its dispersion it maintains memory of the possible range of the time dependent one[ The _rst fundamental question is how to model the size distribution[ We can conclude that\ for m × 09−01 kg\ there is no available data against the simple power law with index a independent of the dust mass m[ Recent DIDSY data _ts show that the bumps of the DIDSY ~uence are perfectly consistent with such a simplistic assumption[ Therefore\ the dispersion of the a values may contain information on the true time variations of the unknown time dependent size distribution[ The second fundamental question is which is the actual a range[ When we exclude surely unrealistic isotropic models\ all in situ and ground!based available data provide the range −2[8 ³ a ³ −1[4[ When we take into account in situ measurements only\ −2[6 ³ a ³ −1[4[ When we take into account ground!based observations only\ −2[8 ³ a ³ −2[9[ A lot of work is required to better constrain such wide ranges[ The upper a limit is par!

ticularly urged\ at least to understand to which sizes most common coma observations are most sensitive[ More! over\ we must stress that the inverse tail model provided much wider range of the time dependent size distribution power index\ −4 ³ a ³ −2\ so that perhaps there is no hope to improve signi_cantly our information on this fundamental parameter describing the cometary dust[

References A|Hearn\ M[F[\ Schleicher\ D[G[\ 0873[ Comet Bowell 0879b[ Astron[ J[ 78\ 468Ð480[ Burns\ J[A[\ Lamy\ P[L[\ Soter\ S[\ 0868[ Radiation forces on small particles in the solar system[ Icarus 39\ 0Ð37[ Campbell\ D[B[\ Harmon\ J[K[\ Shapiro\ I[I[\ 0878[ Radar observations of comet Halley[ Astrophys[ J[ 227\ 0983Ð0094[ Cremonese\ G[\ Fulle\ M[\ 0878[ Photometrical analysis of the neck! line structure of comet P:Halley[ Icarus 79\ 156Ð168[ Cremonese\ G[\ Fulle\ M[\ 0889[ The dust tail of comet Wilson 0876VII[ Astron[ J[ 099\ 0174Ð0181[ Cremonese\ G[\ Fulle\ M[\ 0883[ The dust environment of comet Levy 0889XX[ Planet[ Space Sci[ 31\ 152Ð157[ Crifo\ J[F[\ 0876[ Are cometary dust mass loss rates deduced from optical emission reliable< In] Ceplecha\ Z[\ Pecina\ P[ "Eds[#\ Inter! planetary Matter[ Czech[ Academy of Sciences Report G6\ pp[ 48Ð 55[ Crifo\ J[F[\ 0880[ Hydrodynamic models of the collisional coma[ In] Newburn\ R[L[ Jr[\ Neugebauer\ M[\ Rahe\ J[ "Eds[#\ Comets in the Post!Halley Era[ Kluwer\ Dordrecht\ pp[ 826Ð889[ Crifo\ J[F[\ 0886[ Private communication[ Divine\ N[\ Newburn\ R[L[\ 0876[ Modeling Halley before and after encounters[ Astron[ Astrophys[ 076\ 756Ð761[ Edenhofer\ P[\ Bird\ M[K[\ Brenkle\ J[P[\ Buschert\ H[\ Esposito\ P[B[\ Porsche\ H[\ Volland\ H[\ 0875[ First results from the Giotto radio! science experiment[ Nature 210\ 244Ð246[ Finson\ M[L[\ Probstein\ R[F[\ 0857[ A theory of dust cometsÐI] model and equations[ Astrophys[ J[ 043\ 216Ð241[ Fulle\ M[\ 0878[ Evaluation of cometary dust parameters from numeri! cal simulations] comparison with analytical approach and role of anisotropic emissions[ Astron[ Astrophys[ 106\ 172Ð186[ Fulle\ M[\ 0889[ Meteoroids from short!period comets[ Astron[ Astro! phys[ 129\ 119Ð115[ Fulle\ M[\ 0889b[ A model of the cometary dust trails detected by IRAS[ In] Bussoletti\ E[\ Vittone\ A[A[ "Eds[#\ Dusty Objects in the Universe[ Kluwer Academic\ Dordrecht\ pp[ 062Ð068[ Fulle\ M[\ 0881a[ Dust from short period comet P:Schwassmann!Wach! mann 0 and replenishment of the interplanetary dust cloud[ Nature 248\ 31Ð33[ Fulle\ M[\ 0881b[ A dust tail model based on maxwellian velocity distribution[ Astron[ Astrophys[ 154\ 706Ð713[ Fulle\ M[\ 0883[ Spin axis orientation of 1959 Chiron from dust coma modeling[ Astron[ Astrophys[ 171\ 879Ð877[ Fulle\ M[\ 0885[ Dust environment and nucleus spin axis of comet P:Tempel 1 from models of the IR dust tail observed by IRAS[ Astron[ Astrophys[ 200\ 222Ð228[ Fulle\ M[\ 0886[ Injection of large grains in orbits around cometary nuclei[ Astron[ Astrophys[ 214\ 0126Ð0137[ Fulle\ M[\ Sedmak\ G[\ 0877[ Photometrical analysis of the neck!line structure of comet Bennett 0869II[ Icarus 63\ 272Ð287[ Fulle\ M[\ Cremonese\ G[\ Cimatti\ A[\ 0889[ The dust tail of comet Brad_eld 0876XXIX[ In] Bussoletti\ E[\ Vittone\ A[A[ "Eds[#\ Dusty Objects in the Universe[ Kluwer Academic\ Dordrecht\ pp[ 062Ð068[ Fulle\ M[\ Mikuz\ H[\ Bosio\ S[\ 0886[ Dust environment of comet Hyakutake 0885B1[ Astron[ Astrophys[ 213\ 0086Ð0194[

M[ Fulle : Planetary and Space Science 36 "0888# 716Ð726 Fulle\ M[\ Cremonese\ G[\ Jockers\ K[\ Rauer\ H[\ 0881[ The dust tail of comet Liller 0877V[ Astron[ Astrophys[ 142\ 504Ð513[ Fulle\ M[\ Colangeli\ L[\ Mennella\ V[\ Rotundi\ A[\ Bussoletti\ E[\ 0884[ The sensitivity of the size distribution to the grain dynamics] simulation of the dust ~ux measured by GIOTTO at P:Halley[ Astron[ Astrophys[ 293\ 511Ð529[  Fulle\ M[\ Bohm\ C[\ Mengoli\ G[\ Muzzi\ F[\ Orlandi\ S[\ Sette\ G[\ 0883[ Current meteor production of comet P:SwiftÐTuttle[ Astron[ Astrophys[ 181\ 293Ð209[ Fulle\ M[\ Bosio\ S[\ Cremonese\ G[\ Cristaldi\ S[\ Liller\ W[\ Pansecchi\ L[\ 0882[ The dust environment of comet Austin 0889V[ Astron[ Astrophys[ 161\ 523Ð549[ Fulle\ M[\ Mennella\ V[\ Rotundi\ A[\ Colangeli\ L[\ Bussoletti\ E[\ Pasian\ F[\ 0883[ The dust environment of comet P:GriggÐSkjellerup as evidenced from ground based observations[ Astron[ Astrophys[ 165\ 471Ð477[  Grun\ E[\ 0886[ Private communication[ Hanner\ M[S[\ Newburn\ R[L[\ 0878[ Infrared photometry of comet Wilson "0875l# at two epochs[ Astron[ J[ 86\ 143Ð150[ Harmon\ J[K[\ Campbell\ D[B[\ Hine\ A[A[\ Shapiro\ I[I[\ Marsden\ B[G[\ 0878[ Radar observations of comet IRASÐArakiÐAlcock 0872d[ Astrophys[ J[ 227\ 0960Ð0982[ Jenniskens\ P[\ 0883[ Meteor stream activity*I] the annual streams[ Astron[ Astrophys[ 176\ 889Ð0902[ Jewitt\ D[C[\ 0885[ Debris from comet P:SwiftÐTuttle[ Astron[ J[ 000\ 0602Ð0606[ Jewitt\ D[C[\ Matthews\ H[E[\ 0886[ Submillimeter continuum obser! vations of comet Hyakutake "0885B1#[ Astron[ J[ 002\ 0034Ð0040[ Jorda\ L[\ Colas\ F[\ Lecacheux\ J[\ 0883[ The dust jets of P:SwiftÐ Tuttle 0881t[ Planet[ Space Sci[ 31!8\ 588Ð693[ Kimura\ H[\ Liu\ C[P[\ 0886[ On the structures of cometary dust tails[ Chinese Astronomy 0\ 124Ð153[ Levasseur!Regourd\ A[C[\ Fulle\ M[\ McBride\ N[\ Hadamcik\ E[\ 0886[

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Constraints on the physical properties of cometary dust particles from in situ measurements[ Astron[ Astrophys[ "submitted for pub! lication#[ Levasseur!Regourd\ A[C[\ Bertaux\ J[L[\ Dumont\ R[\ Festou\ M[\ Giese\ R[H[\ Giovane\ F[\ Lamy\ P[\ LeBlanc\ J[M[\ Llebaria\ A[\ Weinberg\ J[L[\ 0875[ Optical probing of comet Halley from Giotto spacecraft[ Nature 210\ 230Ð233[ Mazets\ E[P[\ et al[\ 0876[ Dust in Comet P:Halley from VEGA obser! vations[ Astron[ Astrophys[ 076\ 588Ð695[ McDonnell\ J[A[M[\ Lamy\ P[L[\ Pankiewicz\ G[S[\ 0880[ Physical properties of cometary dust[ In] Newburn\ R[L[ Jr[\ Neugebauer\ M[\ Rahe\ J[ "Eds[#\ Comets in the Post!Halley Era[ Kluwer\ Dordrecht\ pp[ 0932Ð0962[ McDonnell\ J[A[M[\ et al[\ 0882[ Dust particle impacts during the Giotto encounter with comet GriggÐSkjellerup[ Nature 251\ 621Ð 623[   Muller\ M[\ Grun\ E[\ 0886[ An engineering model of the dust and gas environment of the inner coma of comet P:Wirtanen[ ESA!RO!ESC! TA!4490 Report\ pp[ 0Ð49[ Richter\ K[\ Keller\ H[U[\ 0877[ The anomalous dust tail of comet Kohoutek "0862 XII# near perihelion[ Astron[ Astrophys[ 195\ 025Ð 031[ Richter\ K[\ Keller\ H[U[\ 0884[ On the stability of dust particles orbits around cometary nuclei[ Icarus 003\ 244Ð260[ Sekanina\ Z[\ Schuster\ H[E[\ 0867[ Meteoroids from periodic comet d|Arrest[ Astron[ Astrophys[ 54\ 18Ð24[ Sykes\ M[V[\ Hunten\ D[M[\ Low\ F[J[\ 0875[ Preliminary analysis of cometary trails[ Adv[ Space Res[ 5!6\ 56Ð67[ Sykes\ M[V[\ Lien\ D[J[\ Walker\ R[G[\ 0889[ The Tempel 1 dust trail[ Icarus 75\ 125Ð136[ Waniak\ W[\ 0881[ A Monte Carlo approach to the analysis of the dust tail of comet P:Halley[ Icarus 099\ 043Ð050[ Waniak\ W[\ 0883[ Nuclear dust emission pattern of comet Wilson 0876VII[ Icarus 000\ 126Ð134[