On deviations from free-radial outflow in the inner coma of comet 67P/Churyumov–Gerasimenko

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ContentslistsavailableatScienceDirect

Icarus

journalhomepage:www.elsevier.com/locate/icarus

On

deviations

from

free-radial

outflow

in

the

inner

coma

of

comet

67P/Churyumov–Gerasimenko

S.-B.

Gerig

a ,∗

_,

_R.

_Marschall

o

_,

_N.

_Thomas

a

_,

_I.

_Bertini

b

_,

_D.

_Bodewits

c

_,

_B.

_Davidsson

d

_,

_M.

_Fulle

e

_,

W.-H.

Ip

f

,

H.U.

Keller

g

,

M.

Küppers

h

,

F.

Preusker

i

,

F.

Scholten

i

,

C.C.

Su

j

,

I.

Toth

k

,

C.

Tubiana

l

,

J.-S.

Wu

j

_,

_H.

_Sierks

l

_,

_C.

_Barbieri

m

_,

_P.L.

_Lamy

n

_,

_R.

_Rodrigo

o ,p

_,

_D.

_Koschny

q

_,

_H.

_Rickman

r

_,

J.

Agarwal

l

_,

_M.A.

_Barucci

s

_,

_J.-L.

_Bertaux

n

_,

_G.

_Cremonese

t

_,

_V.

_Da

_Deppo

u

_,

_S.

_Debei

b

_,

M.

De

Cecco

v

_,

_J.

_Deller

l

_,

_S.

_Fornasier

s

_,

_O.

_Groussin

w

_,

_P.J.

_Gutierrez

x

_,

_C.

_Güttler

l

_,

_S.F.

_Hviid

i

_,

L.

Jorda

w

_,

_J.

_Knollenberg

i

_,

_J.-R.

_Kramm

l

_,

_E.

_Kührt

i

_,

_L.M.

_Lara

x

_,

_M.

_Lazzarin

m

_,

J.J.

Lopez

Moreno

x

_,

_F.

_Marzari

m

_,

_S.

_Mottola

i

_,

_G.

_Naletto

m ,b

_,

_N.

_Oklay

i

_,

_J.-B.

_Vincent

i a Physikalisches Institut, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland

b Centro di Ateneo di Studi ed Attivitá Spaziali, "Giuseppe Colombo" (CISAS), University of Padova, Via Venezia 15, 35131 Padova, Italy c Department of Astronomy, University of Maryland, Atlantic Building, 4254 Stadium Drive, College Park, MD 20742-2421, USA d Jet Propulsion Laboratory, M/S 183–401, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

e INAF – Osservatorio Astronomico, Via Tiepolo 11, 34014 Trieste, Italy

f Graduate Institute of Astronomy, National Central University, 300 Chung-Da Rd, Chung-Li 32054, Taiwan

g Institute for Geophysics and Extraterrestrial Physics, TU Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germany h Science Operations Department, European Space Astronomy Centre/ESA, P.O. Box 78, 28691 Villanueva de la Canada, Madrid, Spain i Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Rutherfordstraße 2, 12489 Berlin, Germany j Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 30010, Taiwan

k MTA CSFK Konkoly Observatory of the Hungarian Academy of Sciences Budapest, Konkoly Thege M. ut 15–17. H1121 Budapest, Hungary l Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg, 3, 37077 Göttingen, Germany

m Department of Physics and Astronomy, University of Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

n Laboratoire Atmosphères, Milieux et Observations Spatiales, CNRS & Université de Versailles Saint-Quentin-en-Yvelines, Guyancourt, France o International Space Science Institute, Hallerstraße 6, 3012 Bern, Switzerland

p Centro de Astrobiología, CSIC-INTA, European Space Agency (ESA), European Space Astronomy Centre (ESAC), PO Box 78, 28691 Villanueva de la Canada,

Madrid, Spain

q Scientific Support Office, European Space Agency, 2201 Noordwijk, The Netherlands r PAS Space Research Center, Bartycka 18A, PL-00716 Warszawa, Poland

s Observatoire de Paris, PSL Research University, CNRS, Univ. Paris Diderot, Sorbonne Paris Cité, UPMC Univ. Paris 06, Sorbonne Universités, 5 place Jules

Janssen, 92195 Meudon, France

t INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy u CNR-IFN UOS Padova LUXOR, Via Trasea, 7, 35131 Padova, Italy

v UNITN, Universitá di Trento, Via Mesiano, 77, 38100 Trento, Italy

w Laboratoire d’Astrophysique de Marseille, 38 Rue de Frédéric Joliot-Curie, 13388 Marseille Cedex 13, France x Instituto de Astrofísica de Andalucía (CSIC), c/ Glorieta de la Astronomía s/n, 18008 Granada, Spain

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 18 August 2017 Revised 6 March 2018 Accepted 8 March 2018 Available online 13 March 2018

a

b

s

t

r

a

c

t

TheOptical,Spectroscopic,and InfraredRemoteImaging System(OSIRIS)onboard theEuropeanSpace Agency’sRosetta spacecraft acquiredimagesofcomet67P/Churyumov–Gerasimenko(67P)and its sur-roundingdustcomastarting fromMay2014until September2016. Inthispaperwepresent methods and resultsfromanalysis ofOSIRIS imagesregarding thedustoutflow inthe innermostcomaof67P. Theaimistodeterminetheglobaldustoutflow behaviourandplaceconstraintsonphysicalprocesses affectingparticlesintheinnercoma.Westudy thecomaregionrightabovethenucleussurface, span-ningfromthenucleuscentreouttoadistanceofabout50kmcometcentricdistance(approximately25 averagecometradii). Weprimarilyadopt anapproachusedbyThomasand Keller(1990)tostudy the dustoutflow.Wepresenttheeffectsonazimuthally-averagedvaluesofthedustreflectanceofnon-radial

∗ _{Corresponding author.}

E-mail address: selina-barbara.gerig@space.unibe.ch (S.-B. Gerig).

https://doi.org/10.1016/j.icarus.2018.03.010

(DSMC)approachwithdustparticledistributionscalculatedusingatestparticleapproachhasbeenused todemonstratetheinfluenceofacomplexshapeandparticleaccelerationontheazimuthalaverage pro-files.We demonstratethat,whileothereffects suchas fragmentationorsublimation ofdustparticles cannotberuledout,accelerationofthedustparticlesandeffectsarisingfromtheshapeoftheirregular nucleus(non-pointsourcegeometry)aresufficienttoexplaintheobserveddustoutflowbehaviourfrom imagedataanalysis.

Asaby-productofthiswork,wehavecalculated“Afρ” valuesforthe1/rregime.Wefoundapeak inthe comaactivityintermsof Afρ (normalised toaphaseangle of90°) of∼210cm 20 daysafter perihelion.Furthermore,basedonsimplifiedmodelsofparticlemotionwithinboundorbits,itisshown thatlimitsonthetotalcross-sectionalareaofboundparticlesmightbederivedthroughfurtheranalysis. Anexampleisgiven.

© 2018TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The scatteringof sunlightby dust providesa means of study-ingprocessesintheinnercomaeofcomets.Thescatteredlightin theopticalisrelativelybright(comparabletothebrightnessofthe nucleusitselfinthecaseofcomet1P/HalleyduringtheGiotto fly-by -see Keller et al., 1994 ) andcan be observed by highspatial resolutionimagingsystems(Szego et al., 1989 ).Dustemittedfrom thesurfaceofa cometisacceleratedbygasdrag(Gombosi et al., 1985; Crifo and Rodionov, 1997 ). Hence,the dusttraces,albeit in a complex way, the flow of gas from the nucleus. Modelling of thisflow in3D is nowbecoming increasingly common (Tenishev et al., 2008; Marschall et al., 2016 ). This is helpful in studying gassources on the nucleus because the parent molecules inthe gasare only observable through emission bands outsidethe vis-iblewavelength rangeandare muchfainter requiringmonitoring instrumentstohavelowerspatialresolutionand/orlower tempo-ral resolutionthan imagingsystems(cf. forexampleKeller et al., 2007, Coradini et al., 2007 andGulkis et al., 2007 ).

For the dust itself, the size distribution, the composition, the structureandits evolutiononcereleasedfromthenucleusareall of interest. Sizes may be such that the emission and drag pro-cesses are insufficient for the particles to reach escape velocity (Möhlmann, 1994; Crifo et al., 2005; Thomas et al., 2015; Fulle et al., 2016; Agarwal et al., 2016 ).Fragmentationofparticlesmight occur asa result ofimparted stresses (Thomas and Keller, 1990 ) andsublimationcanresultfromtheheatingthatoccursafter par-ticleshavebeenemitted(Lien, 1990; Gicquel et al., 2016 ).In addi-tion,theacceleration oftheparticle isrelatedto theratioof the cross-sectionalareatothemass.Hence,detailedstudyofdust par-ticledynamicsisanimportantpartofthestudyoftheinnermost coma.

Thepurposeofthispaperistoinvestigatetheinnerdustcoma of67P/Churyumov–Gerasimenkousingatechniquefirstelaborated

byThomas and Keller (1990) inwhichconservationofradiance(as aproxyfordustcolumndensity)onconcentricsurfacesisusedto assesswhetherspecificprocessesaredominant.Weshallbe refer-ring todeviations from“1/rdependence” of thescattered bright-nessfromthedustalongaline-of-sightthatwouldbeexpectedfor force-freeradial outflow.Previouspapers(Lin et al., 2015; Gicquel et al., 2016 )havepresentednon1/rdependenciesindustprofiles of67P.However, non-radialoutflow,inhomogeneity ofthesource andthefinitesizeofthesource,allcomplicatetheinterpretation. Here, we address the question ofwhen the dust outflow can be considered to be 1/r and usemodels to show potential explana-tionsforobserveddeviationsfrom1/rclosetothenucleus.

In Section 2 , we will re-iterate and expand upon the tech-nique used by Thomas and Keller (1990) which includes the definition of a quantity we refer to as the “azimuthal aver-age” and which is related to the commonly used definition of “Af

ρ

” (A’Hearn et al. 1984 ). This relationship will be derived in

Section 4.3 .Given theapparent dominance oflarge slow moving particlesintheinnercomaof67P(Fulle et al., 2015 ),wealso pro-vide an assessmentof theinfluence oftheseparticles onthe az-imuthalaverage.InSection 3 ,wewilldescribetheconstructionof adatabaseofOSIRISimagesinwhichquantitiessuchasazimuthal averagevaluesatdifferentimpactparametersarefitandtabulated. In Section 4 , this database will be analysed indifferent ways to provideinsight intothe processesinvolved. Weendwith ashort conclusionsanddiscussionsection.

2. Theazimuthalaverageanditsinterpretation

2.1. Force-freeradialoutflow

canthenbewrittenas nd

(

r

)

= Qd 4

π

r2

_v

d (1) whereQdisthedustproductionrate,vdisthevelocityofthe out-flow and r is the distance from the point source. Integration of this equation along the line of sight (± infinity) produces a col-umndensity,

Nd

(

b

)

=

Qd 4 b

v

d

(2) whereb istheimpactparameter– thedistancetothenucleusin theimageplanepassingthroughthecentreofthenucleus.

Under the assumption that particles are not modified in any way, the radiance from the dust is proportional to the observed column density. This clearlyshows that imaging observations of dust emission froma point sourceshould produce radial profiles that varyinbrightness with1/b.(In theliterature, thisisusually referred toasa“1/rdependence” butweareusingrherefor dis-tances to the nucleusin 3D whereas in the above equation b is the impact parameter. The vectorial distance to the nucleus will notbeusedfurtherhereinandweshallrefertodeviationsfroma “1/rlaw” inthetext forclaritywhileusingthesymbol,b,forthe impactparameterwithinformulae.)Thisapproximationstillholds intheeventthatemissionofthedusthasanangulardependency which one wouldnormally expect to be the case for insolation-driven emission (i.e.emission roughly proportional to the cosine ofthesolarzenithangle).

Clearly, v_d cannot be constant throughoutthe innercoma be-cause the dust particles must be accelerated from rest at the surface of the nucleus to velocities well in excess of escape ve-locity. The gas itself is also accelerating. The computations of

Gombosi et al. (1985) showedthat dustterminal velocities could easily exceed 200 m/s because of gas drag and these velocities areparticlesizedependent(Davidsson et al., 2010 ).Assuch,v_d in

Eqs. (1 ) and(2 ) issize dependent andthe localdensityand col-umn densitythen become integrals over particle size. Inspection of Eq. (2 ) shows that, after emission fromthe surface, the local dustdensitymustdropmoresteeplythana“1/rlaw” inthe accel-eration regionandthus investigationof theinner comaofcomet 1P/Halley wastargetedatlooking forthisbehaviour inthe inner coma. Inthis case, it wasshown (Thomas and Keller, 1990 ) that the scatteredlight from thedust did indeedtend towards 1/r at distances greaterthan50kmfromthenucleusorinotherwords fordistances _>10average Halley-radii.(TheaverageHalley-radius of 5 kmwas chosen such that a sphere withthe mentioned ra-diushasaboutthesamesurfaceareaasthenucleusofcomet Hal-ley, which was for this purpose approximated by an ellipsoidal shape following Keller et al. (1994) ). However, directlyabove the nucleus, profiles of the conserved quantity, I·b (see Section 2.2 ), whereIistheradiancefromthedustcolumn,dropped when ap-proaching the nucleus surface instead of rising. This was recog-nized (Huebner et al., 1988; Reitsema et al., 1989 ) asbeing the resultofthefinitesize ofthe source.Thiswasalsodiscussed ex-tensivelybyBoice et al. (2002) inrelationtoobservationsofcomet 19P/BorrellywiththeDeepSpace1spacecraft.Thespecificationof the point where b = 0 asthe emitting surface wasseen as be-ing a contributorto thisproblembecausethe sourcehas afinite size. Inthe approximationofEq. (1) , ngoesto infinityasb goes tozero.Evenatthistimeitwasrecognizedthatthereseemstobe nophysicalwaytodefinethevalueofbattheemittingsurface.

2.2. Theazimuthalaverage

The idea of defining a surface through which the dustwould passandtherebyapplyingconservationofscatteringareawasfirst

discussedinThomas and Keller (1990) .Accordingtoaformulation ofGauss’theorem



E· dS = constant

a flux E integratedover an arbitrary closed surface around a fi-nitesource multipliedbythesurfacearea (surfaceelementdS) is constant.Thevalueoftheintegralgivesthesourcestrengthofthe enclosed source.It wasshownby Thomas and Keller (1990) that observed reflectance fromimaging data can be related to a dust fluxandthat inthecaseofline-of-sightobservationsintegrations overcylindricalsurfacesenclosingasource(i.e.thecometnucleus) are a good approximation forclosed surfaces. In other words, in thefreeradialoutflow/pointsourceapproximation,theintegralof thecolumndensityalong thelength ofa circleofcircumference,

d(=2

π

b),centredonthesourceisaconstantindependentofthe sizeofthecircle.Thiscanbewrittenintermsofangleas

A

(

b

)

= k  2π 0 N

(

b,

θ

)

d

θ

= k  2π 0 Q

(

θ

)

4 b

v

d

(

θ

)

d

θ

(3)

wherekisa constantofproportionalitythat canbe usedto con-vertthecolumndensityintoanobservedradianceorareflectance and

A

(

b

)

b = constant = 2

π

A¯. (4) ThequantityA(b)isan azimuthallyintegrateddustreflectance andthequantity A¯=Ab/2

π

,wasreferred toastheazimuthal av-erageby ThomasandKeller andhasdimensionsof[m] ifA isin reflectanceunits.Theproductofreflectanceandimpactparameter iscloselyrelatedtothedefinitionofAf

ρ

(A’Hearn et al., 1984 )as willbeshowninSection 4.3 .

Thisapproacheffectivelyeliminatestheprincipalissuewith ra-dialprofiles,namelytheimpossibilityofdefininga1D profile tak-ingintoaccount intersectingdustflows andnon-radialexpansion ofthedustinto3D,andtheproblemwithdefiningthedistanceat theemittingsurface.Here,thesourceiscompletelysurroundedby acylindricalsurface(takingintoaccount thelineofsight)sothat theentiredustfluxpassesthroughthesurfaceasthedustexpands withconstantvelocityintotheinnercoma.Weareineffect apply-inganequationofcontinuity.Inthefollowingdataanalysis,weare interestedindeviationsfromtheconstancyoftheazimuthal aver-age which we expect forfree-radial outflow andwe discuss the physicalprocessesthatcouldexplaintheobserveddeviations.

2.3.Illustrationusingasimplemodelcalculation

Someoftheimportantaspectsmentionedabove(e.g.theeffect of accelerationon the azimuthal average) can be seen clearly in simplifiednumericalmodels.Weusethesemodelstoillustratethe basicphysics of the problem. Amore detailed descriptionof our numericalmodelsisgiveninSection 5 .

Fig. 1 showsa calculationofisotropicgasoutflowina spheri-callysymmetricsystem.Thegasvelocitiesanddensitieshavebeen computedusing theDirect Simulation MonteCarlo (DSMC) tech-nique.(Cercignani, 20 0 0 discussesanalyticalsolutionstothis prob-lem butwe merely use the numerical approach here asa start-ing point before adding furthercomplexity.) The gasdensity has thenbeenconvertedintogascolumndensity.Fig. 2 showsthe be-haviourofanintegralofthegascolumndensity,multipliedbybin thiscase,showing,asexpected,thevalueasymptotically approach-ingaconstantvaluefromhighervaluesclosetothenucleus.

Fig. 1. The column density through a spherically symmetric outflow of water vapour from a simulated, spherical nucleus of 2 km in radius computed using the Direct Simulation Monte Carlo (DSMC) technique.

asthe surfaceareaofthe irregularnucleusitself.This more gen-eralformulationallows easiercomparisonofthedustoutflowfor differentsizedcomets (e.g.comet Halleyandcomet Churyumov– Gerasimenko).

For completeness, we alsoshow computations forthe dust in thissimplecaseusingastandardanalyticalapproachforthedrag. Theaccelerationofthedustinthegasflowfieldiscomputed tak-ing into account the gas drag through a standard drag equation andthegravityfieldfromthesphere

ma = FG + 1

2 CDmgng

σ|

v

g −

v

d

|

(

v

g−

v

d

)

(5) wherethedragcoefficient(usuallysetto2forsmoothspheres)is heresetthroughtheequation

CD = 2 s2_{+ 1} √

π

s3 e −s2 + 4 s4+ 4 s2− 1 2 s4 erf

(

s

)

+ 2

(

1 −

ε

)

√

π

3 s



Td Tg (6) where s=

|

v

_

g −

v

d

|

2kTg mg . (7)

The othervariables inEqs. (5 ) and(6 ) arethemolecular mass ofthe gas (mg), the local density (ng), the cross-section (

σ

), the

Fig. 3. The azimuthal average for different dust particle sizes follows a similar trend to that of the gas. The differences between the different particle sizes are rather small in this case.

fractionofspecularreflection(

ε

) (set to0),andthegasanddust velocities(vg,vd)whichprovidepartoftheforceonthedust parti-cle(ma).Thegastemperature(Tg)istakenfromtheDSMC simula-tionofthegasflowfieldandweassumethedustandgastobein thermalequilibrium(dusttemperatureT_{d =} Tg).Thegravitational accelerations at the dust particle positions were calculated for a centralspherewithradius2kmandabulk densityof462kg/m3

anda constant densityof 440 kg/m3 _{for spherical dust particles}

wasassumed.

InFig. 3 ,wehavecomputedthevalueof ¯Abutnormalisedto1 at10kmdistancefromthenucleuscentre.EvaluatingEq. (2) with accelerationmodifyingthedustvelocityvdabovethesurfaceofthe nucleus,thelocaldustdensitymustdropmoresteeplythana “1/r-law”.Weobservethisdropinoursimulationresultandbecausein thiscaseweonlytakeintoaccountparticleaccelerationcausedby gasdragandgravity,thisshowstheexpectedbehaviourasaresult of acceleration only. At the limb of the spherical nucleus (2 km fromtheorigin),theazimuthal averageisatamaximumanditis afactor of2higherthanatthe edgeofthedomain.The calcula-tionhasbeenrepeatedforseveralparticlesizes.Ascanbeseenin

Fig. 3 ,thedependenceonparticlesizeinthesphericallysymmetric caseisweak.

Unfortunately,theazimuthalaveragedoesnotallow unambigu-ous separationofall ofthephysicaleffects possibleinthe inner-mostcomabutitdoesprovidemoreconstraintsontheproblemby givingthedustoutflowaquantifiableform(theazimuthalaverage profile)andisthusextremelyusefulasatool.

2.4. Non-radialflow,sublimation,fragmentationandopticaldepth Thomas and Keller (1990) showedthatadistributionofsources (rather than the isotropic source in Fig. 1 ), each exhibiting the characteristics of free radial outflow over a finite sized nucleus, wouldleadtotheazimuthalaveragerisingasoneapproachedthe nucleus. Thepossibilityofnon-radialoutflowofdustclosetothe nucleuswasalsoconsidered.Whenviewingthecolumndensity(in theformofaradiance)thiscouldnotbedistinguishedfromthe ef-fectsofaccelerationwithoutfurtherconstraints.Similarbehaviour canariseiftheparticlessublime.After ejection,dustparticleswill absorb solar radiation andheat up. This can lead to sublimation (Lien, 1990 ) andhenceareductionintheeffectivescatteringarea willensue whichcouldresembleacceleration.The positiveaspect herehoweveristhat thereisno reasonwhytheseeffectsshould operateoverthesamescalelengthsothat,inanidealcase,these variablesmightbeseparated.

Ontheotherhand,fortheobservationsofcomet1P/Halley,the azimuthal average decreased on approaching the nucleus. There are twoprocesses thatcan lead tosuch behaviour.Firstly,ifdust particlesfragment,theeffectivescatteringareaincreasesleadingto anincreasedradiancewithdistancefromthenucleus.Onthebasis ofotherobservations,thiswasthepreferredmechanismofThomas and Keller although thereis some room fordoubt. (It should be notedthatfragmentationintolessoptically activeparticleswould providetheopposite effectandcaremustbetakentodistinguish between the two possibilities. This does however requirea sub-stantial numberofthe particlesto be closeto thewavelengthof the observation.)This wassubsequentlyinvestigated inmore de-tailbyKonno et al. (1989) .

Thesecond optionwasthatoptical deptheffectswere becom-ing increasingly significant abovethe surfaceofthe nucleus.This influencestheproportionalityconstant,k,inEq. (3 )andcouldlead totheobservedbehaviour at1P/Halleywithopticaldepthsofthe orderof0.5.Ifweknowtheopticaldepth,

τ

,atanimpact param-eter, b_far, where optical depth effects are negligible,then the in-tensityofthescatteredfluxcanbecalculatedassumingfree-radial outflowusingthestraightforwardequation,

I

(

b

)

= I0

bf ar

b e

−τ

₍

bf ar

)

bf ar/b. ₍₈₎

Given that a straightforward model for particle fragmenta-tion might also be an exponential decay profile (Thomas and Keller, 1990 ),thetwooptionscannotbeseparated.

To summarize, the influence on the azimuthal average of the effectsconsideredbyThomasandKellerisillustratedschematically inFig. 4 .

2.5. Gravitationally-boundlargeparticles

One process that was not considered by Thomas and Keller (1990) wasthepotentialeffectontheazimuthalaverageof large numbers of gravitationally influenced particles. In the case of67P,particlesofthistypehavebeenfrequentlytoutedasmajor contributors tothe dustcoma (Fulle et al., 2015 ) andtheir effect onthe azimuthalaveragehasnot yetbeenassessedalthough ob-viously their influence should tendto zeroasthey leave the Hill sphere at near constant velocity in a nucleus-centric coordinate system.

The existence of significant numbers of large slow moving particles in the vicinity of the nucleus has been established by Rotundi et al. (2015) and Fulle et al. (2015) . Evidence of particles falling back to the nucleus (following prediction by

Möhlmann (1994) andCrifo et al. (2005) ) has been provided by

Agarwal et al. (2016) andLin et al. (2016) andtheexistenceof de-positsarisingfromthisprocesshasbeenprovidedbyThomas et al.

(2015) andKeller et al. (2018) .Thenumbersinboundorbitsoutto the limit of the Hill sphere have not been determined although estimates for the production rates of large particles have been made (Ott et al., 2017 ; see also Drolshagen et al., 2017 ). As has beendemonstratedbyRichter and Keller (1995) ,thecomputation ofboundparticle orbitsinthevicinity ofthenucleusand, inthe presenceofother forcessuchasgaspressure,is acomplex prob-lem.We canhoweverusetwosimplemodelstoillustrate crudely howgravitationally-boundparticlesmightinfluencetheazimuthal average.

Inafirstapproachweassumethatthecontributiontothe scat-teringareabygravitationallyboundparticlesisdominatedby par-ticlesonballistic(returning)trajectories.Inthiscasewe can con-sider using the Chamberlain model for quasi-collisionless atmo-spheres(Chamberlain, 1963; Chamberlain and Hunten, 1987 )asis commonfor the simulation of planetaryexospheres where colli-sionsbetweenparticlesare rare.Themodeldescribesthedensity distributiondominated by returningparticles asa productof the barometricdensityandapartition functionfortheballistic parti-cles,namely,

n

(

r

)

= n0e−(λ0−λ)

ζ

bal

(

λ

)

, (9) with

λ

(r)_{≡ GMM}/(kTsr)thepotentialenergyatdistancer,whereG isthegravitationalconstant,Mthemassofthecometnucleus,M

theparticlemass,k theBoltzmannconstant,Tstheeffective tem-perature at the surface, n0 is the particle density at the surface and

λ

0=

λ

(

R

)

thepotential energyatthe comet surface.

Follow-ingChamberlain (1963) ,thepartitionfunctionforballisticparticles canbewrittenas

ζ

(

λ

)

= 2

π

1/2



γ



3 ₂ ,

λ



−



λ

02−

λ

2

1/2

λ

0 e−ψ

γ



3 2 ,

λ

−

ψ



, (10) where

ψ

=

λ

2_/

₍

_λ

₊

_λ

0

)

and

γ

(

α

,x

)

=0xyα−1e−ydy isthe

incom-plete

function.Theintegralalongthelineofsightasafunction oftheimpact parameter, b,must be solved numerically.In Fig. 5

weshow examplecalculationswiththree differentvaluesforthe gravitationalpotentialenergy

λ

0 atthesurfaceofaspherical nu-cleusof 2kmradius. Wehave multipliedthedensity integralby thedistancetothecentreofthenucleustosimulatetheazimuthal averageandonecanseeinFig. 5 thedifferenteffectonthe prod-uctresultingfromdifferentvaluesofthegravitationalsurface po-tential.Thecombinationoftheexponentialandthemultiplication by b produces a non-linear function which should be evident in theazimuthal average ifparticles inballistic orbitsdominate the comaofthe comet inthe first fewkilometres above thenucleus surface.

Fig. 4. Schematic diagram of how various near-nucleus processes can affect the azimuthal average. Top left: Deviations from point source geometry and non-radial outflow produce a decrease with increasing distance. Top middle: Sublimation reduces the scattering area but the scale length will be different to that of the non-radial outflow and dependent on particle properties. Top right: Acceleration shows a similar effect but here the scale length is a function of the drag coefficient and the production rate. Bottom left: Rapid fragmentation close to the nucleus increases the effective scattering area so that the azimuthal average rises with distance. The form of the curve is unclear but an exponential function is probably reasonable. Bottom middle: Optical depth effects close to the nucleus mask scattering particles. This reduces with distance as the dust expands, increasing the visible scattering area. Bottom right: Ballistic particles remain close to the nucleus and obviously do not conform to the free-radial outflow approximation. The exact form of the azimuthal average is different for different particle sizes but generally decreases with distance from the nucleus. If particles are in bound closed orbits, the azimuthal average can have several different forms but typically increases with distance from the nucleus out to the Hill radius. The effect of gravitationally-bound particles is discussed more closely in Section 2.5 .

Fig. 5. The product of column density and distance for the Chamberlain model for quasi-collisionless atmospheres dominated by ballistic particles. The maximum with distance of all three curves has been normalised to 1.

The conclusion from these calculations is that if gravitation-ally bound particles dominate the scattering area then the az-imuthalaverage willnot be constant withdistancebutwill have a curved form. It is conceivable that some combination of the twogravitationally-boundschemescouldproduceaconstantvalue for the azimuthal average. However, this would need these two schemes to effectively cancel each other out – something which contradictsOckham’srazor.Furthermore,itappearsthatonemight beabletoestablishanupperlimitonthescatteringareafromslow movingparticlesifthevalueoftheazimuthalaverageshows pre-dictablebehaviour.

Fig. 6. Simplified calculation of the dependence of the product of column density and distance for gravitationally bound particles within the Hill sphere of 67P. Two different assum ptions are shown. The solid line gives the product if the periapsides for the particles are all set to the nucleus radius. The dashed line gives the product if the periapsis for the particles is set to 50 km.

2.6. Observationswithintheinnermostcoma

50·103 _{km.Inthesecases,theintegrationofEq. (1 )alongtheline}

ofsight from−∞to +∞ isperfectlyadequate.For67P, however, Rosettawaswellwithintheinnercoma.Forisotropicemission,the finiteintegralcanbecalculated.Observinganimpactparameterof 10kmfromadistanceof100kmfromthenucleus,thedifference between thefinite andthe infiniteintegrals corresponds to ∼3%. Thisobserving geometrycorresponds toa littlelessthanhalfthe field of view of the OSIRIS WAC and is therefore not negligible. Onthe other hand,cometsdo notexhibit isotropic emission.Ifa cosine distributionforemission is assumedandviewedfromthe terminator (a typical geometryfor the Rosetta spacecraft aswill be shown below) then thedifference between thefinite andthe infiniteintegrals is <0.3%andshould be bornein mindthrough thefollowingdiscussion.

Wenowinvestigatetheazimuthalaveragefor67P.

3. Observationsanddatabaseconstruction

3.1. TheOSIRISdataset

The Optical, Spectroscopic, and Infrared Remote Imaging Sys-tem(OSIRIS, Keller et al., 2007 ) wasthescientific camera system onboard the Rosetta spacecraft. It consisted of two cameras, the narrow angle camera (NAC) and the wide angle camera (WAC). Bothcameraswere equippedwithaseriesofband-pass filtersto studythecometindifferentwavelengthranges.The12NACfilters coverawavelengthrangeof250–1000nmandthe14WACfilters arangeof240–720nm.

Thetwocamerashavefieldsofviewof2.20°× 2.22° (NAC)and 11.35° × 12.11° (WAC) andan angular resolution of 18.6μrad/px and 101.0 μrad/px, respectively. The images are acquired using 2048× 2048pixelbacksideilluminated CCDdetectorswith anti-blooming gates, allowing overexposure of the nucleus without leadingtosaturation artefactsinunsaturated pixels.Thisis akey property for enabling studies of the fainter cometary dust coma aroundthebrightnucleus.

The entire OSIRIS dataset of comet 67P contains just over 70,000imagestakenthroughoutthemission.Theimageswere ac-quiredin thecourse ofdifferentimagingcampaigns dedicatedto achievingthescientificobjectivesalthoughsequencescanbeused toaddressmorethanoneobjective.

The rawimagedatafromthecamera wascorrected througha calibration pipeline (Tubiana et al., 2015 ) andorganised into dif-ferentlevelsaccordingtothedegree ofcorrection.Thecalibration andcorrectionpipelineincludescorrectionsforanalogue-to-digital converter (ADC) offset and gain, bias subtraction, high and low spatialfrequencyflatfielding,badpixelandbadcolumnremoval, anexposuretimenormalisation,radiometriccalibrationanda cor-rectionforgeometricdistortion(resulting inlevel3data,inunits of[W m−2 sr−1 nm−1]).Forthe analysispresentedinthispaper, weusedimagesondatacorrectionlevel3B,whichareadditionally transformed from radiometric units [Wm−2 sr−1 nm−1] into di-mensionless reflectanceunits. Thereflectance factoris definedas

R=

π

I

(

i,_Fe,

₍

_λ

α

₎

,

λ

)

, (11) withtheobservedspectralradianceI,thesolarspectral irradiance atheliocentricdistancefromthecometF,theincidenceanglei,the emission anglee,thephase angle

α

,andthewavelength

λ

.Note that thesolarirradiancewascalculatedatthecentralwavelength ofeachfilter.

3.2. Imageselectioncriteria

Since the OSIRIS image campaigns are dedicated to different scientificgoals,notalltheimagesaresuitedtostudyingdust

fea-turesintheinnermostcoma.Toselectasuitabledatasetfor anal-ysis,severalselectioncriteriaweredefined.Theywereasfollows.

-Data level 3B: Not all the OSIRIS imageswere corrected with the full calibrationpipeline. Weexcluded all images fromour analysisthatwerenotprovidedindatalevel3B.

-Full image 2048 × 2048 pixels: Both OSIRIS cameras use a 2048 × 2048pixel CCDdetector, butnot allthe imageswere acquiredwiththemaximumpossiblefieldofview(FOV).Since ouranalysisfocusseson dustdistributioninthecoma,images withthelargestFOVaredesirabletostudychangesinthedust outflowfromthenucleusasfaroutaspossible.

-Good coverage ofthecoma: Inmany cases,OSIRISNAC images showedonlythenucleusoronlyasmallfractionofthecoma. Fortheanalysisapproachwiththeazimuthalaverage,itis crit-ical that the comais visiblearound the whole nucleusin ev-erydirection. Wedefinedthearea ofinterestfordustoutflow as theregion spanningfromthe outer limb ofthe nucleusto aboutaprojectedcometcentricdistance(animpactparameter) of7km.Imagesthatdonot containthewhole nucleusplus a circular area ofsurrounding comaincluding a majorityof the defined region of interest were rejected. Because ofthis con-straint,amajorpartoftheimagesstudiedhavecomefromthe WAC.

-Fieldofviewof_<100km:Experienceshowedthatimagestaken furtherawayandthereforewithalargefieldofviewinthe tar-get plane did not havesufficient resolution to studythe gen-eral dust behaviour in the defined region of interest. There-fore,these imageswithtoolarge a field ofview (correspond-ing toroughly 50m/px irrespective ofthecamera used)were excluded fromanalysis. Mostofthe imagesrejectedwiththis fourthcriteriondateback tothebeginningofthemissionand were acquired while the spacecraft wasfirst approaching the comet.

-Lower boundaryon coma activity:In our analysiswe saw that aninsufficientsignal-to-noiseratioleadstomeaninglessresults. Therefore, we excluded images whichhad avalue forthe az-imuthally averageddustreflectance at3kmimpact parameter of belowA(b)∼ 1.5·10−5 _{inreflectance units(corresponding to}

an azimuthal averageof 4.5cm).This limitsour analysisto a time periodfromFebruary 2015 toNovember 2015,when the cometshowedsignificantcomaactivity.

-Shuttererrors:Duringcertainperiodsofthemissionsome prob-lems withthe mechanical shutters occurred leadingto incor-rect exposure of the detector. OSIRIS images acquired witha malfunctioning shutter were not a priori excluded from the database,butcouldbeidentifiedandwereexcludedfrom anal-ysis. Some recovery of these data might be possible in some casesatalaterdate.

Fig. 7. Left: The original OSIRIS WAC image was taken pre-perihelion on the 05.08.2015 with filter combination 18 and a phase angle of 89.5 °. An over-plotted angle specification shows the allocation of the transformation angle for the polar transformation. Right: Polar coordinate transform. Impact parameter is along the x-axis and the azimuth angle along the y -axis. The nightside is defined as the region inside transformation angle 0 °−180 ° and dayside as 180 °−360 °, because the sun azimuth angle in this image is exactly 270 °.

- Gas emission contribution in OI-filter: In another step we ex-cluded imagesacquiredwiththe WACnarrowband filter com-bination17(OI-filterat630nm).Comparisonsofintensity pro-filesofimagestakenwiththesameobservationgeometrybut with different filters indicate that in F17 images a significant amountoftheintensitysignalcomesfromgasemission contri-butions.AlthoughBodewits et al. (2016) suggestedthatthereis gascontaminationintheF18 (OIcontinuumfilterat610nm), theazimuthalaverageplotsinthatfilterandintheF16(Na fil-ter at589nm)show similarbehaviour whichonlydeviatesin the orderofthe image error.Hence, theF17 imageswere ex-cludedinourdustcomaanalysistopreventbiasfromgas con-tribution.Wereturntothisissuebelow.

Thisselectionleftuswithatotalof3553imagesinthedataset foranalysis,thereof1880WACimagesand1673NACimages.

3.3.Singleimageanalysis

To generate the data included in the database, every suitable imagewaspassedthroughanautomatedimageanalysissequence. Inthisparagraph,weexplainwithanexampleimage(Fig. 7 ) how theimageanalysiswascarriedout.

- Transformationintopolarcoordinates:Inafirststep,the orig-inalOSIRISimagewastransformedintopolarcoordinates(Fig. 7 ).

- Theoriginwassettotheimagecoordinatesofthenucleus cen-treofmass,whichweredeterminedusingtheSPICElibraryand appropriatekernels.Thesameoriginwasalsousedforthe cal-culationofourimpactparameterb (b=0attheimage coordi-natesofthenucleuscentreofmass).

Inthepolartransformedimage,tworegions weredefined:the daysideandthenightside.Bothregionsaredefinedtakinginto account theprojecteddirectiontothesunintheimageplane. The daysideincludesthe regionsdefinedasthe solarazimuth angle ±90° andthenightsideincludestheregions correspond-ing to the complementary 180° in transformation angle. It is important to keep in mind that the definitions for day- and nightsideasintroducedheredonotfollowtheterminator def-initionforthe illuminateddaysideandnon-illuminated night-side exceptinthecasewhen thephaseangletakesavalue of exactly 90°.Butsince about50% of theimages inthe dataset

weretakenwithaphaseanglebetween80°−100°,the nomen-clature is reasonably accurate. This division of the full angle intotwo regions wasintended togain a better understanding ofthedust outflow.As willbe seen,thisdoesindeedprovide someadditionalinsight.

- Averageazimuthaldustreflectance(A/2

π

):Inasecondstep,dust profiles of azimuthally averaged dust reflectance values with increasing impact parameter were generated (Fig. 8 left). The averagedustreflectance value asafunction oftheimpact pa-rameterwas computed by taking the meanvalue of all pixel intensitiesalongeverycolumnofthepolartransformedimage matrix.This is equivalentto an azimuthal integration of dust reflectancealongacircularpatharoundthecometnucleusover afullangleatdifferentimpactparameters(dividedby2

π

ra-dians).Theanalysiswascarriedoutforthreedifferent integra-tionranges:thefullazimuthalangle(FA,0°−360°),thedayside (DS,180°−360°),andthe nightside(NS, 0°−180°).The limbof thenucleus islocated between2.5and 3km fromtheorigin dependingontheorientationofthenucleusintheimage. Inourexampleimage, thevalueofA/2

π

showsasmooth de-cline startingatradial distances larger than about3 km.This correspondsto thepartof theimage wherethemeasured re-flectance comes fromthe light scatteredby dust inthe coma closeto thenucleussurface. Weobservethisbehaviour forall theanalysedimages.

Naturally,theaveragedustreflectanceonthedaysideishigher thantheaveragedustreflectanceoverthefullangle.The aver-agedustreflectanceonthenightsideisconsequentlylower.

- Azimuthalaveragemultipliedbytheimpact parameter:Tostudy thenatureofthedeclineinreflectivitymoreclosely,theA/2

π

valueswere multiplied by theimpact parameter, b,to give ¯A. Theprofileoftheazimuthallyintegrateddustreflectance mul-tipliedby theimpactparameter willbe referredto asthe“ ¯A-curve” in this paper.

im-Fig. 8. Left: Average azimuthal dust reflectance with increasing impact parameter of the example image. The different colours correspond to different integration ranges: full azimuthal angle (0 °−360 °), dayside (180 °−360 °) and nightside (0 °−180 °). Right: Average azimuthal dust reflectance multiplied by the impact parameter b ( ¯A-curves).

Table 1

The slope parameters w and the parameter errors from the fit in units of %/km. And the starting point of linear behaviour of the ¯A –curve for the full angle, the dayside and the nightside.

Linear gradient w Gradient error σ’ w Starting point

Full angle (FA) 0.05%/km 0.02%/km 9.7 km Dayside (DS) −0.21%/km 0.01%/km 10.8 km Nightside (NS) 0.98%/km 0.03%/km 10.5 km

pactparameterof∼10km)representthetypical ¯A-curvebehaviour whichweobservedinouranalysis.Weidentifiedtwodifferent be-haviourregimesintheprofilecurve:thedecreaseofazimuthal av-eragewithdistanceclosetothenucleus,whichwewillrefertoas the “accelerationregion” (althoughacceleration mayonlybe par-tiallyresponsiblefortheobservedeffects),andthe“linearregime” at larger impact parameters. A constant behaviour in the linear regimeofthe ¯A-curvewouldcorrespondtoa1/b-behaviourofdust outflowintherespectiveregion,whichisexpectedforfree-radial outflow in the absence of phenomena modifying the total dust particle scattering cross-section. (This is the expected “1/r law”.) In theexample analysis, the ¯A-curve forthefull azimuthal angle shows a nearly constant behaviour,while the curve for the day-side isslightlydecreasing.Onthenightside weobservea slightly increasing slope. A quantitative analysisofthe curve gradients is giveninTable 1 .Thenoticeabledropattheendofthe ¯A-curvein

Fig. 8 (right)isan artefactfromimagetransformationandhasno physicalmeaning.

Toquantify thelinear gradientsof the ¯A-curves, we fittedthe curvewithafunctionoftheform

f

(

b; u,

v

,w,z

)

= u

bv+ wb+ z, (12)

where b is theimpact parameter, u andv are constants describ-ing thedecreasein ¯Acloseto thenucleus, andw andzare con-stants that determine the behaviour of ¯A in the linearregime of thecurve.Theslopeparameterwdeterminestheslopeofthe lin-earregimeofthe ¯A-curveandzisaconstantthatisrelatedtothe value ofAf

ρ

inunitsof[reflectanceunits· km].Notethat the ac-tual functionalformisunknown althoughwedonot ruleoutthe possibilitythattheconstantsuandvmighteventuallybeinverted in some wayto provide physical insightinto the processes close to the nucleus. To fitour data curves,we used the least-squares routine MPFIT based on the Levenberg–Marquardt algorithm im-plementedbyMarkwardt (2009) .

Fortheoretical force-freeradial dustoutflowwe wouldexpect thelinearslopeparameterwtobeexactly0fromthepointsource outwards. Inthe real casewith a finite nucleusandacceleration we still expect w to be 0 but to reach this value at a distance fromthenucleusdefinedbythegasdrag,gravityandthenucleus shape. A significantly increasing (w > 0) or decreasing (w < 0)

¯A-curve would indicate deviations from free-radial outflow be-haviour,leavingroomforotherdrivingmechanismsatworkinthe innermostcoma(e.g.fragmentationorgravitationallybound parti-cles).It shouldbe clearthatour nullhypothesis isthat including gasdrag,gravityandthesizeofthefinitesourcealoneissufficient tomodeltheobservations.

Theerrorsonthefitparameterswerecalculatedduringthe fit-tingprocesstakingintoaccount theerrorsontheazimuthal aver-age,whichwere estimatedwiththehelpofthebackground stan-darddeviationina10_{× 10}pixelframeonthecalibratedimage.A statisticalanalysisofthegeneralbehaviour ofthedustprofiles is presentedbelow.

With the help of the fit to the ¯A-curve it is possible to de-terminethestartingpoint ofthe linearbehaviour.We define the startingpoint, bstart, asthe impact parameter where the ratio of thelinearcontributiontothetotalfitfunctionvalueexceeds 99%, whichmeans the contributions fromthe power-law hasdropped below1%,i.e. bstartwhere wb+ z u bv + wb+ z = 0 .99

Forourexampleimage, the valuesforw,the errorsinw and thestartingpointsforlinearbehaviourofthe ¯A-curveforfull an-gle, daysideandnightside, respectively, are giveninTable 1 . The lineargradientisspecifiedinunitsof[%/km],whichallowsamore intuitive feelingof thesignificanceofthe slope.Forthat we nor-malisethegradientfromthefitwiththeaveragedvalueofA¯ over thelasttwokilometresintheprofileandmultiplyby100.The er-rorsprovideanassessmentofthesignificanceoftheresult.

3.4.TheOsimanadatabase

The main purpose of the Osiris Image Analysis (Osimana) databaseis to collect the data from thesingle image analysis of thewholedataset.Thefollowingparameterswereproduced.

Averageazimuthaldustreflectance:Fromtheaveragedazimuthal dustreflectanceprofileoverthefull anglefourvaluesatdifferent impactparameters(b=3km,5km,10kmandbmax)weresaved. Themaximum impactparameter bmax isdifferentforeach image anddependsonthefieldofviewandthepositionofthecometin theimageandisincludedinthedatabase.

Fitparameters: Theresults fromthe fits tothe ¯A–curves were savedintheformofthefourfitparametersandthecorresponding parametererrorsforthefull angle,thedaysideandthenightside analysis.

full angle, in (b) we see the same histogram for the analysis on the dayside and in (c) for the nightside. Positive gradients correspond to an increase in the original ¯A-curve, negative gradients to a decrease. A linear gradient of zero indicates a constant behaviour of the ¯A-curve, which directly relates to a 1/r-behaviour for the dust outflow .

Fig. 10. As Fig. 9 but for the NAC images in the database.

All inall,thedatabasecontainsa hugeamountofinformation forthestudyofthedustdynamicsintheinnermostcomaandwe expecta significantamount ofoutputfromdetailedanalysis.The majorpotential ofthedatabaseliesinstatisticalanalysisof differ-entpropertiesofthedustcomaandtheir evolutionoveralonger periodoftime.Here,weprovideaninitialoverviewofsomeofthe keyproperties.

4. Analysis

4.1.Approachto“1/r” behaviour

As illustrated in Section 2 , in the absence of effects from gravitationally-influencedparticles andeffects influencing the to-talparticlescatteringcross-section(e.g.fragmentation),weexpect theazimuthal average toapproach a “1/r law” asthe impact pa-rameterincreasesandthusthelinearslope,w,shouldbezerofor theazimuthalaverage, ¯A.

We haveplottedhistogramsshowingthedistributionofthefit parameterw,whichcorrespondstotheslopeinthelinearregime ofthe ¯A–curve. Thehistograms inFig. 9 areforthe WAC dataset whileFig. 10 showstheanalysisfortheNAC.Thebinsizeineach histogramwaschosentobethemedianvalueoftheuncertainties ofthecorrespondingslopeparameterswin[%/km],

σ

’w.

InFig. 9 a),thecompleteazimuthalaverageisshown.A compar-isonwithaGaussiandistributionshowsthatthederivedvaluesof

wcloselyfollowanormaldistributionaboutthemean.The distri-butionofslopeparameterwpeaksat0.26%/km,about1.5

σ

’wfrom theexpectedvalueofzero.Thereisasmalltailtowardsvaluesof

w> 0indicatingan occasionaltendencyfor ¯Ainsome imagesto decreaselesssteeplythan1/r.

If the angularintegration is confined to the daysideonly, the value of w is significantly lower. The distribution has its

maxi-mum ata value of _−0.26%/km. Wetherefore observea tendency forfallingcurvesonthedaysideasourexampleimageanalysisin

Section 3 alreadysuggested. Iftheangularintegrationisconfined to the nightside, w is higher. In mostof the images we observe rising ¯A-curves on the nightside andthe mean value of the dis-tributionisatabout1.5%/km.Thismeansthat thedustbrightness withimpact parameter is ingeneral decreasingless steeply than 1/ronthenightside.Theerroranalysisindicatesthatthisresultis significant.

Thewidthofthedistribution ofthefull angleanalysisisquite broadwithahalfwidthathalfmaximumbeingmorethan6

σ

’w where the sigma value is derived from the median of the error in w in the least-squares fit. Changes of thismagnitude are not negligible asthey would,if significant, correspond to changes in the effective scattering cross-section of the particles over a dis-tance of about10 km. On the other hand, more than one phys-ical process would needto be at work to produceboth negative andpositivevaluesofw.Toillustratethedifficulty,thedustwould havetobesublimingsomeofthetimeandfragmentinginto opti-callyactivedaughtersatothertimes.Thisisconceivablebutseems rather improbable. This may indicate that our erroranalysis has beenslightlyover-optimistic.

The NACresults show verysimilar trends although thevalues forthepeaksofthehistogramsare slightlyshiftedtowards lower

w compared to the WAC analysis but are identical within error. Forthefullazimuthalaverage,thepeakoftheGaussianfittothe distributionis closerthan1

σ

’w toa value ofzero(Fig. 10 a). For thedaysideanalysis,weseethatthedistributionshowsapeakat about −1%/km. The distribution forthe nightside analysisshows the sameshifttowards lower linear gradients,although its mean valueremainspositiveat0.94%/km.

Fig. 11. Distribution of the slope parameter, w , for images acquired in filter 17 (OI) showing a deviation from 1/r at the 5 sigma level.

indicatesthatgasemissionsarenotofsignificance(withinthe er-rorofthemeasurement)forthisanalysis.

There are potential explanations for the observed effects. The differencebetweenthedaysideandthenightsideslopesare possi-blytheresultoftransportofdustfromthedaysidetothenightside hemisphere, leading to decreasing slopes on the dayside and in-creasingslopesonthenightside.Thiswouldhavealargerrelative effect on the nightside giventhat the nightside is farless active thanthedayside.Weshallshow inthenext sectionthat thisisa plausibleexplanation.An alternativeisthat theshadowingofthe dust in the nightside coma by the nucleusitself has a measure-ableeffect.Thenightsidehemispheresubtends2

π

steradianwhile the shadowed solid angle is a function of the impact parameter. Ifnightsidedustemissionwereisotropicthiscouldproducean ef-fectequivalentto+1.2%/kminthebehaviourof ¯Aandthussmaller than the observed meanvalue. Additionally, thedust is certainly not isotropic and we wouldassume this to be a stringent upper limit. In conclusion, the shadowing fromthe nucleus cannot ex-plain thewhole amount ofrising nightside curves,although itis probablyresponsibleforapartoftheeffect.

It might be thought that radiation pressure effects producing reflectedparticlesmightcontributetothebrightnessinthe night-sidecoma.However,itisfairlysimpletodemonstratethatthis ef-fectsoclosetothenucleusisprobablynegligible.

The differencesinthepeaksofthedistributions betweenNAC and WAC illustrates a small inconsistency. The calibration of the two cameras seems an improbable explanation.However, itis to be noted that, forthe NAC images,the field of view is generally smaller than forthe WAC. Therefore, we are probably biased by imageswhere thedecreaseof the ¯A-curvein theacceleration re-gionstilldominatesthegeneralbehaviourandwherethereare in-sufficient data points in the linear regime to guidethe solution. Thisisalsoapparentfromtheerroronthelinearslopeparameter whichissystematicallyhigherfortheNAC.Thephaseangle distri-bution(fortheNACmostimageswereatphaseanglesbelow90°) mayalsoplayarole.

Basedonourstatisticalanalysisofthelineargradientswe con-cludethatthedustoutflowingeneralconvergestowardsa1/r be-haviour in the inner coma. Although the statistical analysis pro-vides a clear picture, the rather broad distributions leave some room for further investigation of individual images to establish whysomeimagesdeviatefromthemean.

We also note in passing that the azimuthal average inthe OI (630nm)filterdeviatedfromzeroby morethan5

σ

’w indicating

Fig. 12. The resulting Af ρvalues calculated from the constant value, z , from the fit to the azimuthal average with impact parameter, b . The variation with heliocentric distance is to be expected. A phase angle normalization to α= 90 ° has been applied to the data shown in this plot. The green data points mark Af ρvalues calculated for single images and the red data points indicate averaged Af ρvalues over one comet day. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

gascontaminationinthisfilter(Fig. 11 ).Furtherdetailed compari-sonisasubjectoffuturework.

4.2.Aroughestimateontheinfluenceofgravitationallybound particles

Giventhe major discussion on-going about the relative num-ber of large non-escaping particles in the emitted flux, we have usedthecalculationinSection 2.2 tomakeacrudeestimate. Non-escapingparticles are not force-free anddo not followa 1/r law asshowninSection 2.2 .Assuming a

λ

0of5atthesurfaceofthe

nucleusin the Chamberlain model,the brightness decrease with distancefallscorrespondingtow_<−3%/kmoverimpact parame-terscomparabletothoseusedinourdataanalysis.Thehistograms areshowingvaluesof>−1%/km.Thus,ifa

λ

0=5isrealistic,the

proportionalscatteringarea ofparticlesintheseballistic trajecto-riesmust< (1/3)or< 33%.Clearly,theuncertaintyonthe initial ejectionvelocityoftheseparticleswouldmakethisestimatehighly uncertain.Forlowervaluesof

λ

0 atthesurface, theeffectwould

be difficult ifnot impossibleto extract from theprofile cuves in ourapproach. However, dependingon thevalue of

λ

itindicates thatthetendencytowards1/rshownby ouranalysismayplacea rigorousconstraintonthenumbersofreturningparticles.Thiswill bethesubjectoffurtherwork.

4.3.Thevalueoftheconstant,z

The fittingroutine provides, as output, the constant, z, which isthevaluethat ¯Aapproachesatlargedistancesfromthenucleus assumingfreeradialoutflow.zisavaluethatcandirectlybelinked totheAf

ρ

valueintroducedbyA’Hearn et al. (1984) ,whichenables directcomparisonofcomaactivityofcomets.

InFig. 12 ,weseeAf

ρ

expressedasafunction oftime to peri-helionandthereforeheliocentricdistance.Theplotshowshowthe Af

ρ

value wasincreasingasthecomet moved towardsperihelion andbecamemoreactive.

an observer-comet distance of



.Multiplying the observed radi-ance,I,by

π

/F_allowsustoreplacethefluxwiththereflectance sothat

π

Fobs

F_ =

 ₂

_π

_Rb



2 db.

Eliminatingthe

π

,wehaveanexpressionforFobs/Fwhichcan

besubstitutedinA’Hearnetal.’sdefinitionofAf,i.e.

A f = 4



2 b2 Fobs F_ whichgives A f = 8 b2  Rbdb.

Forthe force-freeradial outflowcase, Rbisa constantandits mean value on a circular path centred on the nucleus is our ¯A, orinthiscase thevalue of ourconstant offsetparameter z. This showsthat

A f

ρ

= 8 _A¯ _{= 8 z}

The resultsshowninFig. 12 indicatethatourmaximumvalue ofAf

ρ

isabout210cmoccurring20daysafterperihelion. Thisis inremarkableagreement withAf

ρ

valuesgivenby Agarwal et al. (2007) inthepre-primemissiondustenvironmentmodelfor67P.

Snodgrass et al. (2016) measured 400 cm at a phase angle of 34° (Snodgrass,pers. comm.).Theythen extrapolatedthisto zero phaseangletoget1000cm.Wehavenormalizedthemat90° us-ingBertinietal.’sphasecurve.Oneshouldnotethat90° isalmost aminimumofthephasecurveandisveryappropriatefor normal-izationofOSIRISobservations.Hence,thereisactuallyconsistency – atleastinthetrend– withtheSnodgrassetal.resultbecauseat 90° theirAf

ρ

leadstoavaluethatmustbe<400cm.

Inourdataanalysisweseesignificantdiurnalvariationsinthe observedAf

ρ

values.InFig. 13 ,anexampleoftheobserved varia-tionsisgiven.Thedatawereacquiredbetween04–05.05.2015.The datapointsfollowaperiodiccurve withtwo maxima,one witha largeamplitudeandonewithasmalleramplitude.Thevariationis around30%peak-to-peak.We observeaperiodicityofroughly0.5 Earthdaywhichcorrespondstothespinperiodof67Pof_∼12.4h. Thediurnalvariationsare thusmostprobablyan effectofthe ir-regularshapeandaninhomogeneoussurfaceactivityofcomet67P.

4.4.Derivationofthestartingpointoflinearbehaviour

Thefittingroutinecanalsoprovidean estimateofthedistance fromthenucleusatwhich1/rbehaviourcanbe assumed.We re-ferredtothisaboveasthe“startingpoint”.Wedeterminedstarting pointsforeveryimageintheanalysis(Fig. 14 ).Becauseweobserve significant changes inthe startingpoint dueto different viewing geometries, the major contribution beinga periodic change aris-ingfromcomet rotation,wecalculated theaveragestartingpoint

Fig. 13. Diurnal variations of Af ρ value. The data points were acquired between the 04.05.2015 and the 05.05.2015 with the WAC camera and filter combination 18. The variations are most probably an effect of the complex shape and the inhomo- geneous activity of comet 67P.

Fig. 14. This diagram shows the starting points of the linear behaviour of the single image analysis in green dots. The average starting point over one comet day is plot- ted in red diamonds. The errors on the average starting points arise from statistical standard deviations of the daily changes. In orange the mean starting point over the whole time period is plotted. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

foreach cometdayasthemeanvalueofthestartingpointsofall singleimagesacquiredatthatsameday.Theerrorontheaverage startingpointismerelythestandarddeviationofthesingle start-ingpoints.Themeanvalueofthestartingpointovertheperiodof timecovered byourdatasetwasthen foundtobe11.9 _{± 2.8}km. Thus, in the general case, effects influencing deviations from 1/r closetothenucleusarebecominginsignificantatdistanceslarger thanabout6cometradii fromthe nucleuscentre.However,from theplotwecanseethatthestartingpointisprobablynotconstant withtime.Aminimumisseenclosetoperihelionnearthepeakof productionrate.

5. Modellingofthedustoutflowandcomparisons

5.1. TheDSMCapproach

isacommercializedderivativeofthePDSC++code(Su, 2013 )used inpreviouspapers(e.g.Marschall et al., 2016 ,2018 ).PDSC++isa C++based,parallelDSMCcodewhichiscapableofsimulating2D, 2D-axisymmetric,and3Dflowfields.Thecodehasbeendeveloped over thepast 15years(Wu and Lian, 2003; Wu et al., 2004; Wu and Tseng, 2005 ) andcontainsseveral importantfeatures includ-ing the implementation of 2D and3D hybridunstructured grids, a transient adaptive sub-cellmethod (TAS) fordenser flows, and a variable time-step scheme (VTS). In the parallel version, com-putational tasks are distributed using the Message Passing Inter-face (MPI) protocol. A Domain Re-Decomposition (DRD) method, to optimizetheparallel performance,has alsobeenimplemented (Liao, 2017 ).

In additionto sphericalcases, the3D shape modelofthe nu-cleus can be integrated as the input surface into the computa-tionwiththehelpofagridgenerationprogram.Here,weusethe GRIDGENprogramfromPointwisetomaptheSHAP7shapemodel (Preusker et al., 2018 ) ofthe nucleus of 67P into the simulation code.

To calculate the surface gas flux used for the boundary con-dition in our numerical simulation we impose a simple ther-mal model neglecting thermal conductivity (Gulkis et al. (2015) ;

Schloerb et al., 2015 ) butincludingsublimation ofwaterice. The resultinggasproductionrateisthusdriven onlybytheincidence heat fluxfromthe solarradiationandwethereforecall thistype ofboundaryconditionpurely-insolationdrivenoutgassing.Amore thoroughdescriptionofthesimulationsetupandanextensive pa-rameter study including comparisons of the model results with Rosetta OSIRIS andROSINA (Rosetta Orbiter Spectrometer forIon andNeutralAnalysis)dataispresentedinMarschall et al. (2016) .

Once the gasflow field hasbeen derived, the motion ofdust particles within the flow is calculated usingequations ofmotion basedondragandgravitationalforcesashasalreadybeenshown forthesphericalcaseinSection 2 .Thecomplexshapeofthecomet hasbeentakenintoaccountinthegravitationalmodel.Themodel assumesa totalnucleusmassof1_{× 10}13 _{kg(Sierks et al., 2015 ).}

For the dust particles a constant density of 440 kg/m3 _was

as-sumed.Amoredetaileddescriptionoftheusedgravitational mod-elsisalsopresentedinMarschall et al. (2016) .

The intention of this section of the paper is to demonstrate the validity ofthe use ofthe azimuthal average andto illustrate that a numerical model of the dust outflow successfully repro-duces mostof the features seen in the analysis ofthe data pre-sentedinSection 4 .Wewillnotstressand/oroptimizethe bound-ary conditions of our simulation in detail (as has been done in

Marschall et al. 2016 ).Theaimistoillustratethedeviationsfrom 1/rresulting frombreaking pointsourceorsphericallysymmetric geometryusingarealisticshapemodelandarealisticactivity dis-tribution. Webeginhoweverby returningbriefly to thespherical case.

5.2. Simplifiedcases– azimuthalaverage

Weappliedourmethodofimageanalysistosimulationresults generatedfromnumericalmodelcalculationsofgasanddustflow andanalysedprofilescalculatedastheintegralofthecolumn den-sitymultipliedbytheimpactparameteraroundthefullazimuthal angleon concentriccirclesfromthecometcentre.The latter cor-responds directly to the azimuthal average. The gas calculations havebeencarried outforan insolationdrivencasewithno ther-mal inertiaimplying peak emission fromthesub-solar point and theemissiondecreasingtozeroattheterminator.Theviewing ge-ometry inall fourcasesinFig. 15 correspondto 90° phaseangle observations.InFig. 15 (a),(c)and(d)thesphericalmodelwasused inthesimulationandonlyincase(b)theshapemodelof67Pwas used.Inthissectionweusethemodelcalculationstodemonstrate

thedependencyofthe behaviourofthe azimuthalaverage inthe accelerationregiononparticlesizeandproductionrate.Aswillbe seen,thisshowsthe challengeswe arefacing whenstudying the accelerationregioninremotesensingdata.

TheprofilesinFig. 15 (a)weregeneratedwithagasproduction rateof Qg = 2 kg s−1 andthe dust calculationson the gas flow fieldwereperformedforfivedifferentparticlesizesbetween0.01 and1000μm.We observea dependency ofthe behaviour ofthe “accelerationregion” intheprofilecurvesonthedifferentparticle sizes.The azimuthalaverage ofsimulationresultscalculatedwith largerparticlesshowsatendencytodecreaselesssteeplycloseto thecometsurface.

In Fig. 15 (b) we used the same model specifications but in-stead ofthe sphericalnucleus theshape model of67P was used astheinput boundaryofthesimulation.The profilecurvesshow thesamefundamentalbehaviourseeninFig. 15 (a).Becauseofthe complexshapeofthenucleusof67P,theazimuthalaveragecannot beevaluateddowntothesurfaceandalargepartofthe “acceler-ationregion” islostforanalysis.

Inbothcases,thedifferencesbetweentheprofilecurvesof dif-ferent particle sizes are rather small.The slightly less steep de-creaseinthe“acceleration region” forlarge particles couldbe an effectofgravitywhichistrappingparticlesinballistictrajectories close to the comet and thus causing the dust brightness to de-creaselesssteeply.However,thedifferencesaresosmallonecould probablynotdistinguish betweenthedifferentparticle sizesonly fromthebehaviouroftheazimuthalaverageindataanalysisfrom OSIRISimages.

InFig. 15 (c)we showthedependency ofthebehaviour ofthe azimuthalaverageintheaccelerationregiononchangesinthegas productionrate.Threesimulationswiththreedifferentgas produc-tionrateswereanalysedforaparticlesizeof90μm.Thehigh pro-ductionrates(20and200kgs−1)showverysimilarbehaviourand decrease more steeply in the “acceleration region” than the low productionrate(2kgs−1)case.

InFig. 15 (d)acomparisonofprofilecurvesshowsthatwe con-serve the behaviour of the dust outflow if production rates and particle sizes are scaled proportionally. This can easily be seen from the drag equation. The behaviour holds for particles large enoughsuchthat theirspeedismuchsmallerthan thegasspeed but not as large such that their dynamics is influenced signifi-cantlybygravity.Anysizedistributiondominatedbythis interme-diatesizerangewillthusalsoscaleinthementionedmanner.This also indicates that we cannot determine production rateor par-ticlesize fromthe behaviour of the azimuthal average alone be-causethere are severaldifferent combinations leading to exactly thesamebehaviour.Ontheotherhand,thisgivesusthe opportu-nitytocomparelow productionratesimulationcaseswithimage analysisfromimagesacquiredduringperiheliontransitby simply takingintoaccountthattheresultsapplytodifferentparticlesizes thatscaleproportionallytotheproductionrate.

5.3.Simplifiedcases– dayside–nightsidetransport

IntheanalysisofthelinearslopeparameterwinSection 4.1 we sawastrongindicationforlateraldusttransferinformofa signifi-cantasymmetryintheresultsfordaysideandnightside.By study-ing thisphenomenon innumerical simulations withdifferent se-tups,weshow that theirregularshapeofthe nucleusof67P has an important influence on the rateof dust transfer towards the nightside.

day-Fig. 15. The different plots show profiles of column densities multiplied with impact parameter, which is proportional to the azimuthal average, ¯A , of simulation results. The model uses a sphere with radius 2 km as an input boundary, except in plot (b) for which the shape of 67P was used. The model includes only acceleration of gas and dust and gravity from the nucleus. Some model specifications are given in the plot legends. The plots in panels (a) and (c) clearly show that the dust outflow behaviour in the acceleration region is dependent on both, the particle size and the gas production rate. A comparison with panel (b) shows that the dust outflow behaviour shows similar results for simulations based on the 67P shape model. The plot in panel (d) illustrates that different combinations of particle size and production rate lead to exactly the same behaviour in dust outflow and are thus indistinguishable.

Fig. 16. The azimuthal average for model calculations. Left: For a spherical nucleus. Right: For a 67P shape model. The two calculations have been made for identical conditions. A gas production rate of 2 kg /s has been used. 10 μm particles have been used and the azimuthal average has been computed at a phase angle of 90 °. Plots integrating over the dayside and the nightside separately are shown. The plots have all been normalised at an impact parameter of 10 km. We can see that the irregular shape of the nucleus of 67P is significantly influencing the lateral dust transport to the nightside.

sideandnightsidecontributionshavealsobeencalculatedandall plotshavebeennormalizedto1at10kmforeaseofcomparison. Theazimuthal averagedropsrapidlywithdistanceandreachesits asymptoticvalue around 8–9 km from the centre of the sphere. Thenightsidecomponenttotheazimuthal averageshowsarising valuewithdistance.Thereisnoemissionfromthenightsideofthe sphere butnon-radial flow transfers dust to the nightside hemi-sphereandcanbeseeninthecontributiontoazimuthalaverage.

InFig. 16 (right),thesamecalculationhasbeenperformedbut with the shape model of 67P. One can see that the shape has some influence on the rate at which dust is transferred to the nightsidehemisphere: The irregular shapeof 67P,which features a lot ofsurface area not orthogonal to the radial direction from

thecomet centre,permitslateral dusttransport moreeasily than inthesphericalcasewherethedustisleavingthecomet perpen-diculartothesurface.

Theazimuthalaveragecannotbecomputeddowntothesurface becauseoftheirregularshape.Hence,wemissalotofthe acceler-ationregionandsothemaximumvalue oftheazimuthal average is not as highas in the spherical case. However, the fundamen-talbehaviouroftheazimuthalaverageisidenticaltothespherical case andto the behaviour of the azimuthal average observed in dustoutflowanalysisofOSIRISimages.