Compressive strength and elastic modulus at Agilkia on comet 67P/Churyumov-Gerasimenko derived from the SESAME/CASSE touchdown signals

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Icarus

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Compressive strength and elastic modulus at Agilkia on comet 67P/Churyumov-Gerasimenko derived from the SESAME/CASSE touchdown signals

Diedrich Möhlmann

^a,1

, Klaus J. Seidensticker

^a

, Hans-Herbert Fischer

^b

, Claudia Faber

^a

, Alberto Flandes

^c

, Martin Knapmeyer

^a

, Harald Krüger

^d

, Reinhard Roll

^d

, Frank Scholten

^a

, Klaus Thiel

^e

, Walter Arnold

^f,g,∗

aDLR Institute of Planetary Research, Rutherfordstr. 2, 12489 Berlin, Germany

bDLR Microgravity User Support Center, Linder Höhe, 51147 Köln, Germany

cInstituto de Geofísica, Universidad Nacional Autónoma de México, Coyoacán 04510, Mexico City, Mexico

dMax Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany

eNuclear Chemistry Department, University of Cologne, Zülpicher Str. 45, 50674 Köln, Germany

fSaarland University, Department of Materials Science and Technology, Campus D 2.2, 66123 Saarbrücken, Germany

gI. Physikalisches Institut, Georg-August University, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

a rt i c l e i n f o

Article history:

Received 4 February 2017 Revised 22 September 2017 Accepted 28 September 2017 Available online xxx Keywords:

Comet surface material Compression strength Elasticity

a b s t r a c t

WereportananalysisoftheCometAcousticSurfaceSoundingExperiment(CASSE)accelerationsignalsat Philae’sfirsttouchdownsiteAgilkiaoncomet67P/Churyumov-Gerasimenko.Thesignalsyieldtheforces inthecontactzonefoot-soleandcometsurface,andfromtheseforcesacompressionstrengthofapprox- imately10kPacanbederived.Thesole’scontact-resonancesprovideanelasticmodulusoftheorderof 10MPa.Ourresultsarepartiallybasedoncalibrationexperiments,whicharedescribedintheappendixof thecurrentpaper.Relationsknowninmaterialscience,linkingporositytoelasticityandfractureenergy, allowonetochecktheinterdependencebetweencompressionstrengthandelasticity.

© 2017TheAuthors.PublishedbyElsevierInc.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Elasticity in a highlyporous material is primarily determined by the elasticity of its constituents andits porosity (Pabst et al., 2006). Its mechanical strength is linked to a number of proper- tiessuch asthe adhesionforcesbetweenthe constituentsorpar- ticulates of the material, its porosity, the distribution andorien- tation of cracks, inclusions, and cavities. The elasticity and the mechanicalstrengthofanymaterialandhencealsoofacometary surface material cannot be measured remotely, becauseelasticity andplasticdeformationcanonlybeprobedbyforcesandbymea- suringthe ensuingdeformationsinsitu. Thelanding ofPhilaeon comet 67P/Churyumov-GerasimenkoatAgilkiaprovides directac- cesstodeterminesurfaceelasticityandplasticdeformationbyan- alyzing thedataobtainedby accelerometersmountedonthebot-

∗ Corresponding author at: Saarland University, Department of Materials Science and Technology, Campus D 2.2, 66123 Saarbrücken, Germany.

E-mail address: w.arnold@mx.uni-saarland.de (W. Arnold).

1Deceased.

tomofthe Philaelandersoles. Weuse theaccelerometerdata to derivetheforces,that actedinthecontactzonebetweenthesole andthecomet surfaceatAgilkiacorroboratingtheanalysisofthe bouncing dynamics which allowed a first estimate of the comet surfacecompression strength ofsome kPa atthe ∼10cm to 1m scale(Bieleetal.,2015).Adetailedfiniteelementmulti-bodysim- ulationanalysiscodefromSIMPACK(Version9.5.1)aswellasana- lyticalestimatesbasedonthelanderinternaldynamicsledtosim- ilarvaluesforthecompressionstrength(RollandWitte,2016;Roll etal.,2016).

First,abriefaccountofthelandingeventsisgivenaccordingto Bieleetal.(2015).OnNovember12th,2014,thelanderPhilaewas separated from the Rosetta orbiter. The separation of Philae oc- curredwitharelativevelocityof^v=0.1876m/satabout20.5km altitude at 08:35:00 UTC. After 6:59:04 h of ballistic descent, whichwastrackedbytheCONSERT(CometNucleusSoundingEx- periment by Radiowave Transmission) instrument (Kofman etal., 2015), Philae landed at15:34:03.98±0.10s UTC (first touchdown TD1)at Agilkiaon comet 67P The time refers to the triggersig- nal at the vertical axis of the +Y-foot CASSE accelerometer. The https://doi.org/10.1016/j.icarus.2017.09.038

0019-1035/© 2017 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Please cite this article as: D. Möhlmann et al., Compressive strength and elastic modulus at Agilkia on comet 67P/Churyumov-

(b) (a)

X-foot Leg 2

+Y-foot Leg 1

x-direction y-direction

-Y-foot Leg 3

Transmitter Receiver Ice screw

Fig. 1. (a) Definition of the Philae lander coordinate system (X, Y, Z) and the accelerometer x-, y-, and z-coordinates. Leg 1 holds the + Y-foot, Leg 2 holds the + X-foot, and Leg 3 holds the -Y-foot. The coordinate system of each accelerometer is defined locally. The x-directions of all accelerometers in each of the right foot soles (seen from the outside) point downward. The y-direction is parallel to the legs of the landing gear pointing inward, whereas the z-direction is perpendicular to the x- and y- directions forming a right-handed coordinate system; (b) Detail of one double foot: The left sole houses the CASSE transmitter whereas the right sole houses a B&K tri-axial accelerometer as receiver. The “ice screw” served for fixation of the lander on the comet soil (color online).

descenttrajectorywasperpendiculartothesurfaceofthelanding site.Indeed,thereconstructionshowedthatbeforetouchdownthe anglebetweentheincomingvelocityvectorandthesurfacenormal was0.9°.However,duetoalocalslopeoftheterrain,theincoming velocity was11.5±1° away fromthe local normal.Philaearrived withavelocityvectormainlyparalleltothelanderz-axis,butleft witha lateral velocity to the South-East of magnitude 32.5cm/s.

Thetouchdownspeed relativetothecometsurfacewas1.012m/s (RollandWitte,2016).

AttouchdownthecentraldampingtubeofPhilae’slandinggear waspushedinagainstthemainlanderhead,generatingthetouch- downsignalwhichoccurredat15:34:06.471±1sUTC(Bieleetal., 2015)withafewsecondsdelaytothefirstCASSEsignal.Thestroke length of the damper was 42.6±0.1mm out of a full length of

∼170mm.ThelandinggeardamperdidnotmoveafterleavingAg- ilkia,i.e.thecomponentofforceinlander’szdirectionduringfur- thertouchdownsmust havebeen <30N,due to staticfriction in the mechanism. Because the anchor harpoons did not fire upon touchdownandthe hold-downthrust of thecold gassystem did notwork,thelanderbouncedseveraltimesuntilitcametorestaf- terabout2hofballisticflight.Furtherdetailsofthelandingevents atAgilkiacanbefoundelsewhere(RollandWitte,2016).

Thispaperisorganizedinthefollowingway:First,ashortin- troductiontocontactmechanicsisgivenfollowedbythedescrip- tion ofthe CASSE signals.We then analyzethe signals by treat- ing thefoot soles ascontact oscillators.The surfacecompression strengthandelasticmodulusderivediscomparedtodataobtained fromscalinglawsknowninmaterials scienceforveryporousma- terials. Eventually the data are compared to strength results ob- tainedfromother experiments oftheRosetta mission andtothe dataobtainedbytheKOSI(“Kometensimulation”)experiments.

2. Cometacousticsurfacesoundingexperiment(CASSE)

A double foot is fixed to each of the landing gear legs. One soleofthefootcontainsanaccelerometer andthe othera piezo- electrictransmitter.Viewedfromtheoutside,thetransmitterisin the left sole, whereas the accelerometer is in the rightsole (for thedefinition of Philae’saxes, see Fig. 1). The original operation principleofCASSE wasto insonifythe cometatfrequencies from 0.5kHzto1kHzbythesolescontainingthetransmitterandtore- ceivethesignalsbythesolesinwhichBruel&Kjaer(B&K)triaxial accelerometersweremounted.Detailsofthisset-uparedescribed elsewhere(Seidenstickeretal.,2007).

Additionally,CASSEwasalsousedtomeasuretheaccelerations ofthe landing shockand thesole’s contact-resonanceswhen the feet touchedground the first time on comet 67P at Agilkia. The strength of the landing shock or the force acting between Phi- lae’s landing gear and the comet soil is determined on the one handby thecomplianceofthe materialencountered attheland- ing location and, on the other hand, by the compliance of that part of the landing gear which touches the surface. Besides the absolute value of the force of the landing shock, the forced res- onances of the soles of the feet can be used to obtain informa- tionon thelocalelasticmodulus Eandthe compressionstrength

σ

^{cofthecometsoil.Theseprinciplesandmodeofoperationhave}

beendiscussedinanearlierpaperdescribingtheevaluationofthe landing testsatthe Landing&MobilityTestFacility (LAMATests) (Faberetal.,2015).

3. Philae’slandingandcontactmechanics

Theelastic,anelastic,andplasticinteractionsbetweentheland- ingsolesandthecometbody aredeterminedbythecontactstiff- nessk^∗(inverseofthecompliance):

k^∗=2Er∗

(

^Ac/

π )

^{0.5, (1)}

whereEr∗is theeffective storagemodulus ofthecontacting ma- terials, here the comet soil and that part of the landing gear whichtouches thesurface, andA_c is thecontactarea. The quan- tity Er∗ contains the elastic moduli of both contacting partners (Johnson,1985),herethecometsoilEc andthefootsolesE_sh,and theirPoissonratios

υ

cand

υ

sh:

1/E^∗_r=1/Er,c+1/E_r,sh=

1−

υ

c²

/Ec+

1−

υ

sh2

/E_sh, (2)

where E_r,c andE_r,sh are the reduced elastic moduli of the comet soil and the foot sole, respectively. Eq. (2) is valid for homoge- neous,isotropicandvolumetricmaterials. Thefootsole,however, isathinshellstructure.We willaccommodateEq.(2)tothisfact later on when required. The parameter Er∗ reflects the fact that displacementsoccurinbothcontactingmaterials.Eq.(1)isavery generalrelationthatappliestoanyaxisymmetricindenter.Itisnot limitedtoa specificsimple geometry.Althoughoriginallyderived forelasticcontactsonly,ithassubsequentlybeenshownto apply equally well to elastic–plastic contacts. Small perturbations from an axisymmetric geometryinthe contactingmaterials do not ef- fect k^∗ either.It isalsounaffected by pile-upofthematerial and

sink-inoftheindenter(OliverandPharr,2004).Incaseofasphere againstaflatsurface,thecontactstiffnessisgivenby

k^∗=2acE^∗_r, (3)

where ac is the contact radius. In case of viscous damping or plastic deformation, k^∗ becomes a complex quantity k^∗=kr+ik_i (Yuyaetal., 2008) withk_i=2E_i^∗(Ac/

π

^{)0.5 where E}i∗ isthe corre- spondinglossmodulusandkr=2Er∗(Ac/

π

^{)0.5istherealpartofthe}

contactstiffness(Eq.(1)).Therelationofcontactstiffnesstomate- rial propertieshasbeenexploitedinparticularinnanoscalemea- surements,i.e.forcontactradii lessthan100nm both innanoin- dentation experiments (Landman et al., 1990; Oliver and Pharr, 2004) andinatomicforcemicroscopy (Arnold,2012).Inourcase, asitwillbecomeclear,thecontactradiusaccanbeaslargeasthe geometricalradiusofthesole,i.e.r=5cm.

Inordertounderstandtheoccurrenceofcontactresonancesin Philae’s sole, let us consider at the basic behavior of a contact- oscillator. The angular contact-resonance frequency

ω

^{ct of two}

freely supported bodies of massesm₁ andm₂ can be written as (Johnson,1985):

ω

ct² =k^∗×m₁+m₂

m1m2 . (4)

The masses m_1,2 are effective masses anddepend on the ge- ometryoftheoscillatorandontheexcitationfrequency.Inpartic- ular Philae’ssoles are mass-distributedoscillators. Therefore,dis- persion oftheresonancefrequencyversus contactstiffnessorre- ducedelasticmodulusofthecontactingpartnersistobeexpected.

Ifthereisfrictioninthecontactvolumeorinanyotherpartofthe contact oscillator, k^∗=kr+ik_i becomes complex. This entails the complexreducedmodulusmentionedabove.

The basic physics of contact vibrations, in particular the behavior at large amplitudes or static loads are discussed in Nayak (1972). Further engineering applications of the contact- resonancetechniquearediscussedinLange(1994).Inatomicforce acousticmicroscopythecantilevercontact-resonancesledtoanew modeofoperation(Arnold,2012;Marinelloetal.,2013).

During the development andqualification of CASSE, which is part of the Surface Electric Sounding and Acoustic Monitoring Experiment (SESAME), eigenmodes and eigenfrequencies of the soles were characterized. A detaileddiscussion ofthe results ob- tainedcanbefoundelsewhere(Arnoldetal.,2004;Schieke,2004).

A flight spare sole with mounted accelerometer showed reso- nancefrequencies ofapprox. 600Hzand650Hzatroom temper- ature(Krause,2007).Thedoubleresonancemaybecausedbythe mounting of the accelerometer relative to the orientation of the glass-fiberfabricofthecompositematerial(GFC=glass–fibercom- posite)thesoleismadeof,liftingthedegeneracyofthefirsteigen- mode of the sole due to the elastic anisotropy of the GFC (the mass-distribution of the accelerometer is not axially symmetric).

This is typical forgeometrical oscillators that are not completely symmetric, and similar phenomena were observed for the sole’s highereigenmodes (Arnoldetal., 2004; Schieke,2004). Thedou- bleresonanceswerealsoobservedwithadifferentflightsparesole usedforcalibrationpurposes,asreportedintheAppendixA5.As said above, the possibility to exploit the contact-resonances was evaluated by landing testsattheLanding &MobilityTest Facility (LAMATests)(Faberetal.,2015).Itisthecompressionstrength

σ

^c

andthe reducedmodulus Er,c ofthe comet soil,that we wantto extract fromthelandingshockandthecontact-resonancesofthe Philae’slandergear’ssoles,respectively.

4. CASSEsignalsrecordedatAgilkia

We discussfirstthesignalsrecordedby theCASSE accelerom- etersatAgilkia.The B&Kaccelerometerswere mountedsuch that

thex-directionpointsperpendiculartothesolesurfaceandhence intothe comet soil, the y-direction parallel tothe corresponding legintothelanderbody,andthez-directionperpendicular tothe x-andy-direction,forminga right-handedcoordinatesystem, see Fig.1.ThedesignationsofthefeetcanbeseeninFig.1aswell.

The accelerometer signals in x-direction of the three feet and their Fourier transforms are shown in Figs. 2–4. The ori- gin of the time axes correspond to 15.34:03.78±0.1s UTC, see Biele et al. (2015). From the appearance and the amplitude of the signals (Figs. 2a–4a) it becomes clear that the +Y-foot or leg 1 encountered the comet surface in a different way from that of the+X-foot and the –Y-foot. In order to obtain a first overview, we can see that the +Y-foot underwent negative (up- wardmovement)andpositive amplitudes(downward movement) ofupto±140m/s²(whenthedigitizersaturated)whereasthe+X- footandthe–Y-foot experiencedvalues ofmaximalaccelerations in x-direction up to±40m/s². Also the Fourierspectra are quite distinct.Additionally,allthreefeetshowlow-frequencysignalsbe- low 300Hz, there are frequency groups above noise at 0.67kHz andat1.3kHzfor the+Y foot(Fig.2b), whereas forthe+X-foot (Fig.3b)andthe−Y-foot(Fig.4b)therearetwogroupsoffrequen- ciesabovethenoiseat1–1.2kHz.Thefactthatbothsignsoccurin thesesignalsmeansthatthesolesofthePhilaefeet,whichhouse theaccelerometer, underwentoscillations when encountering the cometsurface.Theseoscillationsareanalyzedinthefollowingsec- tions.TheCASSEsignalsinallthreefeetinthey-andz-direction arediscussedelsewhere(Seidenstickeretal.,inpreparation).

5. FeaturesoftheCASSEsignalsrecordedatAgilkia

Before we discuss the details of the various signals, let us discuss how Philae encountered the comet surface (Biele et al., 2015). In Section 1, a first description ofthe landing events has been given. In addition, OSIRIS Narrow Angle Camera (NAC) im- agestakenfromtheRosetta orbiterwere usedtogether withRO- LIS (Rosetta Lander Imaging System) descent images in order to determine theactual landing coordinates andattitudeat Philae’s first landing site, Agilkia (Fig. 5). Altitude, attitude with respect to the surface, and the rotational state were derived indepen- dently from thisanalysis. The motion of the lander close to the surfaceof67P comprises, besidesthe verticaldescent witha ve- locityof v ≈ 1m/s, also a counter-clockwise rotation around the mainaxisofthelanderintopview withanangularfrequencyof

ω

≈ 1.26×10⁻² s⁻¹ (0.72°/s) (Biele etal., 2015). Because of the strongsignalsrecordedbythesole’saccelerometermountedinthe +Y-foot,itistemptingtopostulatethat the+Y-leghitthelargest boulderwith its foot soles or with a strut of the leg as seen in Fig.5.Possibleencounters aredepictedschematically inFig.6.In view oftheuncertainty inspatial(±10cm) andangularpositions (±1°) ofthePhilaelegsrelative tothesurface, let usbeginwith thehypothesis that theboulder washit severaltimes by various partsofthelandinggear’sleg1withtheattached+Y-footwithits sole.Wewilldiscusswhetherthishypothesisisinagreementwith observations.

Philae’s first contact with the nucleus of 67P at Agilkia was a short “touch” of the +Y-foot, see signals at t=160 – 230ms (Fig. 2a). The signal strengths of approx.±10m/s² are compara- tivelyweakinviewofthelatervaluesatt>450ms.Theforcesact- ingonthe solesinitiatedmomentainfluencingthefurthermove- mentsof Philaebytilt rotationandnutation ina rathercomplex way (Roll and Witte, 2016). A detailed account of this signal as well asthe time-sequence ofthetouchdown signals ofthe three feet of Philae will be discussed elsewhere (Seidensticker et al., inpreparation).

Att=437ms,withadelayofmorethan260msfromthisfirst signal,strongsignalssetinforthe+Y-foot (Fig.2a). Theonsetof

Fig. 2. (a) Acceleration data and (b) its frequency spectrum in x-direction for the + Y-foot.

Fig. 3. (a) Acceleration data and (b) its frequency spectrum in x-direction for the -Y-foot.

Fig. 4. (a) Acceleration data and (b) its frequency spectrum in x-direction for the + X-foot.

thesesignals isseenin they- andz-directionsoftheaccelerom- eterswithin 1ms, i.e.almostsimultaneously. Almost atthe same time,signalsinthe+X-footsetin(Fig.4a),whereasthemainsig- nalsinthe–Y-footarefurtherdelayedbyabout200ms(Fig.3a).

AscanbeseenfromFigs.2ato4a,thesignalsappearindistinct groups. We firstanalyze the variousparts of thesignals intime, amplitude, andspectral components.Although the Fouriertrans- formsofthesesignals,Figs.2b–4b,seemtoindicatethatthereare continuousbandsoffrequencies,thisisnotthecase.Ifoneexam- inesthetimeseriespiecewisearound thesignal groupsbycount- ingtheoscillationsmaximaandbyshorttimeFouriertransforma- tion(STFT),muchnarrowerfrequencypeaksbecomevisiblewhich arelistedinTables 1–3.Inthesetables,themaximal acceleration valuesinthetimewindowsaregiven,aswellasthedominantfre- quencypeaks.

The spectral components contain dominantly frequencies at 1.3kHz, 1.1kHz, 670Hz, 400Hz and 300Hz and an almost con- tinuous spectrum of signals below 300Hz. These frequencies have bandwidths of the order of 50–100Hz. If this bandwidth is ascribed to a damping mechanism, the Q-value would be Q=

ω

^/

ω

≈ 20−10 which indeed corresponds roughly to the

value measured witha sparesole(see Appendix A5). One hasto becareful, however;itmightalsobe dueto adistributionofres- onance frequenciescausedbyvarying elasticboundaryconditions whenthesoletouchesthesurface,whichentailsthattherearein- homogeneouslybroadenedresonance-curves.

We propose that thesegroupsreflect individual contactswith the comet surface due to the motion of the lander, the induced oscillations ofthe landing leg, andthelocal terrain, inparticular the multi-asperity contactsof the lander soles withpebbles of a size distribution down to cm (Mottola et al., 2015). Some of the signals seen in the +X-foot and the –Y-foot may stem from the oscillatory excitation ofthe whole lander structureby the strong signalfromthe+Y-foot.

The total stiffness kt of the spring arrangement, represent- ing one leg ofthe landing gear, withouttaking into account the Coulombfrictiondampingintheicescrews(seeFigs.1band8),is givenby

1 kt=1

k^∗+ 1 k_sh+ 1

kLG+ 1

i

ω γ

^{D (5)}

Here,k^∗isthecontactstiffnessofthe sole-cometsurfacecon- tact,k_LG=13.3kN/misthestiffnessofonelegofthelandinggear,

Table 1

Features of the signals of the + Y-foot; x-direction: Amplitudes, main frequency components deter- mined by the time differences in signal maxima and/or by short time Fourier transforms (STFT).

Time interval (s) Amplitude (m/s ²) Frequencies (kHz) Frequencies STFT (kHz)

0.15–0.25 + 18/ −12 0.38/1.2 0.35/1.2/

0.43–0.47 ±32 1.1 0.7/1.1/1.2

0.47–0.5 + 120/ −81 0.68 0.68/1.3

0.5–0.6 + 140/ −140; Clipped signals 0.67/1.05/1.3 0.67/1.05/1.3

0.6–0.9 + 25/ −33 0.43/1.1 0.35/0.75/1.1

0.9–1.6 + 7/ −15 0.335/1.1 0.34/1.1

Table 2

Features of the signals of the −Y-foot; x-direction: Amplitudes, main frequency compo- nents determined by the time differences in signal maxima and by short time Fourier transforms (STFT).

Time interval (s) Amplitude (m/s ²) Frequencies (kHz) Frequencies STFT (kHz)

0.3–0.6 5/ −5 0.38/1.03 0.34/1.02

0.6–0.8 23/ −32 0.35/1.05 0.35/1.01

0.8–1.0 40/ −30 0.47/0.98 0.42/0.96

1.0–1.6 8/ −7 0.41/1.1 0.43/1.1

Table 3

Features of the signals of the + X-foot; x-direction: amplitudes, main frequency compo- nents determined by the time differences in signal maxima and by short time Fourier transforms (STFT).

Time interval (s) Amplitude (m/s ²) Frequencies (kHz) Frequencies STFT (kHz)

0.3–0.6 20/ −12 0.38/1.1 0.38/1.16

0.6–0.75 18/ −18 1.1 1.1/1.2

0.75–1.0 11/ −15 1.1 1.1/1.2

1.0–1.6 8/ −10 1.05 0.68/1.05

Fig. 5. Extrapolated lander position and orientation during Philae’s first contact with the comet at the site Agilkia. The lander legs are to scale and superimposed on merged and rectified ROLIS images. North is up. The thin yellow line indicates the lander balcony edge. The landing gear was rotated during decent by 11 °. The + Y leg hits the edge of the boulder seen in the center of the image. The resolution of the ROLIS image is 2 cm/pxl (full-resolution) and the positional uncertainty of Phi- lae relative to the surface is ±10 cm. This figure is an updated version of the one shown in Biele et al. (2015) (color online).

andk_sh=270kN/mis the stiffnessofthe sole (see Appendix A3, Fig. A3.2). Due to the damper in the landing gear the stiffness of the landing gear becomes complex with the imaginary part k_i=i

ωγ

D (

γ

D=567N/m). It is easy to see from Eq.(5) that the deformation of thelander structure on a givenmaterial isdeter-

mined by that partof thelanding gear withthe highestcompli- anceorloweststiffness.

6. Evaluationofthedatameasuredoncomet67P 6.1. Signaltypes

Thereareanumberofindividualsignalsinthetimeseriesfrom comet 67P which look very similar to the touchdown signals in theLAMAtests(Faberetal.,2015).TwoofthemmeasuredatAg- ilkiaareshowninFigs.7aandb.Theyarecharacterizedbyatime- lengthcalledcontacttimeTc inHertziancontactmechanics,here T_c=7ms(Fig.7a)andT_c=8ms(Fig.7b).Othersignalsoccurinthe +Y-footinx-directionat0.735swithTc=22msandat0.98swith Tc=8ms both at a contactresonance of 1.2kHz. Further signals canbeseeninthe–Y-foottimeseriesinx-directionat0.65swith Tc=10ms andfc=1kHz,at0.823s withTc=8ms andfc=1kHz;

inthe+X-foot,x-directionat0.429s,Tc=7msandfc=1.1kHzand at0.89swithTc=5ms andfc=1.2kHz. Theircommonsignatures arecontacttimesT_c ≈ 10ms,negativewave-formcloseto ahalf- sinusoidalshape and amplitude-modulated witha frequencysig- nalcloseto1.1kHz, seeTables1–3.We interpretthemodulation signalsascontact-resonancesofthesoleinanalogytothesignals measuredintheLAMAtests.Inthefollowingsections,wediscuss thehalf-wavesignalsaswellasthecontact-resonances.

Letusmakesome estimatesbeforeweanalyzethedata.From thetime-of-flightdataoftheMUPUShammeringsignal,theelastic surfacemodulusatthefinallandingsiteAbydosmustbeoftheor- derof10MPa(Knapmeyeretal.,2015).Letusassumethatthereis thesamesurfacemodulusatAgilkia.Themagnitudeoftheforces exertedby thefootsolesonthecomet soilwere P≈34NatAg-

Fig. 6. Possible scenarios of the touchdown for the + Y-foot at the landing site Agilkia. The scenario (c) agrees with most of our finding; see text.

ilkia(Rolletal.,2016).Beforeplasticdeformationofthecometsur- facesetsin,therewasanelasticdeformationandthisallowsusto estimatethecontactradiusa_c,

ac=

(

^3PR/4E∗

)

^{1/3, (6)}

based on Hertzian contact mechanics (Johnson, 1985) to be ac ≈ 8mm using E^∗=10MPa as effective modulus between the soleandthecometsurfacematerial.Theelasticdeformationdepth is

δ

=a_c²/R ≈ 0.3mm. Then we obtain a contact stiffness of k^∗=2acE^∗=160kN/m,which is much larger than thestiffness of onearm ofthelanding gear(k_LG=13.3kN/m),butcomparableto thestiffnessofthesolek_sh=270kN/m.Forasignallengthofabout Tc=0.1s (lengthof thestrongsignal of the+Y-foot), the angular frequencywould be

ω

=2

π

^/Tc=2

π

^{/0.1 s−1}=62.8 s⁻¹ andhence

ωγ

D ≈ 35.6kN/m.In addition, for the first contacts the damper ofthelanding gearwasnot activated(Roll andWitte,2016), and thereforeweneglectthecontributionoftheimaginaryparttothe complianceofthelandinggear.

If the corresponding foot was fixed rigidly to the leg of the landinggear,therestoringforcewouldbedeterminedbyitscom- plianceand henceno informationcould be gained onthe elastic modulusofthesurfacematerial,seeFaberetal.(2015).However, iftheicescrewmechanisminthefootisoperating,thefootisde- coupledduringthistime fromthe landinggear legandthe stiff- ness of thesole determines the reaction to thecontact forcesat Agilkia. The ice screws (Fig. 1b) were actuated at least partially uponthefirstlandingofPhilae.Inthelaboratorytests,thisperiod lastedsome100mswithaforceof≈20N(Faberetal.,2015).

Thesignalsofthe+Y-footarethestrongest,exceedingthoseof theother twofeetbyafactorof≈5.Alsoonlyinthe+Y-footac- celerometer,therearesignalsrecordedat1.3kHzinthex-direction which are clearly above the noise level. As said in Section 5, the positional accuracy of the reconstruction of the landing site is±10cm.Thereforeitisquitefeasiblethatastrutofleg1ofthe landinggearhittheboulderatitsedgeandnotthesole(scenario (c)inFig.6).Then,thedanglingfootcausedthefreeoscillationsof thesoleat670HzandthesignalsareamplifiedbytheQ-factorof the resonance,whichwasmeasured on thesparefoot soleto be Q=9.Due tothe nonlinearbehavior ofthe stiffnessofthespare soleversus loading force(see Appendix A3,Fig.A3.2),the gener- ation ofthe second harmonic,i.e.thesignal frequency at1.3kHz isquite natural. Ina numberoftest experiments,itwaschecked whethersuchascenarioispossibleandthisseemstobethecase, seeApendixA4,Fig.A4,andthevideointhesupplementarymate- rial.Inseveralofthesetests,itwaseasytogeneratesignalampli- tudes comparableto andlarger thanthesignal strengthobserved oncomet67Pwhenthestruthitastyrofoamplate(E_r,cal≈20MPa) atitsedge.

6.2. Half-wavesignals

Let usconsider the half-wave sinusoidal signals as a periodic input disturbance ofthe oscillator shownschematically in Fig.8, which depictsthe mechanical equivalentmodel forthe Coulomb friction force inthe fixation icescrew mechanism, inaddition to theinertiaforcebythemassofthefoot.Thedryfrictionelement operates along the guidance systemof rectangular tubes holding thefootsoles.Itactsaswellontheicescrewfixture,whichforces thescrewrotation whenthesolesare pushedupward.Hence the dryfrictionattacksalongadistributedmassandthereforetheme- chanicalequivalentmodelinFig.8isasimplification.There area numberofpaperspublishedinthe1960sand1970sdiscussingos- cillatorswithviscousandCoulombfrictionelementswhoseresults are now summarized in textbooks; see for example Nayfeh and Mook(1995).ThepaperbyLevitan(1960)isparticularlysuitedto describethemainfeaturesofthepresentproblem.Treatingtheos-

Fig. 7. (a) Signal at 437 ms of the + Y-foot, x-direction. The overall shape of the signal is sinusoidal with one negative half-wave of length T c= 7 ms. Counting the number of oscillations in the signals, a contact-resonance at ∼1.1 kHz ±0.1 kHz becomes noticeable; (b) Signal at 980 ms of the + Y-foot, x-direction. The overall shape of the signal is sinusoidal with one negative half-wave of length T c= 8 ms. Counting the number of oscillations in the signals, a contact-resonance becomes noticeable at 1.2 ±0.1 kHz.

mL

ɣD

kLG

Coulomb friction F0 in the screw mechanism

mfoot

ksh

k^* y

x z=y-x

Fig. 8. Mechanical equivalent model for the Coulomb friction force in the fixation ice screw mechanism, in addition to the inertia force by the mass of a foot. The dash-pot element represents the viscous damping of the sole and the comet soil, here 7.1 Ns/m. Furthermore, m Lis the mass of the lander, γDis the damping con- stant of the landing gear ( Roll et al., 2016 ), k LGis the stiffness of the landing gear, k shis the stiffness of the sole, and k ^∗is the contact stiffness of the comet surface.

The damping element parallel to k shand k ^∗is the damping of the sole determined by the Q-value of the contact-resonances; see text.

cillatorofFig.8asa point-masssystem,thecorrespondingdiffer- ential equationforthemotionofthemassm_foot withthecoordi- natez relative(z=y– x) tothebaseplatewiththecoordinatex (representingthecometsurface)is:

m_foot¨z+cz˙+kz−F

(

^t

)

^=−mfoot¨x=m_foot

ω

^2cos

( ω

^t+

φ )

⁽⁷⁾

for a sinusoidal excitation x=x₀cos(

ω

^t+

ϕ

^{). Here F}(^t)= (^4F0/

π

)n(^sin(ⁿ

ω

^t)/n) ^{is the Fourier series} representation ofthesquarewavefrictionalresistancecurveF₀×sign(z˙)andF₀= 2-20Nisthemagnitudeofthefrictionalforce(measuredinspare footassembliesinthecourseoftheLAMAtests(Faberetal.,2015).

Eq.(7)hasbeensolvedanalyticallyforasteadystatesolution.The series z(^t)=

n

{

^Ancos(ⁿ

ω

^t)+B_nsin(ⁿ

ω

^t)

}

^{(Levitan, 1960) was}

truncated at n=5 resulting in an error of less than 1% for the amplitudez.Thefollowingparametersdeterminethesolution: (i) thefrictiondampingfactorC_f=F₀/kx₀ (with1/k=1/k^∗+1/ks),(ii) viscousdampingratio

α

=c/c₀ (withc₀=2m_foot

ω

0),(iii)ratio

β

^of

the excitation frequency/eigenfrequency, here (

π

^/Tc/

ω

0), and (iv) x₀=forcedexcitationamplitudeofthebaseplatex₀×cos(

ω

^t).The

parametersare

α

=9.26×10⁻²≈ 0.1;

β

=7.48×10⁻².TheratioC_f is difficultto estimate. Because theaccelerometer in each soleis firmly attachedto thesoles, wecan equatethe soles’amplitudes x₀=b/

ω

^{2 withthe excitation amplitude x}0. Here, a typical value isb≈20m/s²,seeTables1–3,and

ω

^/2

π

^isthecontact-resonance frequency ≈ 1.1–1.2kHz. This yields x₀ ≈ 0.4μm. The stiffness of the springs k^∗ and k_s in series is then k ≈ 150kN/m (see Section 6.3.2). Finally with F₀ ≈ 20N, one obtains C_f ≈ 330 as an upper value and with F₀ ≈ 2N C_f ≈ 33 as a lower value.

ReadingtheparameterizeddatashowninFig.4ofLevitan(1960), we conclude that the peak transmissibility,i.e.the ratioof y₀/x₀ is close to 1. The oscillating forces acting on the sole are not influenced by theicescrew mechanismbecause they aresmaller than the friction forces,and hence the soles can be regarded as an isolated system as long as the screws are not stopped. This finding allows us to apply the theory of Faber et al. (2015), to invertthedatawherethecomplianceofthefootsolecompensates the restoringforce when contactingthe cometinstead ofthe leg ofthe landing gear.Inthat caseF(x)istherestoringforce inthe contactsole-cometsurface,i.e.F(x)=(4/3)R^1/2E_r^∗x^3/2foranelastic Hertzian contact. For a plastic contact, this force is given in a first approximation by F(x)=A×

σ

^{c where A is the contact area}

and

σ

c is the compression strength. In view of the permanent indentations Philae left behind at Agilkia, it is appropriate to assume that the elastic-plastic yield limit wasexceeded by each

foot and that the further deformation occurred then only by plasticdeformation,i.e.thatthesurfaceofthecometmaterialwas loaded so that the individual signals are caused by the crushing ofthe porousstructure similar to theresults ofthe KOSIexperi- ments(Grünetal.,1993;Kochanetal., 1998;Kochanetal., 1989;

Thomas et al., 1994). Here, the simplified assumption is made that the compression strength is a constant. Applying Eq. (3) in Faberetal.(2015)yields:

|

^bmax

|

^=K0

1 m_foot −

ω

²

k_sh

2

+

ω

²

γ

sh²

=K0

1 m²_foot

1−

ω

²

ω

²0

2

+

ω

²

γ

sh²

= K₀ m_foot

1−

ω

²

ω

²0

2

+

ω

^2Q2

ω

²0

. (8)

In laboratory tests the Q-value of a spare sole’s free oscilla- tionswasmeasuredtobeQ=9,anditremainedunchangedwhen the sole was firmly immersed in Aerosil (appendix, Table A5) as an analog material for the loose comet soil. In Eq. (8), the angular frequency

ω

^is

ω

=

π

^/Tc where T_c is the contact time,

ω

0=2

π

×670 s⁻¹=4.2×10³ s⁻¹ isthe free resonancefrequency ofthesole,andm_foot=0.82kgisthemassofafoot.Furthermore, thedamping constant of thesole is

γ

sh=k_sh/Q

ω

0=7.1 Ns/m. As saidabove,thecontacttimesareoftheorderofTc ≈10msyield- ing

ω

=

π

^/Tc=3.1×10² s⁻¹. Inserting these parameters and be- cause

ω

<<

ω

0,Eq.(8)leadsto:

K0≈

|

^bmax

|

^×mfoot

1.2 =A×

σ

^{c. (9)}

The acceleration values bmax are between −10 and −34m/s². Thisyields magnitudes ofimpact forces K₀ between7 and23N.

Thisforce correspondstothemeasurementoftheforcesobtained in a sparefoot, see above, andalso to the estimate ofthe force exertedby theeffectivemass ofalanding gearlegattouchdown (Rolletal.,2016).

Forpredominantlyplasticdeformation,themaximalcontactra- diusisgivenbytheradiusofthesoleprovidedthatthewholesole surfacecontactsthe cometsurface. Thisyieldsa maximalcontact area of A ≈ 0.016 m² if the two soles of a foot touch down at the sametime. Hence thecompression strength

σ

^c=K₀/A isbe- tween 0.43 and1.5kPa and will be correspondingly larger if the contact area is smaller given by the particles of the regolith at Agilkia with diameters down to cm size (Mottola et al., 2015).

Thisyieldsa contactarea ofA=2×10⁻³ m² andhencecompres- sion(orcrushing)strengthsbetween3.5kPaand12kPa.Theseval- ues are indeed comparable to the estimates of

σ

c=2kPa given in Roll et al. (2016), where the magnitudes of the forces were estimated that the lander soles exerted onto the comet surface basedonadetailedmechanicalmodelofPhilae.Here,wederived theforcesfromtheacceleration dataatthePhilaefeetmeasured insitu.

Insuch a scenario theCASSE signalsare caused bythe forces exerted on the sole when crushing and pushing the assembly of the individual particles of the comet regolith further into the cometsurface. Oneshouldkeepinmind thatEqs.(7)and(8)are basedonasimplifiedmodelanditisre-assuringthatallthreeap- proachestodeterminethecompressionorcrushingstrengthatthe Agilkiasiteyieldthesameresultwithinafactorofthree.

6.3.Philae’ssoleresonances

Beforeweanalyzethecontact-resonancedatafromcomet67P, we discuss the free and the contact-resonances of the sole of a sparefootwhichwere determinedbycalibrationexperimentsand aredescribedindetailinAppendixA5.

6.3.1. Freeresonances

The freeresonanceofasparefootsolewasmeasuredatroom temperatureby tapping onthelid ofthe sole,recordingtheout- putof the accelerometer either with a digital oscilloscopeor by theB&K “Pulse” device withcorresponding software which pow- ered the sensor and which recorded the signals as well, see AppendixA5.Thefreeresonanceaveraged over10measurements was540±14Hz, a value obtainedrepeatedly in many additional individualmeasurements.

FromEq.(A5)(AppendixA5),itcanbeseenthattheresonance frequencyisproportionaltothesquarerootoftheYoungmodulus ofthesole,i.e.

ω

∝E_sh^0.5.TheYoungmodulusofthepolymercom- ponentofthecompositesmayincreaseconsiderablywithdecreas- ingtemperature, dependingonwhetherthetemperaturerangeis closetotheglasstransitiontemperatureofthepolymer.According tothehousekeeping dataof theCASSE sensors,thetemperatures of the feet were roughly −100°C. Literature data show (Amash andZugenmaier,1997;BansemirandHaider,1998;Shokriehetal., 2012) that the elastic modulus of GFC materials increases by a factor of ≈1.7 for a temperature of T ≈ 200K compared to the valueatroomtemperature.Therefore,onecanexpectthatthefree resonance-frequencyincreasesbyafactorof√

1.7≈1.3relativeto roomtemperature,i.e.fromf_f=540Hztoabout700Hz.

6.3.2. Contact-resonances

From theFourierspectraoftheaccelerometersignals,onecan clearlyseefrequencybandsbelowandabove680Hz;seeFigs.2b–

4bandTable 1. As said above, we interpretthese frequencies as contact-resonancesofthefootsoleswithvariationsinthebound- aryconditionsduetothevariationsinthecontactmechanicswhen the soles encountered the surface of 67P at Agilkia. The oscilla- tionsofthesoleswereexcitedbythetime-varyingforcesofmulti- asperity contacts with the pebbles of the comet surface and/or withthesurfaceoftheboulderhit,orbythestruthittingtheedge oftheboulder.

Eq.(4)describesinaquitegeneralwaythecontact-resonances

ω

thasafunctionofthecontactedmassesinvolvedandofthecon- tactstiffnessk^∗.Mass loadingof thesoles leadsto adecrease of thecontact-resonanceswhereasstiffeningbyrestoringforcesleads to their increase (Vidic etal., 1998). Calibration curves were ob- tained by measuring the contact-resonance of a spare foot-sole restingonmaterialsofknownreducedelasticmoduli.Thecontact- resonances were excited by tapping on top of the lid with the sole having contact to the material tested or by fall tests; see AppendixA5.

Material constants, i.e. Young’s modulus E, shear modulus G, and the Poisson ratio

υ

^{of the} calibration materials were measured by ultrasonic velocity measurements and by density measurements. From these data the reduced Young’s modulus E_r,cal=E/(1−

υ

^{)2 ofthe materialwas}determined;see TableA5in theappendix.Thenormalizedcontact-resonance frequencyfc/f_f is plottedasafunctionofthereducedmodulusofthecalibrationma- terials,E_r,cal;seeFig.9.

FromTables1to3,wereadthatmostofthecontact-resonances measuredon comet 67Pextend from≈1.1to 1.2kHz. Thatcorre- spondsto fc/f_f=1.2/0.68 ≈ 1.8, andfrom Fig.9 one infers a cor- responding reducedmodulus E_r,cal ≈ 2GPa. In order to calculate thecorresponding contactstiffness, one needs the contactradius aswell, asseenin Eq.(3).In ourcase R=0.2m isthe curvature radiusof the footsole, P=4.92Nis the load, i.e.the staticforce exertedby theweight ofoneleg ofthelanding gearonthe cali- brationsampleviathesole(housingtheaccelerometer),whichwas measuredbyaidofabalance.InEq.(3),Er∗istheeffectivemod- ulusofbothcontactingstructures,thefootsolewithitsmeasured stiffnessandthecalibrationmaterialsthat areassumedtobe ho- mogeneous.Thisshouldholdforthematerialswithasmoothsur-

Fig. 9. Normalized contact-resonance frequency f c/f f versus reduced elastic mod- ulus E r,calof the calibration materials. Squares represent contact-resonance values obtained by tapping, circles are data for porous materials, and triangles represent data from fall tests. The data are listed in Table A5 in Appendix A5 . The steep in- crease in contact stiffness occurs when the contact stiffness is comparable to the sole stiffness (color online).

face.Letusfindanequivalentvolumetricmaterialwithareduced modulus E_equi that developsthesame contactradiusasthe shell structure of the foot sole when pressed into the calibration ma- terial.ObviouslyE_equi=k_sh/2ac musthold (Eq.(3)).UsingEq. (2), i.e.1/E_r,eff^∗=2ac/k_sh+1/Er,_calandinserting1/E_r,eff^∗intoEq.(6),one obtainsanalgebraicequationforthecontactradius.Thesolutionof thisequationyieldsac≈3mmandthecorrespondingcontactstiff- ness at f_c/f_f=1.8 is k^∗=2a_cE_r,eff^∗ ≈ 157kN/m (E_r,eff^∗=26.3MPa).

That means that the maximal change in the dispersion curve of Fig.9occurswhenk_sh ∼k^∗,whichisphysicallyplausible.

Letusapply thisfindingtothesituationonthecomet.Weob- tainthesameresultifthecontactradiusremainedthesame.How- ever, this cannot be expected. For the contactradius one has to takeratherthesoleradius,i.e.afactor50/3≈16.7largerwhereas thesolestiffnessshouldincreaseto≈270kN/m(seeAppendixA3).

Thistranslatesintoasurfacemodulusofabout3MPa.

Incaseofmulti-asperitycontacts,likeonanensembleofpeb- bles onthe comet,orin caseofvery porousandrough surfaces, likevolcanic rock,the contactradiuscan no longerbe calculated withEq.(6),andasaresulttheinversionofthedataasdoneabove isno longerpossible.Thissituation isevenmore pronouncedfor the fall tests on such materials (data marked with triangles in Fig.9).

7. Discussion

Inporousmaterials,boththeelasticmoduliaswellasthefrac- turemechanicalparameters such astensilestrength, compressive strength,andfracture toughnessarereduced.Duringtheprepara- tionphaseoftheRosettamissionaneffortwasundertakentoesti- matethecompressivestrengthofcometsbyusingtheratioofthe strength/elastic modulus structure for a foam-like structure with opencells(GibsonandAshby,1997;Möhlmann,1996):

σ

c/Ec=0.03

( ρ

p/

ρ

s

)

²

1+

ρ

p/

ρ

s

²

. (10)

Here,

σ

^{c is the elastic stress for} compression failure and

ρ

^p

and

ρ

s are the mass densities of the comet nucleus materials withporosityandfortheporefreematerial,respectively.Ec isthe Young’smodulus ofthecometmaterial.Inserting forthe porosity (1−

ρ

p/

ρ

s)=0.7–0.8, one obtains 2.5×10⁻³<

σ

c/E_c<6.5×10⁻³ and for

σ

^c=8kPa, this yields 1.2MPa<Ec<3.2MPa. This range showsthelargeinfluenceoftheporosity.

Similar expressions to Eq. (10) have been derived for cellular materials (Ashby, 1983; Gibson and Ashby, 1997). Based on the proposalthat the gluebetweenthe particulatesof thecomet re- golithmightbeice(Greenbergetal.,1995;Möhlmann,1996) and considering thehighporosity,thecometsurfacematerial maybe viewed as a cellular material with the ice as the wallsbetween

the regolith particles. Let us briefly discuss the consequences of such an approach. There are experimentally determined master curves for cellular materials for the ratio of the elastic collapse strength/elastic modulus of the cells by cell-wall buckling. i.e.

crushing(Ashby,1983).Againforaporosityof0.7–0.8,thisratiois 1.7×10⁻³<

σ

^c/Ec<7.9×10⁻³i.e.forEs=8GPa(ice)(Fellin,2013), we obtain 13MPa<

σ

c<63MPa. For stresses beyond 4MPa, the deformationofthematerialbecomesnon-linear,butitisstillelas- tic,i.e.thestrainisrecoveredwhenunloaded.

If the cell walls can be deformed plastically, which ice can undergo, there is a yield strength

σ

pl that depends on poros- ity. Again there are master curves (Ashby, 1983), and for a porosity of 0.7–0.8, the relative plastic yield strength

σ

pl/

σ

w is 2.4×10⁻²<

σ

pl/

σ

^w<5.7×10⁻². It is not straightforward to se- lect the correct parameter for the strength

σ

^{w of the ice walls,}

because they willundergo both shearing andcompression when loaded. From the tables in Fellin (2013), we find the lower limit

σ

^w ≈ 1MPa and

σ

^w ≈ 10MPa as an upper limit. This entails

σ

pl ≈24kPaasalower limitand

σ

pl≈ 57kPa asanupperlimit for

σ

^w=1MPaand

σ

^w=10MPa,respectively.

Independent of the question whether ice serves as glue be- tween theregolith particles,brittle cellularporous materialsmay failincompressionbycrushing.Afterelasticdeformationfollowed by crushing,there isaplateauregime ofstress versusstrain dur- ing which energy is absorbed upon further straining the mate- rial and eventually densification sets in at larger strains

ε

^(see

Fig. A6).Such a scenario hastaken placeaccording tothe analy- sisoftheenergybalanceofthelandingevents(Bieleetal., 2015;

Rolletal., 2016). Asan exampleforporousceramicAl₂O₃ witha porosity of 80%, theratio is2×10⁻³<

σ

^c/

σ

^w<10⁻² (Dam etal., 1990), which entails a crushing strength of 20kPa<

σ

^c<100kPa for

σ

w≈10MPa,and2kPa<

σ

c<10kPafor

σ

w≈1MPa.

The fracture ofa material is sensitive to defects such aspre- existing cracks, inclusions or large cells or voids in cellular ma- terials. This holds in particular for the tensile strength of mate- rials. Defects act as stress concentrators. This is taken into ac- count by the concepts of fracture mechanics (Lawn, 1993) which are also applied in rock mechanics, see for example Hatzor and Palchick,1997.Againtherearemastercurvesforthefracturestress

σ

^{forporousmaterials (Ashby,1983),whichforaporosityof80%}

give

σ

≈10⁻²

σ

w/

d/Lc. (11)

Here,

σ

^{is thefracturestress, Lc is thelength ofthecellsand}

d is the linear defect size. Eq. (11) holds for d>L_c. The small- est cell size forthe cometregolith is Lc=0.4mm (Mottola etal., 2015). Let us assume we have d=10cm. This yields a fracture stress of 6.5kPa if we assume again that the cell walls are ice, i.e.

σ

^w ≈ 10MPa. It should be mentioned that Greenberg et al., 1995,statedthatafracturemechanicsapproachwouldbethecor- rectwayofdescribingthefracturestrengthofcometmaterials.

Similarconsiderationsandrelationsexistfortheelasticmoduli andfracturestrengthofceramicsbeforesinteringwhichareeither slurries or dry-pressed powders, so-called green bodies (Kendall etal.,1987;Kendalletal.,1986).Greenceramicbodieshaveanap- preciableamountofporosityandalso,likeassumedforthecomet material, the individual particles in the unfired material contact eachotheratcertainpoints,whereinteratomicforcesholdtheen- semble of particles together (Tabor, 1977). They deform the par- ticles atthecontactpointselastically.The energyneededto sep- aratetwoparticles isa_cp²× ^whereacpistheradius ofthecon- tactpointsand^{isthesurfaceenergy.Whenthetwoparticlesare} separated,theelasticenergyisreleasedwhichreducesthetotalen- ergyneededforseparation.Thenumberofcontactsintheensem- ble withsurfacearea

π

^{acp2 dependson the overall porosity.Ex-}

perimentally thisdependencewasdetermined to be proportional

to (

ρ

^p/

ρ

^{s)z with z being2, 3,and 4 (Carneim andGreen, 2001)}

insteadof thealmost squaredpower dependenceof Eq.(10)and themastercurvescitedabove.Thismakestherelations verysen- sitiveto theactual porosity atsmallvaluesof

ρ

p/

ρ

s (Arnoldand Möhlmann,2009).

In summary, it is the porosity which determines the elastic- ityandstrengthforsmall

ρ

p/

ρ

s (Ashby,2006; Leite andFerland, 2001). The material of the cell wallsand the binding by surface forces,i.e.thesurfaceenergy,playaminorrole.

8. Comparisontootherrelevantdata

Largecracks andfeaturesof irregularpolygonalstructurewith asizeoftheorderofsomemtosome100mhavebeenobserved oncomet67P(El-Maarryetal.,2015).Smallercracksofsizescom- parabletoandsmallerthanthediameterofPhilae’ssolesexistas well.Infactfracturesandcracksareubiquitousatgrainandboul- derscales(Bibringetal.,2015).

Tensileorfailurestrengthsforcomet67Pof≈100Pahavebeen reported for dust overhangs (Thomas et al., 2015) much smaller thantheestimateforthetensilestrengthestimatedintheprevious section (6.5kPa),which mightbe due tothe still higherporosity inthedustoverhang.Groussinetal.(2015)discussedindetailthe variousstrength valuesderived from theairfall oftheoverhangs, shearstrengths fromadhesionandrollingofboulders(4 Pa),and eventually thecompression strength derived fromthe depression depth the lander soles left behindat Agilkia(≈16kPa). One also finds in this paper a table summarizing tensile strengths, shear strengths and compressive strengths for cometary materials and thecorrespondingreferences.

Atensile strengthof 10Paalready influenced thefurther mo- tion ofPhilaeat Agilkiain modeling the landing event(Roll and Witte,2016).Estimatingtheforcesactingonthesolesupon land- ing,Rolletal.(2016) deriveda compressivestrengthatAgilkiaof 2kPa.Finally,ahardnessvalueof4MPahasbeenreportedforthe MUPUSexperimentcarriedoutatAbydos(Spohnetal.,2015).

Thus,the experimentaldata reportedsofar forcomet67P in- dicatethatthetensilestrengthsareoftheorder≈100Pa,whereas compressive strengths are much larger, i.e. ofthe order ofup to 10kPa andat Abydos up to some MPa, seealso Basilevsky etal.

(2016).

Let us compare these data with the KOSI laboratory experi- ments (Kochan et al., 1998; Thomas et al., 1994). The crushing strength ofa porous ice–mineral composite with an olivine con- tent of3.4% wasexamined.Mostinteresting wasthe observation thatthecrushingstrengthincreasedwithsinteringtimereachinga valueof≈0.5MPaafter≈3daysat253K.Withincreasingmineral contentthecrushingstrength increasedslightly.The neckgrowth betweentheparticlesandhencethestabilityofthesinteredma- terial wasestimated aswell. Inabout 10⁶ years the neck radius wouldbe about equal tothe particle radius, andthe strength of thecompositewouldthenbefullydeveloped.Differentspecimens wereproducedwithgrainsizesaround1mm,andtheycontained H₂OandCO₂icewith15%weightpercentage.Themineralsadded were olivine and montmorillonite. The overall porosity was 49–

63%. The measured hardness was 0.15–1MPa. As said earlier, in thesemeasurementsgradualcrushingoftheKOSImaterialwasob- servedwhichmanifested itselfby oscillatingstrengthvalueswith increasing penetration by the indenter. That fracture mechanical conceptsshouldbeappliedtodescribethestrengthofcometana- loguematerialsandeventuallycometmaterial,wasclearlyspelled outina numberofpapersgivenattheInt. WorkshoponPhysics andMechanicsofCometaryMaterialswhichtookplaceinMünster, Germanyin1989(ProceedingsVol.SP-302,ESA,Paris,France).

A lot of data have been published concerning the proper- ties of ice and snow, see for example data in Fellin (2013).

Petrovic gave a detailed account of the mechanical properties of ice (Petrovic, 2003). The tensile strength of ice varies from 0.7 to3.1MPaandthe compressivestrength variesfrom5to 25MPa overthe temperature range−10 °C to −20°C. The ice compres- sivestrengthincreaseswithdecreasingtemperatureandincreasing strainrate,buticetensile-strengthisrelativelyinsensitivetothese variables.

There exist quite alot oflaboratory experimentsandtheoret- icalestimatestogain insightinto thestrengthofdust aggregates relatedtocomets.Onefindsvaluesfortensilestrengthsoftheor- derofsome10Pawithexceptionsofuptosome10kPa.However, we refrainfrom summarizingthese papersin detail andrefer to Luding(2008),Meisneretal.(2012),Musioliketal.(2017),Skorov andBlum(2012),andreferencestherein.

9. Summaryandconclusion

The analysisoftheCASSEaccelerationdatayieldscompression strengthsof≈10kPa forthecometsurfacematerial.Thisisinline withthedatapreviously publishedconsideringtheforcesexerted by the lander on the comet surfaceat Agilkia (Roll et al., 2016).

Inouranalysis, theseforceswerederived fromthesignalsof the accelerometersmountedinthefootsolesofPhilae’slanding gear.

Assumingthatthesamerelationsknownforhighlyporousnatural andengineeringmaterials applyto thesurfacematerial ofcomet 67P, its elastic modulus should be of the order of a few MPa at Agilkia.Suchavaluewasindeedfoundbytheevaluationoftime- of-flightdataduringtheMUPUShammeringatPhilae’sfinalland- ingsiteAbydos(Knapmeyeretal.,2015).

It is the porosity which determines the elasticity and the strengthofaporousmaterialwithlargeporosity.Ontheonehand, theporosity reducesthe number ofcontacts orthe coordination numberbetweentheindividualconstituentsofthecometagglom- eratematerial,andontheotherhandlargeporesactasstresscon- centratorinitiating failure upon loading. Thisconcept isalso ap- pliedinrockmechanics(HatzorandPalchick,1997).Thelargevari- ationsof strengthvaluesreportedon comet67P froma fewtens ofPa (Groussin etal., 2015; Thomas etal., 2015) to several MPa (Spohnetal., 2015)can beexplainedbylocalvariationsinporos- ity.

Acknowledgments

With shock and grief the co-authors learnt that Prof. Dr. D.

Möhlmannpassed away on September 28th 2016, 2 days before the end of the Rosetta mission. In his active time at the DLR- InstituteofSpaceSimulationinCologne,Germany,hewasrespon- sibleasthefirstoneofthethreeSESAMEPIsfortheplanningand integrationof thethree instruments CASSE,DIM and PP intothe PhilaeexperimentSESAME.Theco-authorsexpresstheirdeepgrat- itudeforhisoutstandingcontributionstoourcommonproject.

WethankR.BirringerandA.Steinbach,DepartmentofPhysics, Saarland University for the measurement of the stiffness of a flight-sparefoot-sole.We alsothankG. Falk andW.Possart, both Department of Materials, Saarland University, for providing the tuffstone (with theporosity value)and thepolyurethane calibra- tionsamples,respectively.Likewise,wearegratefulfortheporous rocksampleswith porosity valuesprovided by M.Prasad, School ofMines, Golden,CO,USA.W.A.thanks K.Samwer,I.Phys.Insti- tut,Georg-AugustUniversitätGöttingen,andJ.Turner,Department ofMechanicalEngineering,UniversityofNebraska,Lincoln,NE,for manyhelpfuldiscussionsonvariousaspectsofcontact-resonances, andalso G. H. Schwehm,ESA-ESTEC, Noordwijk, Netherlands for helpful discussions on the results of the KOSI experiments. We thankN.A.Burnham,WorcesterPolytechnicInstitute,MAforacrit- ical reading of the manuscript. Last but not least, we thank W.

Gebhardt andR. Lichtfortheir invaluablecontributions in devel- opingtheCASSEinstrumentatFraunhoferIZFP.

Rosetta isanESAmissionwithcontributionsfromitsmember statesandNASA. Rosetta’sPhilae landeris providedby a consor- tiumledbyDLR,MPS,CNESandASI.SESAMEisanexperimenton the Rosetta lander Philae. It consistsofthree instruments CASSE, DIMandPP,whichwereprovidedbyaconsortiumcomprisingDLR, MPS,FMI,MTAEK,FraunhoferIZFP,Univ.Cologne,LATMOSandES- TEC.Alldatawillbemadeavailable viatheESAPlanetaryScience Archive.

Appendix

A.1. Elasticmodulusoftheglass–fibercompositesole

The elastic parameters of the GFC sole were obtained from ultrasonic velocity measurements at frequencies 2–3MHz us- ing broadband excitation. They yielded a longitudinal veloc- ity of v_L=3330m/s. With a density of

ρ

=2.3×10³kg/m³, this givesL=25.55GPa,whereL=vL2×

ρ

^{istheso-called}longitudinal modulus. Likewise, the measurement of the shear velocity gave v_S=1910m/s. This yields a shear modulus G=v_S²×

ρ

=8.4GPa and with

υ

=[(L − 2G)/2(L−G)] a Poisson ratio of

υ

=0.255.

Therefore, the E-modulus is

υ

=2(1+

υ

^)G=21.1GPa and the re- ducedmodulusisE^∗=E_sh^∗=22.6GPa.Thesevaluesareclosetolit- eraturevaluesofE=20GPa(Schwartz,1992).Duetohighattenua- tioncausedbyscatteringofultrasoundatthecompositestructure, itwasnot possibletodeterminetheanisotropy factoroftheGFC materialintheplaneofthewovenfabric.

A.2. Heightoftheboulder

Thelength oftheshadowoftheboulderhit isonits left side (West)aboutw≈ 0.4m;seeFig.5.Thesunstoodatan elevation of ϑ=40° to the plane of the landing site (Mottola et al., 2015).

Thisyieldsamaximalheightoftheboulder

H=w×tan

ϑ

^{≈0.34m (A1)}

Since the edge of the boulder is rounded and flattened from WesttoEast(leftto rightinFig.5) andeventuallydives intothe ground, there is no singleheight. At the part where the landing gearprobablyhittheboulder,theheightisonly15–20cm.

A.3. Solestiffness

Analyticalestimateofthesolestiffness

The stiffnessofthesole includingthe footstructurewasesti- mated to be 173kN/m in the appendix (i) in Faber et al.(2015), basedonthecollectionofequationsinYoungandBudynas(2002). An analytical expression for the stiffness of a spherical shell is givenbyMansoor-BaghaeiandSadegh(2011):

k_sh=2.3E_sht²_sh/R_sh 1−

υ

sh². (A2)

Here, E_sh is the elastic modulus of the shell, t_sh is its thick- ness, and R_sh is the curvature radius of the shell. Inserting for the glass-fiber E_sh=21.1GPa with a Poisson ratio of

υ

sh=0.26, t_sh=1mm,andR_sh=0.2m,yieldsk_sh=251kN/m.Inanadditional work (Mansoor-BaghaeiandSadegh, 2015), thestiffnessofan el- lipsoidalshellwasgivenas

k_sh=8

(

^DbE_sht_shK

)

^{1/2. (A3)}

Here,KistheGaussiancurvatureinloadinglocationandD_b is thebendingstiffnessoftheshelldefinedas

D_b=E_sht³_sh/12 1−

υ

sh². (A4)