The Rosetta mission orbiter science overview: the comet phase
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B. J. Buratti | N. Altobelli | M. G. G. T. Taylor | B. Buratti | N. Altobelli | M. Choukroun | Matthew Taylor | M. Choukroun | M. Taylor
[1] S. Fuselier,et al. Rosetta observations of solar wind interaction with the comet 67P/Churyumov-Gerasimenko , 2015 .
[2] J. Lunine,et al. The presence of clathrates in comet 67P/Churyumov-Gerasimenko , 2016, Science Advances.
[3] K. Glassmeier,et al. Two-point observations of low-frequency waves at 67P/Churyumov-Gerasimenko during the descent of PHILAE: comparison of RPCMAG and ROMAP , 2016 .
[4] Eric Schindhelm,et al. Far-UV phase dependence and surface characteristics of Comet 67P/Churyumov-Gerasimenko as observed with Rosetta Alice , 2015 .
[5] J. Lebreton,et al. Evolution of the ion environment of comet 67P/Churyumov-Gerasimenko - Observations between 3.6 and 2.0 AU , 2015 .
[6] S. Debei,et al. Orbital elements of the material surrounding comet 67P/Churyumov-Gerasimenko , 2015 .
[7] T. Owen,et al. Molecular nitrogen in comet 67P/Churyumov-Gerasimenko indicates a low formation temperature , 2015, Science.
[8] U. Fink,et al. Exposed water ice on the nucleus of comet 67P/Churyumov–Gerasimenko , 2016, Nature.
[9] J. Lebreton,et al. Observation of a new type of low-frequency waves at comet 67P/Churyumov-Gerasimenko , 2015, 1505.06068.
[10] D. Bramich,et al. Beginning of activity in 67P/Churyumov-Gerasimenko and predictions for 2014–2015 , 2013, 1307.7978.
[11] C. Walsh. Chemical complexity in protoplanetary disks in the era of ALMA and Rosetta , 2015, 1605.07825.
[12] Gregory A Petsko. The Rosetta Stone , 2001, Genome Biology.
[13] S. Debei,et al. Fractures on comet 67P/Churyumov‐Gerasimenko observed by Rosetta/OSIRIS , 2015 .
[14] E. Palomba,et al. GIADA: shining a light on the monitoring of the comet dust production from the nucleus of 67P/Churyumov-Gerasimenko , 2015 .
[15] Regional surface morphology of comet 67P/Churyumov-Gerasimenko from Rosetta/OSIRIS images: The southern hemisphere , 2016 .
[16] S. Debei,et al. Search for satellites near comet 67P/Churyumov-Gerasimenko using Rosetta/OSIRIS images , 2015 .
[17] M. Kaasalainen,et al. A Portrait of the Nucleus of Comet 67P/Churyumov-Gerasimenko , 2007 .
[18] J. Berthelier,et al. Composition-dependent outgassing of comet 67P/Churyumov-Gerasimenko from ROSINA/DFMS - Implications for nucleus heterogeneity? , 2015 .
[19] T. Owen,et al. Detection of argon in the coma of comet 67P/Churyumov-Gerasimenko , 2015, Science Advances.
[20] J. Berthelier,et al. HIGH-TIME RESOLUTION IN SITU INVESTIGATION OF MAJOR COMETARY VOLATILES AROUND 67P/C–G AT 3.1–2.3 au MEASURED WITH ROSINA-RTOF , 2016 .
[21] K. Glassmeier. Interaction of the solar wind with comets: a Rosetta perspective , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[22] D. Plettemeier,et al. Properties of the 67P/Churyumov-Gerasimenko interior revealed by CONSERT radar , 2015, Science.
[23] J. Berthelier,et al. ROSINA/DFMS and IES observations of 67P: Ion-neutral chemistry in the coma of a weakly outgassing comet , 2015 .
[24] Martin Rubin,et al. Inventory of the volatiles on comet 67P/Churyumov-Gerasimenko from Rosetta/ROSINA , 2015 .
[25] Giampiero Naletto,et al. EVOLUTION OF THE DUST SIZE DISTRIBUTION OF COMET 67P/CHURYUMOV–GERASIMENKO FROM 2.2 au TO PERIHELION , 2016 .
[26] Y. Langevin,et al. Typology of dust particles collected by the COSIMA mass spectrometer in the inner coma of 67P/Churyumov Gerasimenko , 2015 .
[27] Giampiero Naletto,et al. OSIRIS observations of meter-sized exposures of H2O ice at the surface of 67P/Churyumov-Gerasimenko and interpretation using laboratory experiments , 2015 .
[28] Wlodek Kofman,et al. The Philae lander mission and science overview , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[29] S. Debei,et al. Large-scale dust jets in the coma of 67P/Churyumov-Gerasimenko as seen by the OSIRIS instrument onboard Rosetta , 2015 .
[30] Karl Battams,et al. SOHO comets: 20 years and 3000 objects later , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[31] A. Fitzsimmons,et al. The nucleus of Comet 67P/Churyumov-Gerasimenko. A new shape model and thermophysical analysis , 2012 .
[32] M. T. Capria,et al. Photometric properties of comet 67P/Churyumov-Gerasimenko from VIRTIS-M onboard Rosetta , 2015 .
[33] S. Debei,et al. Characterization of the Abydos region through OSIRIS high-resolution images in support of CIVA measurements , 2016 .
[34] S. Debei,et al. Spectrophotometric properties of the nucleus of comet 67P/Churyumov-Gerasimenko from the OSIRIS instrument onboard the ROSETTA spacecraft , 2015, 1505.06888.
[35] Hans Rickman,et al. Comets as collisional fragments of a primordial planetesimal disk , 2015, 1504.04512.
[36] H. Hsieh,et al. Asteroid–comet continuum objects in the solar system , 2016, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[37] S. Debei,et al. Rotating dust particles in the coma of comet 67P/Churyumov-Gerasimenko , 2015 .
[38] A. Hubault,et al. Rosetta following a living comet , 2016 .
[39] A. Hubault,et al. Rosetta operations at the comet , 2015 .
[40] S. Debei,et al. Physical properties and dynamical relation of the circular depressions on comet 67P/Churyumov-Gerasimenko , 2016 .
[41] Giampiero Naletto,et al. Morphology and dynamics of the jets of comet 67P/Churyumov-Gerasimenko: Early-phase development , 2015 .
[42] Ian Wright,et al. Low CO/CO 2 ratios of comet 67P measured at the Abydos landing site by the Ptolemy mass spectrometer , 2015 .
[43] J. Berthelier,et al. Rosetta mission results pre-perihelion Special feature Comparison of 3 D kinetic and hydrodynamic models to ROSINA-COPS measurements of the neutral coma of 67 P / Churyumov-Gerasimenko , 2015 .
[44] A. Ercoli Finzi,et al. Philae's First Days on the Comet , 2015, Science.
[45] K. Meech. Setting the scene: what did we know before Rosetta? , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[46] G. Clark,et al. Suprathermal electron environment of comet 67P/Churyumov-Gerasimenko: Observations from the Rosetta Ion and Electron Sensor , 2015 .
[47] S. Debei,et al. Size-frequency distribution of boulders ≥7 m on comet 67P/Churyumov-Gerasimenko , 2015 .
[48] Hans-Herbert Fischer,et al. Dust Impact Monitor (SESAME-DIM) on board Rosetta/Philae: MIllimetric particle flux at comet 67P/Churyumov-Gerasimenko , 2016, 1605.06291.
[49] S. Debei,et al. Geomorphology and spectrophotometry of Philae’s landing site on comet 67P/Churyumov-Gerasimenko , 2015 .
[50] E. Grün,et al. Unexpected and significant findings in comet 67P/Churyumov–Gerasimenko: an interdisciplinary view , 2016 .
[51] A. Fitzsimmons,et al. Distant activity of 67P/Churyumov-Gerasimenko in 2014: Ground-based results during the Rosetta pre-landing phase , 2016, 1602.01493.
[52] N. Biver,et al. The composition of cometary ices , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[53] S. Debei,et al. Observations and analysis of a curved jet in the coma of comet 67P/Churyumov-Gerasimenko , 2016, 1605.02095.
[54] E. Grün,et al. DENSITY AND CHARGE OF PRISTINE FLUFFY PARTICLES FROM COMET 67P/CHURYUMOV–GERASIMENKO , 2015 .
[55] F. Scholten,et al. The structure of the regolith on 67P/Churyumov-Gerasimenko from ROLIS descent imaging , 2015, Science.
[56] S. Debei,et al. Gravitational slopes, geomorphology, and material strengths of the nucleus of comet 67P/Churyumov-Gerasimenko from OSIRIS observations , 2015, 1509.02707.
[57] S. Debei,et al. Dust measurements in the coma of comet 67P/Churyumov-Gerasimenko inbound to the Sun , 2015, Science.
[58] Harry Lehto,et al. Comet 67P/Churyumov-Gerasimenko sheds dust coat accumulated over the past four years , 2015, Nature.
[59] E. Grün,et al. High-molecular-weight organic matter in the particles of comet 67P/Churyumov–Gerasimenko , 2016, Nature.
[60] S. Debei,et al. The morphological diversity of comet 67P/Churyumov-Gerasimenko , 2015, Science.
[61] Nicolas Thomas,et al. REDISTRIBUTION OF PARTICLES ACROSS THE NUCLEUS OF COMET 67P/CHURYUMOV-GERASIMENKO , 2016 .
[62] N. Thomas,et al. Permittivity measurements of porous matter in support of investigations of the surface and interior of 67P/Churyumov-Gerasimenko , 2015, Astronomy & Astrophysics.
[63] N. Thomas,et al. PITS FORMATION FROM VOLATILE OUTGASSING ON 67P/CHURYUMOV–GERASIMENKO , 2015, 1510.07671.
[64] T. Cravens,et al. Charge exchange in cometary coma: Discovery of H− ions in the solar wind close to comet 67P/Churyumov‐Gerasimenko , 2015, Geophysical research letters.
[65] Giampiero Naletto,et al. Shape model, reference system definition, and cartographic mapping standards for comet 67P/Churyumov-Gerasimenko Stereo-photogrammetric analysis of Rosetta/OSIRIS image data , 2015 .
[66] D. Plettemeier,et al. CONSERT suggests a change in local properties of 67P/Churyumov-Gerasimenko's nucleus at depth , 2015 .
[67] S. Debei,et al. Possible interpretation of the precession of comet 67P/Churyumov-Gerasimenko , 2016 .
[68] J. Lebreton,et al. The Rosetta Ion and Electron Sensor (IES) measurement of the development of pickup ions from comet 67P/Churyumov‐Gerasimenko , 2014 .
[69] Jean-Pierre Lebreton,et al. Birth of a comet magnetosphere: A spring of water ions , 2015, Science.
[70] H. Keller,et al. The changing rotation period of comet 67P/Churyumov-Gerasimenko controlled by its activity , 2015 .
[71] I. I. Shapiro,et al. The first results , 1979 .
[72] F. Scholten,et al. The landing(s) of Philae and inferences about comet surface mechanical properties , 2015, Science.
[73] D. J. Andrews,et al. CHO-bearing organic compounds at the surface of 67P/Churyumov-Gerasimenko revealed by Ptolemy , 2015, Science.
[74] T. Gombosi,et al. Observation of charged nanograins at comet 67P/Churyumov‐Gerasimenko , 2015 .
[75] J. Gerring. A case study , 2011, Technology and Society.
[76] S. Debei,et al. The primordial nucleus of comet 67P/Churyumov-Gerasimenko , 2015 .
[77] U. Fink,et al. The organic-rich surface of comet 67P/Churyumov-Gerasimenko as seen by VIRTIS/Rosetta , 2015, Science.
[78] Eric Schindhelm,et al. First extreme and far ultraviolet spectrum of a Comet Nucleus: Results from 67P/Churyumov-Gerasimenko , 2015 .
[79] S. Erard,et al. Refractory and semi-volatile organics at the surface of comet 67P/Churyumov-Gerasimenko: Insights from the VIRTIS/Rosetta imaging spectrometer , 2016 .
[80] K. Glassmeier,et al. Dynamical features and spatial structures of the plasma interaction region of 67P/Churyumov–Gerasimenko and the solar wind , 2015 .
[81] H. Keller,et al. What drives the dust activity of comet 67P/Churyumov-Gerasimenko? , 2015, 1506.08545.
[82] S. Debei,et al. Sunset jets observed on comet 67P/Churyumov-Gerasimenko sustained by subsurface thermal lag , 2016 .
[83] B. Marty,et al. Origins of volatile elements (H, C, N, noble gases) on Earth and Mars in light of recent results from the ROSETTA cometary mission , 2016 .
[84] Hans-Herbert Fischer,et al. Electrical properties and porosity of the first meter of the nucleus of 67P/Churyumov-Gerasimenko - As constrained by the Permittivity Probe SESAME-PP/Philae/Rosetta , 2016, 1604.03678.
[85] M. A’Hearn,et al. Comets: looking ahead , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[86] T. Encrenaz,et al. Subsurface properties and early activity of comet 67P/Churyumov-Gerasimenko , 2015, Science.
[87] F. Scholten,et al. Modelling observations of the inner gas and dust coma of comet 67P/Churyumov-Gerasimenko using ROSINA/COPS and OSIRIS data: First results , 2016 .
[88] S. Debei,et al. Insolation, erosion, and morphology of comet 67P/Churyumov-Gerasimenko , 2015 .
[89] M. Banaszkiewicz,et al. Thermal and mechanical properties of the near-surface layers of comet 67P/Churyumov-Gerasimenko , 2015, Science.
[90] I. Richter,et al. Mass loading at 67P/Churyumov‐Gerasimenko: A case study , 2016, 1805.05587.
[91] S. Debei,et al. Comparative study of water ice exposures on cometary nuclei using multispectral imaging data , 2016 .
[92] S. Debei,et al. Rosetta mission results pre-perihelion Special feature Comet 67 P / Churyumov-Gerasimenko : Constraints on its origin from OSIRIS observations , 2015 .
[93] S. Erard,et al. Three-dimensional direct simulation Monte-Carlo modeling of the coma of comet 67P/Churyumov-Gerasimenko observed by the VIRTIS and ROSINA instruments on board Rosetta , 2016 .
[94] Jean-Michel Reess,et al. First observations of H2O and CO2 vapor in comet 67P/Churyumov-Gerasimenko made by VIRTIS onboard Rosetta , 2015 .
[95] N. Edberg,et al. Suprathermal electrons near the nucleus of comet 67P/Churyumov‐Gerasimenko at 3 AU: Model comparisons with Rosetta data , 2016 .
[96] K. Glassmeier,et al. The Rosetta Mission: Flying Towards the Origin of the Solar System , 2007 .
[97] M. Hilchenbach,et al. COSIMA calibration for the detection and characterization of the cometary solid organic matter , 2015 .
[98] Giampiero Naletto,et al. The rotation state of 67P/Churyumov-Gerasimenko from approach observations with the OSIRIS cameras on Rosetta , 2014 .
[99] J. Berthelier,et al. Solar wind sputtering of dust on the surface of 67P/Churyumov-Gerasimenko , 2015 .
[100] J. De Keyser,et al. Abundant molecular oxygen in the coma of comet 67P/Churyumov–Gerasimenko , 2015, Nature.
[101] J Ginsburg,et al. Looking Ahead , 2011, Metabolism: clinical and experimental.
[102] Eric Schindhelm,et al. THE NATURE AND FREQUENCY OF THE GAS OUTBURSTS IN COMET 67P/CHURYUMOV–GERASIMENKO OBSERVED BY THE ALICE FAR-ULTRAVIOLET SPECTROGRAPH ON ROSETTA , 2016, 1606.05249.
[103] I. Mann. Comets as a possible source of nanodust in the Solar System cloud and in planetary debris discs , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[104] C. Snodgrass,et al. Optical observations of comet 67P/Churyumov-Gerasimenko with the Nordic Optical Telescope - Comet activity before the solar conjunction , 2015 .
[105] Giuseppe Piccioni,et al. Water and carbon dioxide distribution in the 67P/Churyumov-Gerasimenko coma from VIRTIS-M infrared observations , 2016 .
[106] C. Andrews,et al. The Rosetta Stone , 1981 .
[107] S. Debei,et al. The dust environment of comet 67P/Churyumov-Gerasimenko from Rosetta OSIRIS and VLT observations in the 4.5 to 2.9 AU heliocentric distance range inbound , 2016, 1602.01965.
[108] W. Ip,et al. Distribution of water around the nucleus of comet 67P/Churyumov-Gerasimenko at 3.4 AU from the Sun as seen by the MIRO instrument on Rosetta , 2015 .
[109] Claire Vallat,et al. The science planning process on the Rosetta mission , 2017 .
[110] E. Neefs,et al. 67P/Churyumov-Gerasimenko, a Jupiter family comet with a high D/H ratio , 2015, Science.
[111] S. Debei,et al. Summer fireworks on comet 67P , 2016, 1609.07743.
[112] Giampiero Naletto,et al. The 2016 Feb 19 outburst of comet 67P/CG: an ESA Rosetta multi-instrument study , 2016 .
[113] Claire Vallat,et al. Rosetta science operations in support of the Philae mission , 2016 .
[114] H. Leroux,et al. Variations in cometary dust composition from Giotto to Rosetta, clues to their formation mechanisms , 2016 .
[115] T. Guillot,et al. A PROTOSOLAR NEBULA ORIGIN FOR THE ICES AGGLOMERATED BY COMET 67P/CHURYUMOV–GERASIMENKO , 2016, 1604.08827.
[116] Umd,et al. Measurements of the near-nucleus coma of comet 67P/Churyumov-Gerasimenko with the Alice far-ultraviolet spectrograph on Rosetta , 2015, 1506.01203.
[117] Zhong-Yi Lin,et al. 67P/Churyumov-Gerasimenko activity evolution during its last perihelion before the Rosetta encounter , 2011 .
[118] S. Debei,et al. 67P/Churyumov-Gerasimenko: Activity between March and June 2014 as observed from Rosetta/OSIRIS , 2015 .
[119] Andrew Steele,et al. Organic compounds on comet 67P/Churyumov-Gerasimenko revealed by COSAC mass spectrometry , 2015, Science.
[120] E. Kührt,et al. Time variability and heterogeneity in the coma of 67P/Churyumov-Gerasimenko , 2015, Science.
[121] S. Debei,et al. Temporal morphological changes in the Imhotep region of comet 67P/Churyumov-Gerasimenko , 2015, 1509.02794.
[122] Paul Hartogh,et al. Spatial and diurnal variation of water outgassing on comet 67P/Churyumov-Gerasimenko observed from Rosetta/MIRO in August 2014 , 2015 .
[123] Sukhan Lee,et al. Rosetta mission results pre-perihelion Special feature Dark side of comet 67 P / Churyumov-Gerasimenko in Aug . – Oct . 2014 MIRO / Rosetta continuum observations of polar night in the southern regions , 2015 .
[124] K. Manos. In the Southern Hemisphere , 1993 .
[125] J. Bertaux. Estimate of the erosion rate from H 2 O mass-loss measurements from SWAN/SOHO in previous perihelions of comet 67P/Churyumov-Gerasimenko and connection with observed rotation rate variations , 2015 .
[126] H. Keller,et al. MIRO observations of subsurface temperatures of the nucleus of 67P/Churyumov-Gerasimenko , 2015 .
[127] Masanori Kobayashi,et al. Dust Impact Monitor (SESAME-DIM) Measurements at Comet 67P/Churyumov-Gerasimenko , 2015, 1510.01563.
[128] J. Lebreton,et al. RPC observation of the development and evolution of plasma interaction boundaries at 67P/Churyumov-Gerasimenko , 2016 .
[129] T. Owen,et al. Prebiotic chemicals—amino acid and phosphorus—in the coma of comet 67P/Churyumov-Gerasimenko , 2016, Science Advances.
[130] S. Debei,et al. On the nucleus structure and activity of comet 67P/Churyumov-Gerasimenko , 2015, Science.
[131] S. Debei,et al. Are fractured cliffs the source of cometary dust jets ? insights from OSIRIS/Rosetta at 67P/Churyumov-Gerasimenko , 2015, 1512.03193.
[132] F. Scholten,et al. A homogeneous nucleus for comet 67P/Churyumov–Gerasimenko from its gravity field , 2016, Nature.
[133] C. Russell,et al. The nonmagnetic nucleus of comet 67P/Churyumov-Gerasimenko , 2015, Science.
[134] Paul Hartogh,et al. Ocean-like water in the Jupiter-family comet 103P/Hartley 2 , 2011, Nature.
[135] C. Pilorget,et al. 67P/Churyumov-Gerasimenko surface properties as derived from CIVA panoramic images , 2015, Science.
[136] S. Debei,et al. Aswan site on comet 67P/Churyumov-Gerasimenko: Morphology, boulder evolution, and spectrophotometry , 2016 .
[137] S. Debei,et al. Rosetta mission results pre-perihelion Special feature Regional surface morphology of comet 67 P / Churyumov-Gerasimenko from Rosetta / OSIRIS images ? , 2015 .
[138] E. Grün,et al. Characteristics of the dust trail of 67P/Churyumov-Gerasimenko: an application of the IMEX model , 2015 .
[139] S. Erard,et al. The diurnal cycle of water ice on comet 67P/Churyumov–Gerasimenko , 2015, Nature.
[140] S. Debei,et al. Large heterogeneities in comet 67P as revealed by active pits from sinkhole collapse , 2015, Nature.
[141] E. Cupido,et al. Evolution of the plasma environment of comet 67P from spacecraft potential measurements by the Rosetta Langmuir probe instrument , 2015 .
[142] Luigi Colangeli,et al. COMET 67P/CHURYUMOV–GERASIMENKO: CLOSE-UP ON DUST PARTICLE FRAGMENTS , 2016 .
[143] Anders Eriksson,et al. First detection of a diamagnetic cavity at comet 67P/Churyumov-Gerasimenko , 2016 .
[144] S. Debei,et al. Geomorphology of the Imhotep region on comet 67P/Churyumov-Gerasimenko from OSIRIS observations , 2015 .
[145] J. Berthelier,et al. Sulphur-bearing species in the coma of comet 67P/Churyumov–Gerasimenko , 2016 .
[146] M. Horányi,et al. Negatively charged nano-grains at 67P/Churyumov-Gerasimenko , 2015 .
[147] K. Varmuza,et al. Mechanical and electrostatic experiments with dust particles collected in the inner coma of comet 67P by COSIMA onboard Rosetta , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[148] Daniel J. Scheeres,et al. Fission and reconfiguration of bilobate comets as revealed by 67P/Churyumov–Gerasimenko , 2016, Nature.
[149] S. Debei,et al. Photometry of dust grains of comet 67P and connection with nucleus regions , 2016 .
[150] J. Kissel,et al. Searching for calcium‐aluminum‐rich inclusions in cometary particles with Rosetta/COSIMA , 2016 .
[151] S. Debei,et al. The southern hemisphere of 67P/Churyumov-Gerasimenko: Analysis of the preperihelion size-frequency distribution of boulders ≥7 m , 2016 .
[152] Eric Quémerais,et al. The water production rate of Rosetta target Comet 67P/Churyumov–Gerasimenko near perihelion in 1996, 2002 and 2009 from Lyman α observations with SWAN/SOHO , 2014 .
[153] T. Owen,et al. D2O and HDS in the coma of 67P/Churyumov–Gerasimenko , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
[154] S. Debei,et al. Variegation of comet 67P/Churyumov-Gerasimenko in regions showing activity , 2016 .
[155] J. Lebreton,et al. Spatial distribution of low‐energy plasma around comet 67P/CG from Rosetta measurements , 2015, 1608.06745.
[156] S. Debei,et al. Two independent and primitive envelopes of the bilobate nucleus of comet 67P , 2015, Nature.
[157] R. Schulz,et al. The 67P/Churyumov–Gerasimenko observation campaign in support of the Rosetta mission , 2017, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.