Varying water activity and momentum transfer on comet 67P/Churyumov-Gerasimenko from its non-gravitational forces and torques

As onomy & As ophysics A&A, 690, A82 (2024) h ps://doi.o g/10.1051/0004-6361/202450728 © The Au ho s 2024 Va ying wa e ac i i y and momen um ans e on come 67P/Chu yumo -Ge asimenko om i s non-g a i a ional o ces and o ques N. A ee1,2,⋆, P. Gu ié ez1, O. G oussin3, J. Bü ge 2, H. U. Kelle 4, T. K ame 5, R. Lasagni Manghi6, M. Läu e 7, P. Lemos2, J. Ma kkanen2, R. Ma schall8, and C. Schucka 2 1Ins i u o de As o ísica de Andalucía - CSIC, Glo ie a de la As onomía s/n, 18008 G anada, Spain 2Ins i u ü Geophysik und Ex a e es ische Physik, Technische Uni e si ä B aunschweig, Mendelssohns . 3, 38106 B aunschweig, Ge many 3Aix Ma seille Uni , CNRS, CNES, Labo a oi e d’As ophysique de Ma seille, Ma seille, F ance 4DLR Ins i u ü Plane en o schung, Ru he o ds aße 2, 12489 Be lin, Ge many 5Ins i u e o Theo e ical Physics, Johannes Keple Uni e si y Linz, Aus ia 6Alma Ma e S udio um - Uni e si à di Bologna, Dipa imen o di Ingegne ia Indus iale, Via Fon anelle 40, 47121 Fo lì, I aly 7Zuse Ins i u e Be lin, 14195 Be lin, Ge many 8CNRS, Labo a oi e J.-L. Lag ange, Obse a oi e de la Cô e d’Azu , Boule a d de l’Obse a oi e, CS 34229 - 06304 NICE Cedex 4, F ance Recei ed 15 May 2024 / Accep ed 19 Augus 2024 ABSTRACT Aims. We in es iga e he abili y o a simul aneous i ing o come 67P/Chu yumo -Ge asimenko’s non-g a i a ional o ces, o ques, and o al wa e -ou gassing a e, as obse ed by Rose a, o cons ain complex he mophysical models o come a y ma e ial. Me hods. We ex end he p e ious wo k o i ing geog aphically de ined su ace ou gassing models o he Rose a obse a ions by es ing he e ec s o a mo e de ailed geomo phological mapping, he esolu ion o he shape-model used, sel -hea ing by neighbou ing ace s on he shape-model, he mal ine ia in he ou gassing solu ion, and he a ia ion in he momen um coupling be ween he gas and he nucleus. We also di ec ly compa e he non-g a i a ional accele a ion cu es a ailable in he li e a u e. Resul s. We co ec an e o in he calcula ion o pole-o ien a ion in he p e ious pape . We ind ha , unde he assump ions o he model, non-g a i a ional o ces and o ques a e d i en by wa e sublima ion om he nucleus, he mal ine ia and sel -hea ing ha e only mino e ec s, spa ially uni o m ac i i y canno explain 67P’s non-g a i a ional dynamics, spa ially uni o m momen um ans e canno explain 67P’s non-g a i a ional dynamics, and di e en e ain ypes ha e di e en ins an aneous esponses o insola ion. Conclusions. Consolida ed e ain acing sou h on 67P/Chu yumo -Ge asimenko has a high ou gassing lux, a s eep esponse o insola ion, and a la ge gas momen um ans e coe icien . Ins ead, ha acing no h beha es di e en ly, p oducing li le o no wa e ou gassing, and wi h a lowe momen um ans e e iciency. Dus y e ain also has a lowe ou gassing a e and momen um ans e e iciency, and ei he deple es i s ola ile componen o is bu ied in all-back as he come app oaches he Sun. Momen um ans e appea s co ela ed wi h insola ion, likely due o an inc eased enhancemen in he gas empe a u e as he dus i lows h ough is hea ed. Key wo ds. me hods: da a analysis – celes ial mechanics – come s: gene al – come s: indi idual: 67P/Chu yumo -Ge asimenko 1. In oduc ion Come a y ac i i y, comp ising he sublima ion o ola ile ices and he ensuing ejec ion o non- ola ile dus pa icles, is con- olled by he composi ion and s uc u e o he su ace ma e- ial, which is po en ially amongs he mos p imi i e accessible in he Sola Sys em. The e o e, he s udy o ou gassing a es o e he su ace o a pa icula come , alongside he esul - ing non-g a i a ional o ce and o que on he come ’s nucleus, p o ides impo an insigh s in o he e a o plane o ma ion, as well as he subsequen e olu ion o he su ace. Fo come 67P/Chu yumo -Ge asimenko (he ea e 67P), he in es iga ion o ESA’s Rose a spacec a o e a pe iod o wo yea s has esul ed in la ge amoun s o da a being ga he ed, bu no uni- ied model o come a y ac i i y can explain all he obse a ions. I is he e o e wo h e-examining he a ailable da a in o de o ⋆Co esponding au ho ; [email p o ec ed] place u he cons ain s on he mophysical models o come a y ma e ial. The back- eac ion o ce gene a ed by he ou gassing p o- duces a ne accele a ion (non-g a i a ional accele a ion, NGA) ha can be de ec ed in 67P’s ajec o y om he g ound (e.g. Da idsson & Gu ié ez 2005;Gu ié ez e al. 2005) and by Rose a (Mo ola e al. 2020), as well as a non-g a i a ional o que (NGT) ha changes i s spin pe iod (Kelle e al. 2015) and pole-axis o ien a ion. P e ious wo ks (e.g. A ee e al. 2019; K ame e al. 2019;K ame & Läu e 2019) ha e a emp ed o i hese obse ed changes, alongside he o al wa e ou gassing a e, o simple models o su ace ac i i y, bu no single model is able o explain all he da a. Likewise, mo e complex he - mophysical models, such as ha o Da idsson e al. (2022) and Fulle e al. (2020), he la e as applied by A ee e al. (2023), also s uggle o ep oduce he NGA cu es wi hou pos hoc op imisa ion o he pa ame e s. A82, page 1 o 11 Open Access a icle, published by EDP Sciences, unde he e ms o he C ea i e Commons A ibu ion License (h ps://c ea i ecommons.o g/licenses/by/4.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. This a icle is published in open access unde he Subsc ibe o Open model.Subsc ibe o A&A o suppo open access publica ion. A ee, N., e al.: A&A, 690, A82 (2024) Fig. 1. Ex ac ed 67P non-g a i a ional accele a ion cu es, in o al magni ude and he h ee componen s o he come o-cen ic RTN ame, om Lasagni Manghi e al. (2024) (labelled Bologna), Fa nocchia e al. (2021); and K ame & Läu e (2019). The g ey bands show he o mal 10σunce - ain y es ima es om Lasagni Manghi e al. (2024), while he o he models’ unce ain- ies a e omi ed o cla i y. The magni ude plo also shows he o al wa e -p oduc ion a e o Läu e e al. (2020) scaled as desc ibed in he ex . As a b ie desc ip ion o he p ocedu e in ol ed in ou s udy, in ou p e ious pape (A ee e al. 2023), a simple su ace sublima ion model was pa ame ised in e ms o he geog aphic egions de ined on 67P’s su ace (Thomas e al. 2018) in o de o y and ma ch he NGA ex ac ed by Lasagni Manghi e al. (2024); he NGT; and he o al wa e -p oduc ion a e o Hansen e al. (2016). In ou p ocedu e, pole-o ien a ion e olu ion is in en ionally le ou o he i ing p ocess, bu i is used o e - i y he quali y o ou i by compa ing he modelled change de i ed om he solu ion wi h i s ac ual obse a ional change. The i showed an imp o emen o e p e ious a emp s, bu s ill s uggled o ep oduce he non- adial componen s o he NGA simul aneously wi h he NGT. Addi ionally, se e al ac- o s we e no conside ed in de ail, such as he esolu ion o he shape-model used, he e ec s o su ace sel -hea ing and he mal ine ia, and a po en ial a iabili y in he momen um coupling be ween he gas and he nucleus. Addi ional da ase s o he NGA (K ame & Läu e 2019;Fa nocchia e al. 2021), wa e -p oduc ion a e (Läu e e al. 2020), and su ace geo- mo phological mapping (Bi ch e al. 2017) a e also a ailable, bu we e no p e iously used. We he e o e ex end he p e i- ous wo k by inco po a ing he abo e ac o s, while seeking o a ind a s ill ela i ely simple model o pa ame ising su ace ac i i y ha can i he obse a ions and be used as a con- s ain o a ge o he ou pu o mo e complex he mophysical models. The es o he pape is o ganised as ollows. We i s e iew and compa e he a ious ex ac ions o 67P’s non-g a i a ional accele a ions om i s ajec o y in Sec ion 2. Nex , in Sec ions 3 and 4we examine he e ec s o shape-model esolu ion, sel - hea ing, he mal ine ia, and a ying he momen um ans e coe icien on he modelling o 67P’s NGAs and NGTs. In pa icula , in Sec ion 5we p esen ou bes - i ing su ace ac i - i y model. We discuss he implica ions o hese esul s o models o come a y ma e ial in Sec ion 6, and conclude in Sec ion 7. 2. 67P’s non-g a i a ional accele a ion Be o e op imising he NGA models, we i s e-examined he da a ha hey would be compa ed o. The non-g a i a ional o ques ha e been desc ibed elsewhe e (A ee e al. 2019; K ame e al. 2019); below we p esen a b ie compa ison o he accele a ions ex ac ed by h ee di e en au ho s: Lasagni Manghi e al. (2024), Fa nocchia e al. (2021), and K ame & Läu e (2019). As desc ibed in he e e ences, Fa nocchia e al. (2021) i a o a ing-je model o a limi ed se o highly accu a e Rose a adio- anging da a and g ound-based as om- e y collec ed be o e and a e he pe ihelion, while K ame & Läu e (2019) i smoo hed cu es o he esiduals o he a ailable SPICE ke nels om he ESA ligh dynamics solu- ion. Con e sely, Lasagni Manghi e al. (2024) uses a s ochas ic accele a ion model and p ocesses he ull se o Rose a adio- me ic da a ( anging and di e en ial one-way anging, ∆DOR) and g ound-based as ome y a ailable on he Mino Plane Cen- e . To accoun o deg aded accu acy and possible measu emen biases close o pe ihelion, a weigh ing scheme based on he spacec a - o-come dis ance was applied. The Lasagni Manghi e al. (2024) accele a ions ha e been upda ed since being i s p esen ed in Lasagni Manghi e al. (2021) and used in A ee e al. (2023), and we he e o e compa e he h ee cu es he e, wi h hei h ee componen s in he come o-cen ic RTN ame plo ed, along wi h he o al magni ude, in Figu e 1. Also plo - ed o compa ison is he o al wa e -p oduc ion a e, QH2O, om Läu e e al. (2020), scaled o he NGA magni ude by he equa ion o Jewi e al. (2020), among o he s: NGA =kR a QH2O M67P .(1) He e he o bi ally a e aged momen um ans e coe icien o 67P is es ima ed o be kR≈0.5by Jewi e al. (2020), and we use an o bi ally a e aged gas- eloci y o a =500 ms−1(a he mal A82, page 2 o 11 A ee, N., e al.: A&A, 690, A82 (2024) eloci y co esponding o a ound 200 K), along wi h he known come mass M67P(Pä zold e al. 2016). Using hese easonable alues, he NGA magni ude scales well wi h he heliocen ic e olu ion o he wa e -p oduc ion a e, sugges ing ha mos o he NGA is p oduced by wa e sublima - ing om he come a y su ace (ex ended sou ces o wa e in he coma would no con ibu e a back- eac ion o ce on he nucleus). The ag eemen be ween he NGA magni ude and he wa e - p oduc ion a e also suppo s he idea ha NGAs om o he gas species a e negligible (Da idsson e al. 2022;A ee e al. 2023), wi h he possible excep ion o CO2, which exceeds wa e p oduc ion a e he second equinox. The Lasagni Manghi e al. (2024) NGA cu e, al hough small in o e all magni ude he e, sligh ly exceeds he scaled wa e p oduc ion om 200 days a e pe ihelion, possibly hin ing a a CO2ou gassing con ibu ion. None heless, we concen a e o he emainde o his pape on H2O emission ha domina es he la ge magni ude accele a ions ound a ound pe ihelion. The o al magni udes o he h ee NGA ex ac ions a e e y simila . The NGA ec o gene ally ag ees ac oss he h ee wo ks, wi h some mino di e ences. The cu es o K ame & Läu e (2019), o example, con ain mo e noise han he smoo h o a ing-je model o Fa nocchia e al. (2021), while also peaking sligh ly la e a e pe ihelion in he ans e se and no mal com- ponen s. The o a ing-je model has sligh ly mo e accele a ion in hese wo componen s han he o he wo models, al hough i s o e all magni ude is s ill simila . O e all hough, he e is good ag eemen be ween he ex ac ions, while he unce ain y bounds on he Lasagni Manghi e al. (2024) i seem well es ima ed as hey gene ally encompass he o he cu es. We he e o e eel con iden in ocusing on he Lasagni Manghi e al. (2024) accele a ion and unce ain y es ima es in he ollowing sec ions, al hough we ha e also pe o med es -op imisa ions o he ac i - i y model (see below) o all he a ailable accele a ion cu es wi hou d as ic changes in conclusion. 3. Resolu ion, sel -hea ing, and he mal ine ia In an e o o imp o e he ma ch o p e ious models o he a ail- able da a, we now in es iga e some backg ound aspec s in he modelling ha ha e, un il now, mos ly been assumed. These a e he e ec s o shape-model spa ial esolu ion, ace sel -hea ing, and he mal ine ia. Resolu ion was discussed in A ee e al. (2019) and K ame e al. (2019), bu no comp ehensi e analysis o he e ec s o di e en shape-model esolu ions on he NGAs and NGTs has been ca ied ou . In an e o o ind any sys ema ic ends, we he e o e an he exac same analysis me hod wi h ou di e - en e sions o he same shape-model (SHAP7; P euske e al. 2017) wi h esolu ions: 10, 30, 100, and 125 housand ace s ( he las being he same as used in A ee e al. 2019,2023). Iden i- cal wa e -p oduc ion a e cu es we e ound, while NGAs and, in pa icula , NGTs we e seen o a y sligh ly. In Figu e 2we show he esul ing pole-o ien a ion e olu ions, ollowing he algo i hm o A ee e al. (2023) and K ame e al. (2019), o models wi h uni o m ac i i y and an e ec i e ac i e ac ion (EAF) o 8% ( op), and he Fulle e al. (2020) model (bo om). We ocus he e on he pole-o ien a ion because i is e y sensi i e o he shape o he nucleus. None heless, no signi ican ends a e obse ed beyond some sca e in he esul s. Nex , we u ned on sel -hea ing by neighbou ing ace s on he 100 k shape model, using he algo i hm desc ibed in de ail in A ee e al. (2019) o compa ison wi h he esul s o ha pape . Fig. 2. Obse ed pole-o ien a ion e olu ion (RA and Dec) as com- pa ed o esul s o di e en shape-model esolu ions, wi h and wi hou sel -hea ing (SH) and he mal ine ia. Top: o a uni o m EAF o 8%; Bo om: o he model o Fulle e al. (2020). The obse a ions ha e been smoo hed and e o ba s a e plo ed, bu a e small a his scale. Some changes a e seen in he pole e olu ion, pa icula ly o he 8% EAF model; ins ead, he esul s o he Fulle e al. (2020) model, which p oduces a much close i o he obse ed wa e p oduc ion, a e igh ly clus e ed. Finally in his analysis, we included he mal ine ia by using a di e en he mal model (G oussin e al. 2013) ha sol es he 1D hea anspo equa ion o each ace o a simple bulk ma e- ial wi h a he mal ine ia equal o 50 Jm−2s−0.5K−1, ypical o 67P (Gulkis e al. 2015). Due o he long compu ing imes needed o gene a e a hea conduc i i y map o e a comple e o bi al e - olu ion, we used he shape model wi h en housand ace s. As be o e, his was un once wi h no sublima ion, o simula e a pu e dus su ace, and once wi h sublima ion a he su ace, o simula e pu e-ice, and he esul s we e scaled by a cons an 8% EAF. This he mal model is no compa ible wi h ou implemen- a ion o he Fulle e al. (2020) model. Figu e 2( op) shows ha he e ec s o he mal ine ia on he pole e olu ion o he 10k model a e limi ed, wi h only a mino o a ion o he ajec o y o he pole o ien a ion be ween he cases wi h and wi hou he mal ine ia. We a e he e o e con iden ha spa ial esolu ion and he p esence o absence o sel -hea ing and a limi ed he mal ine ia do no ha e a la ge e ec on he pole-o ien a ion e olu ion. Du ing he cou se o his in es iga ion, an e o in he code used o compu ing he pole-o ien a ion e olu ion in A ee e al. (2023) was disco e ed. The e o has been co ec ed and he pole-o ien a ion e olu ion o he bes - i ing solu ions om ha pape is ecompu ed below. A82, page 3 o 11 A ee, N., e al.: A&A, 690, A82 (2024) Fig. 3. Obse ed pole-o ien a ion e olu ion (RA and Dec) compa ed o he co ec ed solu ions o models C, D, and E and he wo e sions o model F om A ee e al. (2023). The hickness o he model lines is due o he daily oscilla ions. Compa e o Figu es 9 and B4 in A ee e al. (2023). Figu e 3shows he esul s o h ee di e en spli s o he su - ace (labelled C, D, and E; see A ee e al. 2023 o de ails) wi h a su ace ene gy-balance he mal model and spa ially a ying and ime- a ying e ec i e ac i e ac ions (EAFs). I can been seen ha he pole-o ien a ion e olu ion is quali a i ely he same as in A ee e al. (2023), Figu e 9, wi h only mino di e ences in he alues. Gi en he inhe en modelling unce ain ies discussed abo e, he p e ious conclusions ega ding hese bes - i models a e unchanged. On he o he hand, he implemen a ion o he di - e en ac i i y model o Fulle e al. (2020) as model F now shows a quali a i ely di e en beha iou , wi h he pole o ien a ion ol- lowing a di e en ajec o y. Compa ing Figu es 2and 3, we see ha a spa ially uni o m ac i i y pa e n will always esul in he pole o ien a ion mo ing in he same di ec ion. This esul can be checked by compa ison wi h he uni o m ac i i y model o K ame e al. (2019) (Fig. 6, g een cu e), eassu ing us ha he upda ed code is co ec . To conclude his p elimina y wo k, we see ha a ia ions in he shape-model esolu ion and he p esence o absence o sel - hea ing and a mode a e he mal ine ia ha e e y li le e ec on he o e all shape o he NGA and NGT cu es. We also no e ha uni o m ac i i y dis ibu ions always p oduce oughly he same pole-o ien a ion e olu ion, and ha his e olu ion is no compa ible wi h he obse a ions. We now mo e on o mo e complica ed, spa ially a ying, ac i i y models. 4. Momen um coupling As desc ibed abo e, p e ious s udies ha e s uggled o simul- aneously ep oduce he obse ed non- adial componen s o he NGA wi h he NGTs a pe ihelion. We ha e seen ha a ying sel -hea ing and he mal ine ia wi h a uni o m su ace ac i - i y has li le e ec on he pole e olu ion, and so, as in p e ious s udies, we now conside ac i i y models ha di ide he su - ace in o supe - egions, each wi h hei own ime- a ying EAF. Fu he , we now in es iga e a a iable momen um ans e coe - icien , η, which was p e iously assumed o be cons an o e he come ’s su ace, being ixed a he beginning o each un. All op imisa ions in his sec ion a e pe o med wi h he algo i hm desc ibed in he p e ious wo pape s (A ee e al. 2019,2023), wi h da a weigh s adjus ed o gi e each da ase app oxima ely equal con ibu ions o he o e all i as be o e. The momen um ans e coe icien , η, desc ibes he collima- ion o he gas low, ela i e o sublima ion om a la pu e-ice su ace, whils also conside ing he amoun o backwa ds, e u n lux as e-condensa ion om he Knudsen laye (C i o 1987). The amoun o collima ion and e u n lux should, in p inciple, depend on he exac loca ion o he gas sublima ion and how i lows h ough he ma e ial (i.e. a he su ace o a dep h, and be ween o wi hin pebbles). Meanwhile, he o al gas momen- um is hen he p oduc o his ac o wi h he mass low- a e and he gas eloci y, which is con olled by i s empe a u e. We p e iously assumed his gas empe a u e o be equal o he calcu- la ed su ace empe a u e o a pu e dus su ace (Tdus in Eq. (5) o A ee e al. 2019), and assumed a ixed η alue co espond- ing o su ace sublima ion. Conside ing he a ia ion in su ace ex u e ac oss he come , and he limi a ions o he su ace subli- ma ion model, i is likely ha bo h o hese ac o s should a y. A p ope ea men o his p oblem equi es a ull he mophysical model, bu he e we can conside a simple modi ica ion o make ou desc ip ion mo e ealis ic. We i s spli he e ec s desc ibed abo e in o a ma e ial-dependen , and an insola ion-dependen componen . The i s akes in o accoun in a ough way he dep h s uc u e o he ou gassing ma e ial, which likely a ies ac oss he come , by simply pa ame ising a di e en momen um ans- e coe icien o each supe - egion. The second pa accoun s o a po en ially enhanced insola ion dependence o he emi - ed gas- empe a u e compa ed o a simple su ace-sublima ion model. Fo example, Sko o e al. (2024) show ha gas lowing h ough a po ous dus -laye can acqui e a empe a u e some- whe e be ween i s ini ial sublima ion empe a u e, Tice, and he dus empe a u e, wi h he oughly linea dependence on he insola ion. We he e o e ew i e ou exp ession o he magni ude o he ou gassing o ce pe ace as F=η·EAF ·Z 8RgasTice πM· Tdus Tice !,(2) whe e ηand EAF a e he local ma e ial momen um ans e coe - icien and a ime-dependen e ec i e ac i e ac ion (gi en by Eq. (3) below), Zis he local sublima ion a e de e mined by he ene gy balance, and Rgas and Ma e he gas cons an and molecula mass. The gas and dus empe a u es a e sol ed o pu e wa e -ice and pu e dus su aces as desc ibed in he p e i- ous pape s, esul ing in a ac o in b acke s anging be ween 1 and ∼1.9and co ela ed wi h inc easing insola ion (see Fig. 4). We can conside he p oduc o his ac o wi h ηas he ull momen um ans e coe icien in ou model, while bea ing in mind ha i is dependen on he speci ic ene gy-balance model A82, page 4 o 11 A ee, N., e al.: A&A, 690, A82 (2024) 400 300 200 100 0 100 200 300 400 Days om Pe ihelion 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Mean daily Tdus / Tice Dus y N Dus y S Rocky N Rocky S 400 300 200 100 0 100 200 300 400 Days om Pe ihelion 0 100 200 300 400 500 Mean daily insola ion (W m 2) Fig. 4. Fac o s ela ing o momen um enhancemen , a e aged o e each o he ou supe - egions used in ou bes - i solu ions. Top: mean daily momen um enhancemen ac o (Tdus /Tice, as used in Eq. (2)), and Bo om: mean daily insola ion. used he e and ha compa isons wi h alues de i ed using o he he mophysical models a e complica ed. We e- an he bes - i solu ions o A ee e al. (2023) (solu- ions D and E) wi h he upda ed momen um coupling model, while also modi ying he way EAF le els a e pa ame ised. Ins ead o he p e ious ime- a ia ion model ha used a base and a peak EAF le el, linked by hal -Gaussian p o iles, we now mul iply he peak EAF by a gene alised asymme ic Gaussian in ime (e.g. Nace eddine e al. 2010), EAF( ) =         EAFpeakexp h− 0− si, o < 0, EAFpeakexp − − 0 s, o > 0,(3) whe e each supe - egion is hen pa ame ised by i s peak EAF le el, peak ime 0, wid hs o i s ise- ime , and all- ime , and a shape ac o ha allows he p o ile o be adjus ed o be mo e o less s eep han he s anda d s=2Gaussian. Alongside he spa ially a ying momen um ans e coe icien , his leads o 6 ee pa ame e s pe each o he i e o six supe - egions, o a o al o 30 o 36 pa ame e s ( o compa ison, K ame e al. 2019 also ha e 36 pa ame e s). Some small imp o emen s o he i we e achie ed, pa icu- la ly in he R and T componen s o he NGA, as he asymme ic shapes and a ying momen um ans e coe icien s elaxed he limi s on p e ious solu ions. Howe e , simila p oblems emained, as discussed p e iously, in e ms o simul aneously i ing he NGA and NGT. In pa icula , he ac ha he supe - egions de ined in he A ee e al. (2023) models D and E Fig. 5. Supe - egions used in he bes - i solu ions, shown wi h hei EAF as a unc ion o ime o solu ion 1 and mapped on o he shape model wi h he co esponding colou s. co e a eas o bo h come a y hemisphe es limi s hei abili y o gene a e he la ge non- adial NGA peaks seen a pe ihelion. A his poin we also a emp ed a i using he pe ace geological mapping o Bi ch e al. (2017), ins ead o he a he b oad geog aphic egions de ined by Thomas e al. (2018). The mapping o Bi ch e al. (2017) di ides he su ace in o se en e - ain ypes, which b oadly ollow he p e ious classi ica ion in o gene ally dus y and gene ally ocky e ains, wi h some mo e p ecise di e en ia ion o di e en ypes o dus y plain and alus ma e ials. Un o una ely, nei he i s o he ull se en e ain ypes (7×6=42 ee-pa ame e s), no a g ouping in o ‘Dus y’ and ‘Rocky’ e ains (2×6=12 ee-pa ame e s) could p oduce sub- s an ial imp o emen s o he esul s abo e, o o he esul s o ou p e ious wo k. Looking a he maps in Figu e 7 in Bi ch e al. (2017), i can be seen ha many e ain ypes ex end o e bo h come hemisphe es, in pa icula he ‘cli ’ e ain ha domi- na es ac i i y a pe ihelion. We hus e u n o he same p oblem discussed abo e. 5. Bes - i ing su ace models Based on he esul s o Sec . 4, we decided o pu sue a simple model, spli ing e ains ha c oss bo h hemisphe es in o no h- e n and sou he n a eas. We i s combined he Thomas e al. (2018) Dus , B i le, Smoo h, and Dep ession e ains in o a sin- gle Dus y supe - egion. This Dus y, and he emaining Rocky, supe - egions we e hen spli by hei NGA no mal componen a pe ihelion in o no he n and sou he n supe - egions, esul ing in a o al o ou supe - egions. This spli ing by an accele a ion componen (i.e. ace o ien a ion a he han geog aphic la i ude) was mo i a ed by he ac ha i is su ace o ien a ion no la i ude ha ac ually con ols insola ion and ou gassing om a pa ic- ula ace . The esul ing supe - egion de ini ion is mapped in Figu e 5, and he model con ains 4×6=24 ee-pa ame e s. A82, page 5 o 11 A ee, N., e al.: A&A, 690, A82 (2024) Fig. 6. Resul s o he bes - i solu ion (solu ion 1) and he no malised esid- uals o ( om op le ): o al wa e - p oduc ion a e, adial NGA compo- nen , angen ial NGA componen , no - mal NGA componen , z componen o he o que, and pole o ien a ion. Indi- idual con ibu ions om each supe - egion a e shown alongside he o als and he obse a ional da a (see ex in his pape , A ee e al. 2019, and A ee e al. 2023 o e e ences). Figu e 6shows he esul s o he i agains he obse ed o al wa e -p oduc ion a e, adial, angen ial, and no mal NGA com- ponen s ( om Lasagni Manghi e al. 2024), and z componen o he o que, as well as he ma ch o he pole o ien a ion. As a eminde , his las da ase is no included in he i ing ou- ine, bu is ou pu by he model o compa ison. As can be seen, his model p o ides a signi ican imp o emen o e he p e i- ous solu ions, wi h an o e all excellen ma ch o all he i ed da ase s, and a easonable ma ch o he pole o ien a ion. The o al o a ion-pe iod change o 22.5 minu es is also e y close o he measu ed 21 minu es (Mo ola e al. 2014;Jo da e al. 2016). Fo he pole-o ien a ion e olu ion, he aw da a a e plo ed along- side he smoo hed da a used in he p e ious plo s, demons a ing ha , while ou compu ed solu ion displays signi ican diu nal oscilla ions and is no a pe ec ma ch o he da a, i does show oughly he co ec magni ude o change and a simila sca e . Conside ing he mino o a ion o he pole-o ien a ion e olu ion achie ed in Sec ion 3abo e, we e- an he i using a he mal ine ia o I=50 he mal ine ia uni s on he 10k shape-model. Only e y mino di e ences we e obse ed in he NGA and wa e -p oduc ion cu es, bu he pole-o ien a ion e olu ion was indeed o a ed sligh ly, as shown in he bo om igh o Fig. 6, now coming e y close o he obse a ions. The emaining di - e ences be ween he modelled and obse ed pole-o ien a ion e olu ions may be explained by a ia ions in his he mal ine - ia, pa icula ly spa ially. Mino albedo o emissi i y a ia ions may also play a ole, as will he pa icula i ies o disc e ising he egions on he shape model and he ini ial o a ion condi ions. F om he wa e -p oduc ion plo in Fig. 6, i can be seen ha he no he n dus y e ain domina es he o al wa e ou gassing un il oughly he inbound equinox; ins ead, a e he equinox, he ocky sou he n e ain akes o e and domina es un il he end o he Rose a pe iod. A ound he equinox i sel he e is a b ie pe iod when he ocky no he n e ain p o ides he la ges con- ibu ion. In e ms o he NGAs and NGTs, he ocky sou he n e ain comple ely domina es (in ag eemen wi h ou p e ious wo k, as well as o he s udies: K ame e al. 2019), while he addi ion o a ela i ely ac i e no he n dus y e ain p oduces he nega i e peak in o que seen be o e pe ihelion. O e all, he sou he n dus y e ain con ibu es li le o he p oduc ion, NGA, A82, page 6 o 11 A ee, N., e al.: A&A, 690, A82 (2024) Table 1. Momen um ans e coe icien s, η, in he bes - i solu ions. Solu ion Dus y N Dus y S Rocky N Rocky S One 0.56 0.15 0.12 0.62 Two 0.85 0.10 0.10 0.57 Fig. 7. EAF as a unc ion o ime in bes - i solu ion 2. o NGT, in line wi h i s small geog aphical co e age. As shown in Figu e 5, he abo e esul s a e achie ed by a high ac i i y peak o EAF ≈15% in he sou he n ocky e ain a pe ihelion, while all o he e ains peak a a ound 2−4%, wi h no he n egions cons ained o all o in EAF be o e pe ihelion. This educ- ion in ela i e ac i i y o he no h as he come app oaches he Sun is a common ea u e o model i s he e, and is impo - an o simul aneously i ing he peak NGA, NGTs, and wa e p oduc ion. In e ms o momen um ans e , Table 1shows ha he ocky sou he n e ain has he highes basic alue o η, o go along wi h i s high EAF. The dus y no he n e ain has simila i sligh ly lowe η; he wo o he egions mus be much less e icien a ans e ing momen um. I mus be emembe ed, howe e , ha his base alue is mul iplied by he empe a u e-dependen ac- o (in b acke s in Eq. (2)) when calcula ing he o ce pe ace , enhancing ηby a ac o ha peaks a pe ihelion and in he day- ime, concen a ing NGAs and NGTs diu nally and seasonally, as shown a he op o Figu e 4. A hei mos ac i e imes, when hei EAF cu es peak, he diu nally a e aged o al η al- ues o no he n dus and sou he n ock a e a ound 0.75 and 1, espec i ely, wi h he o he wo egions much lowe . Conside ing he esiduals o he wa e -p oduc ion cu e ( op le in Fig. 6), we see ha he e a e wo ou lie poin s loca ed nea pe ihelion and he i s equinox. The ela i ely small e o ba s o hese poin s combined wi h hei high alues d i e he o e all i s a is ic, limi ing he abili y o he i ing ou ine o imp o e he model. The e o ba s ake in o accoun only he limi ed su ace co e age o he Rose a O bi e Spec ome e o Ion and Neu al Analysis (ROSINA) ins umen (Läu e e al. 2020), and he e- o e may unde es ima e he ac ual unce ain y, which is isible in he sca e o he o he da a poin s. We he e o e e- an he i wi h hese wo p oduc ion poin s emo ed in o de o e alua e hei e ec . Figu es 7and 8show he EAF cu es and i o he da a o his new solu ion, which we e e o as solu ion 2. The ma ch o he NGAs and NGTs is e y simila in his case, while he p o- duc ion esiduals a e imp o ed, leading o an o e all somewha imp o ed i s a is ic. The o a ion-pe iod change is now e en close o he measu emen s a 21.4 minu es. Figu e 7shows ha solu ion 2 has a sligh ly highe sou he n ocky ac i i y (≈18%), and a much highe sou he n dus y ac i i y han be o e, along- side ze o EAF in he ocky no he n e ain. The no he n dus alls o as be o e. The o al wa e p oduc ion is hen domina ed by he no he n dus , smoo hly ansi ioning o sou he n ock a he equinox, wi h a mino con ibu ion om sou he n dus , again due o i s small a ea. Table 1shows ha he momen um ans e coe icien s ollow a simila pa e n o be o e, bu now wi h he no he n dus y e ain ha ing he highes alues wi h espec o he sou he n ock. When mul iplied by he enhancemen ac o o Eq. (2), he o al momen um ans e coe icien s a e a ound 1.1 and 0.9 o hese wo egions, espec i ely. The ocky no he n e ain’s momen um ans e coe icien has li le meaning due o i s negligible EAF. Simila ly o abo e, e- unning solu ion 2 wi h he mal ine ia led o almos exac ly he same NGA cu es and z componen o he o que, while he pole-o ien a ion e olu ion was sligh ly o a ed (bo om igh panel o Fig. 8). Mino a ia ions in hese i s could be achie ed by a ying he weigh ing and e o ea men o he a ious da ase s, bu he compe ing es ic ions placed by he NGA and NGT cu es ensu ed ha no signi ican imp o emen s could be ound using his model. Finally, we hus conside his ou bes - i model o 67P’s wa e ou gassing pa e n; solu ion 2 is ou p e e ed choice, due o i s imp o ed i s a is ic. 6. Implica ions o come a y ac i i y models In gene al, he e is a dis inc change in he ac i i y a equinox, om being d i en by he no he n hemisphe e o being d i en by he sou he n, esul ing in di e en beha iou (as has also been no ed in o he wo ks, e.g. Da idsson e al. 2022). Rega d- ing NGAs, he accele a ions a e small in magni ude ou side o he equinoxes, and a e he e o e no use ul in cons aining ac i - i y a la ge heliocen ic dis ances, bu a e comple ely domina ed by ac i i y be ween he wo equinoxes. This means ha hey a e mos sensi i e o he sou h (which domina es ac i i y be ween he equinoxes); howe e , he high no mal and angen ial NGA componen s cons ain no he n ac i i y o be low du ing his pe iod, while simul aneously cons aining he sou he n ac i - i y o be ela i ely e y high and a he e icien a ans e ing momen um. Con e sely, NGTs a e sensi i e o bo h no he n and sou h- e n ac i i y du ing he Rose a mission p oximi y ope a ions, o example he i s peak (which is nega i e) in he z o que compo- nen (co esponding o an obse ed inc ease in he spin-pe iod) is d i en by ac i i y om he no h a he i s equinox, whe eas he second peak (which is posi i e, and co esponds o he o e - all la ge dec ease in spin-pe iod) is d i en by sou he n ac i i y (speci ically, he no he n dus and he sou he n ock, espec- i ely, in ou bes - i model). Likewise, he pole-o ien a ion e olu ion appea s sensi i e o all a eas o he come a y su ace. Finally, he wa e -p oduc ion a e cu e o Läu e e al. (2020) equi es high ela i e ac i i y a pe ihelion (al hough peak p oduc ion is sligh ly less han es ima ed by Hansen e al. 2016; see discussion in Läu e e al. 2020), bu also an asymme- y, wi h highe p oduc ion o a pa icula ime p e-pe ihelion as o he co esponding pos -pe ihelion ime. F om he abo e, and he p e ious h ee sec ions, a numbe o conclusions ega ding come a y ac i i y on 67P can be made: 1. Wa e sublima ing om he nucleus is he p ima y d i e o non-g a i a ional o ces and o ques. The analysis o he magni udes o he NGA and wa e -p oduc ion cu es in A82, page 7 o 11 A ee, N., e al.: A&A, 690, A82 (2024) Fig. 8. Resul s o he bes - i solu- ion (solu ion 2) and he no malised esiduals ( om op le ): o al wa e - p oduc ion a e, adial NGA compo- nen , angen ial NGA componen , no - mal NGA componen , z componen o he o que, and pole o ien a ion. Indi- idual con ibu ions om each supe - egion a e shown alongside he o als and he obse a ional da a (see ex in his pape , and A ee e al. 2019 and A ee e al. 2023 o e e ences). Sec ion 2is consis en wi h wa e sublima ing om he su - ace, a he han o he gas species o wa e om ex ended sou ces, being he main d i e o 67P’s non-g a i a ional dynamics. 2. The mal ine ia and sel -hea ing ha e only mino e ec s on non-g a i a ional o ces and o ques. In some p e ious s udies, he mal ine ia is hough o d i e he non- adial componen s o he NGA, bu , a leas o ou bes - i ing model o 67P, he e ec s o a mode a e he mal ine ia on NGA and NGT a e limi ed o a mino o a ion o he pole-o ien a ion e olu ion. 3. Spa ially uni o m ac i i y canno explain 67P’s non- g a i a ional dynamics. The analysis om Sec ion 3, oge he wi h p e ious wo ks, shows ha when he esponse o he su ace o insola ion is he same e e ywhe e on he come , hen he esul ing pole-o ien a ion e olu ion does no ma ch he Rose a obse a ions. This is he case e en when he insola ion esponse has a non-linea dependence. A eas o he come wi h o he wise simila ins an aneous illumina ion le els (see A ee e al. 2023, Fig. 12, and ou Fig. 4) mus supply di e en amoun s o ou gassing in o de o gene a e he obse ed NGTs. We no e ha his does no comple ely ule ou models such as ha o Fulle e al. (2020), bu imposes s ong cons ain s on hem in e ms o he su - ace esponse o sola ene gy inpu : some a eas mus show an enhanced ou gassing compa ed o o he s, by ma e ial di - e ences o by a eedback e ec om long- e m insola ion di e ence o all-back, o example. 4. Spa ially uni o m momen um ans e canno explain 67P’s non-g a i a ional dynamics. Likewise, i appea s impossible o simul aneously i he non-g a i a ional o ces and o ques using models wi h a single ixed alue o η, he momen um ans e coe icien . Ins ead, ηmus a y ac oss he su ace and/o wi h insola ion, wi h di e en e ain ypes being mo e o less e icien a collima ing he ou gassing, and a s ong enhancemen wi h insola ion being a ou ed. 5. Di e en e ain ypes ha e di e en ins an aneous esponses o insola ion. Imp o emen s o he i o A ee e al. (2023) a e achie ed by ha ing he sou h- acing ocky e ain exhibi a di e en ac i i y esponse o he A82, page 8 o 11 A ee, N., e al.: A&A, 690, A82 (2024) no h- acing. Only in his way can he peak NGAs a pe i- helion be i . This is he case whe he spli ing he su ace by b oad geog aphic egion, as in Thomas e al. (2018), o by ea u e-scale geomo phology, as in Bi ch e al. (2017). In bo h cases, e ain ha appea s isually he same beha es di e en ly depending on whe he i aces gene ally no h o sou h. The second and las poin s can be mo e easily isualised when plo ing he a e aged daily insola ion cu es o each supe - egion, as done in Figu e 4. Sou he n- acing e ains clea ly ecei e he mos insola ion, bu he loca ion o some no he n ocky e ains means ha hey oo ecei e a la ge ene gy inpu a pe ihelion. Insola ion o he no he n dus peaks a ound he i s equinox and hen begins o decline o pe ihelion. When compa ed o ou bes - i ing EAF cu es (Figs. 5and 7), he e is a gene al co ela ion be ween inc easing insola ion and peak EAF (i.e. blue hen o ange hen ed cu es, om lowes o high- es o bo h pa ame e s), bu he g een ocky no h cu e does no ollow he end. Likewise, he shapes o he wo dus y cu es a e di e en om he ocky ones in solu ion 2, and bo h solu ions ha e peak EAF in he dus y no h eached be o e peak insola ion, ollowed by a decline whils insola ion emains high. A simila bu in e se decoupling o insola ion and he gas emission a e was obse ed by Läu e e al. (2022) o CO2. In hei no he n pa ch egions close o he pe ihelion passage, he diu nally a e - aged insola ion dec eases and he gas p oduc ion inc eases a he same ime. O e all hen, he ques ion a ises o wha his implies o he su ace ma e ial o 67P. Ou bes - i model ein o ces he idea ha sou he n ocky e ain is composed o ela i ely p is ine, ola ile- ich ma e ial (see e.g. Cambianica e al. 2020; He ny e al. 2021;Da idsson e al. 2022) ha p oduces a high wa e -ou gassing a e wi h a e y s eep inc easing esponse o insola ion. This may be simila o he wa e enhanced blocks (WEBs) composed o pebbles (as p oposed by Fulle e al. 2019,2020; and Cia niello e al. 2021,2023), o may e lec an enhancemen in ou gassing om a mo e deeply pene a ing hea -wa e due o he cons an illumina ion o pola day (Sko o e al. 2020). This la e sugges ion is en a i ely suppo ed by he ac ha sou he n ocky e ain mus also ha e a ela i ely la ge momen um ans e coe icien , especially in solu ion 1 (i.e. a sublima ion ha is e y collima ed, possibly because i happens a a deep le el o is o he wise e icien ly channelled). In ou p e e ed solu ion 2, howe e , no he n dus has an e en highe momen um ans e , which complica es his in e p e a- ion. The gene al imp o emen o he i s using an insola ion dependen momen um ans e , as desc ibed by Eq. (2), implies a co ela ion be ween inc easing sola inpu and inc easing gas empe a u e ha is g ea e han he simple ene gy balance model. This enhancemen is less p onounced in he Fulle e al. (2020) model, due o i s lowe su ace empe a u es compa ed o a pu e dus su ace. Su aces ha a e smoo h on a local scale migh also be expec ed o ha e a highe momen um ans e o e all han ough su aces, whe e pa s o he gas low will cancel ou . This would sugges highe ηin he smoo h dus y plains, which is seen in solu ion 2, bu no in solu ion 1. On he o he hand, ough su aces may be ho e han smoo h ones, depending on he scale o he oughness. Ou gassing speed, mass, and momen um lux a e clea ly coupled o he ma e ial s uc u e and su ace shape in a complex way ha equi es u he he mophysical modelling o disen angle. No h- acing ocky e ains, al hough simila in isual appea ance, beha e di e en ly o sou h- acing ones, wi h much lowe peak ou gassing and momen um ans e . In ou solu ion 1, peak EAF o no h- acing ock is simila o ha o sou h- acing dus , whe eas solu ion 2 p e e s no ou gassing a all om he o me egion. Bo h solu ions ag ee, howe e , ha his e - ain p oduces less ou gassing han sou h- acing ock. This may e lec di e ing he mal his o ies in nea - e ical cli s e sus la plains in he no he n and sou he n hemisphe es, espec i ely, o i may ep esen a s uc u al o composi ional di e ence. In he la e case, cli s such as hose in Geb would ha e di e en p op- e ies o nea by a eas on bo h lobes such as Nei h, Wos e , and Bes, while pa s o Anhu di e ed om one ano he . I is di i- cul o u he cons ain hese di e ences wi h he a ailable da a, howe e , while ou gassing and o he Rose a measu emen s gen- e ally do no sugges such sha p composi ional dis inc ions (see e.g. Läu e e al. 2020;Pä zold e al. 2016;G oussin e al. 2019). Dus y ma e ial has a di e en ly shaped ou gassing cu e; in pa icula , no h- acing dus inc eases i s EAF wi h inc easing insola ion, bu eaches lowe peak alues han he sou he n ocky e ain, and hen begins o decline, whils insola ion emains high. The no he n plains’ con ibu ion o he o e all wa e p oduc ion hen alls apidly and becomes negligible owa ds pe ihelion, p o iding an e en smalle con ibu ion han no h- e n pola win e would sugges , while i also does no eco e a e he second equinox, leading o he asymme ic p oduc ion cu e men ioned abo e. This may be because no he n dus y plains a e composed o all-back ma e ial ha has been deple ed in ola iles (Cambianica e al. 2020) and i begins o un ou o ice wi h inc easing insola ion. Al e na i ely, he ongoing all- back o ma e ial om he mo e ac i e sou h may be bu ying he plains in an e e deepe d y-dus laye as i s ac i i y dec eases. The ac ha 67P’s ac i i y appea s ela i ely consis en be ween o bi s, bo h in e ms o i s ou gassing cu e (Snodg ass e al. 2017) and o a ion- a e changes (Mo ola e al. 2014;Jo da e al. 2016), means ha ei he way he no he n plains mus be esup- plied by esh all-back wi h a leas some ola ile con en in o de o epea he cycle (Kelle e al. 2017). Con e sely, i mus no be eac i a ed a e pe ihelion in his same o bi despi e he ac ha insola ion condi ions a e simila o he inbound o bi (Fig. 4). Da idsson e al. (2021) and Da idsson e al. (2022) oge he sugges a possible explana ion whe eby app oxima ely cen ime e-sized all-back pa icles de elop an insula ing man- le du ing hei ligh - ime, allowing hem o main ain 85−95% o hei o e all wa e ice con en . Then, ei he he sligh ly di - e ing insola ion condi ions combined wi h he man le es ic s hei ou gassing un il a mo e a ou able insola ion inbound nex o bi a ou s gas elease o hey su e subs an ial al e - a ion du ing aphelion, due o he mal a igue p ocesses. This al e a ion o he cha ac e is ics could acili a e he elease o ola ile ma e ial on he nex inbound o bi . Kelle e al. (2017) ins ead explain he eac i a ion as being limi ed o Hapi, which emains shadowed and cold e en a e he es o he dus y no h begins o be illumina ed again a e pe ihelion, allowing ola ile- ich all-back pa icles o su i e he e un il he nex appa i- ion. The iming and magni ude o he dec easing ou gassing esponse should be s udied in mo e de ail o cons ain hese scena ios. Sou h- acing dus y e ain co e s a e y small a ea, and hence i is ha d o cons ain i s ac i i y. Ou p e e ed solu- ion (solu ion 2), howe e , has i beha ing in a simila way o no h- acing dus , bu wi h he mo e in ense insola ion lead- ing o a highe peak EAF. Once again hough, EAF alls o be o e ecei ing peak insola ion, which may e lec he same deple ion o ola iles in he pa ially dehyd a ed all-back ma e- ial as abo e. Once again, his ma e ial mus be eac i a ed a he nex pe ihelion passage o main ain 67P’s epea ing A82, page 9 o 11