Characterization of the Ejecta from the NASA/DART Impact on Dimorphos: Observations and Monte Carlo Models

Cha ac e iza ion o he Ejec a om he NASA/DART Impac on Dimo phos: Obse a ions and Mon e Ca lo Models Fe nando Mo eno 1 , Ad iano Campo Baga in 2,3 , Gonzalo Tanc edi 4 , Jian-Yang Li 5 , Alessand o Rossi 6 , Fabio Fe a i 7 , Masa oshi Hi abayashi 8 , Eugene Fahnes ock 9 , Alain Mau y 10 , Robe Sandness 10 , And ew S. Ri kin 11 , Andy Cheng 11 , Tony L. Fa nham 12 , S e ania Soldini 13 , Ca mine Gio dano 7 , Gianma io Me isio 7 , Paolo Panicucci 7 , Ma ia Puglia i 7 , Albe o J. Cas o-Ti ado 1,14 , Emilio Fe nández-Ga cía 1 , ignacio Pé ez-Ga cía 1 , S a o I ano ski 15 , An i Pen ila 16 , Ludmilla Kolokolo a 17 , Ja ie Licand o 18,19 , Olga Muñoz 1 , Zu i G ay 20,21 , Jose L. O iz 1 , and Zhong-Yi Lin 22 1 Ins i u o de As o ísica de Andalucía, CSIC Glo ie a de la As onomía, s/n, E-18008 G anada, Spain 2 Ins i u o de Fìsica Aplicada a las Ciencias y las Tecnologías (IUFACyT), Uni e sidad de Alican e, San Vicen del Raspeig, E-03690 Alican e, Spain 3 Depa amen o de Física, Ingenie ía de Sis emas y Teo ía de la Señal Uni e sidad de Alican e, San Vicen del Raspeig, E-03690 Alican e, Spain 4 Depa amen o de As onomía, Facul ad de Ciencias, Iguá 4225, 11400 Mon e ideo, U uguay 5 Plane a y Science Ins i u e, Tucson, AZ, USA 6 FAC-CNR, Via Madonna del Piano 10, I-50142, Ses o Fio en ino, I aly 7 Depa men o Ae ospace Science and Technology, Poli ecnico di Milano, Milano, I aly 8 Aubu n Uni e si y, Aubu n, AL, USA 9 Je P opulsion Labo a o y, Cali o nia Ins i u e o Technology, Pasadena, CA, USA 10 SPACEOBS, San Ped o de A acama, Chile 11 Johns Hopkins Uni e si y Applied Physics Labo a o y, Lau el, MD, USA 12 Uni e si y o Ma yland, Depa men o As onomy, College Pa k, MD, USA 13 Depa men o Mechanical, Ma e ials and Ae ospace Enginee ing, Uni e si y o Li e pool, Li e pool, UK 14 Unidad Asociada al CSIC, Depa amen o de Ingenie ía de Sis emas y Au omá ica, Escuela de Ingenie ías, Uni e sidad de Málaga, Málaga, Spain 15 INAF—Osse a o io As onomico di T ies e, Via G.B. Tiepolo, 11, T ies e, I aly 16 Depa men o Physics, P.O. Box 64, FI-00014, Uni e si y o Helsinki, Finland 17 Depa men o As onomy, Uni e si y o Ma yland, College Pa k, MD, USA 18 Ins i u o de As o ísica de Cana ias C/Vía Lác ea s/n, E-38205 La Laguna, Spain 19 Depa amen o de As o ísica, Uni e sidad de La Laguna, E-38206 La Laguna, Tene i e, Spain 20 A magh Obse a o y & Plane a ium, College Hill, A magh, BT61 9DG, UK 21 Mulla d Space Science Labo a o y, Depa men o Space and Clima e Physics, Uni e si y College London, Holmbu y S . Ma y, Do king, Su ey RH5 6NT, UK 22 Ins i u e o As onomy, Na ional Cen al Uni e si y, No. 300, Zhongda Road, Zhongli Dis ., Taoyuan Ci y 32001, Taiwan Recei ed 2023 June 2; e ised 2023 July 7; accep ed 2023 July 16; published 2023 Augus 7 Abs ac The NASA Double As e oid Redi ec ion Tes (DART)spacec a success ully c ashed on Dimo phos, he seconda y componen o he bina y (65803)Didymos sys em. Following he impac , a la ge dus cloud was eleased, and a long-las ing dus ail de eloped. We ha e ex ensi ely moni o ed he dus ail om he g ound and he Hubble Space Telescope. We p o ide a cha ac e iza ion o he ejec a dus p ope ies, i.e., pa icle size dis ibu ion and ejec ion speeds, ejec ion geome ic pa ame e s, and mass, by combining bo h obse a ional da a se s and using Mon e Ca lo models o he obse ed dus ail. The size dis ibu ion unc ion ha bes fi s he imaging da a is a b oken powe law ha ing a powe index o –2.5 o pa icles o 3 mm and –3.7 o la ge pa icles. The pa icles ange in size om 1 μm up o 5 cm. The ejec a is cha ac e ized by wo componen s, depending on eloci y and ejec ion di ec ion. The no he n componen o he double ail, obse ed since 2022 Oc obe 8, migh be associa ed wi h a seconda y ejec ion e en om impac ing deb is on Didymos, al hough is also possible ha his ea u e esul s om he bina y sys em dynamics alone. The lowe limi o he o al dus mass ejec ed is es ima ed a ∼6×10 6 kg, hal o his mass being ejec ed o in e plane a y space. Unified As onomy Thesau us concep s: As e oid dynamics (2210) 1. In oduc ion The Double As e oid Redi ec ion Tes (DART)is a NASA mission ha impac ed a spacec a on he su ace o Dimo phos, he sa elli e o he p ima y as e oid (65803) Didymos (Cheng e al. 2018). On 2022 Sep embe 26, 23:14 UT, DART impac ed in a nea ly head-on configu a ion on Dimo phos’s su ace, gi ing ise o a as ejec ed ma e ial (plume; speed o ≈2kms −1 )whose spec um consis s o emission lines o ionized alkali me als (Na I, K I, and Li I; Shes ako a e al. 2023). This plume was clea ly obse ed in images ob ained om Les Makes Obse a o y (G aykowski e al. 2023) igh a e impac ime and was also seen in he ea lies images du ing he Hubble Space Telescope (HST) moni o ing (Li e al. 2023). A wide ejec ion cone o dus pa icles and me e -sized boulde s was moni o ed by he Ligh I alian CubeSa o Imaging o As e oid (LICIACube; Do o e al. 2021; Fa nham e al. 2023), which pe o med a as flyby o he sys em. Apa om he plume, a ac ion o he ejec ed mass was emi ed a significan ly lowe speeds, o ming he ejec a pa e n and ail ha could be seen on he ea lies images acqui ed om g ound-based obse a o ies (Bagnulo e al. 2023; Opi om e al. 2023)and he HST (Li e al. 2023). Ou pu pose is o cha ac e ize he dus p ope ies o his mos ly slow-mo ing ejec a, he ejec ion eloci ies, he size dis ibu ion, and he The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus h ps://doi.o g/10.3847/PSJ/ace827 © 2023. The Au ho (s). Published by he Ame ican As onomical Socie y. O iginal con en om his wo k may be used unde he e ms o he C ea i e Commons A ibu ion 4.0 licence. Any u he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)and he i le o he wo k, jou nal ci a ion and DOI. 1 ejec ed mass using Mon e Ca lo models o simula e he mo ion o he pa icles in he spa ial egion nea he bina y sys em. A e desc ibing he ejec a obse a ions in Sec ion 2,in Sec ion 3, we in oduce he Mon e Ca lo models used o calcula e he syn he ic ail b igh ness and hei ime e olu ion and discuss he esul s ob ained. In Sec ion 4, we p o ide a compa ison o he DART esul s wi h hose ob ained om he obse a ion o na u al ac i e as e oids. Finally, he conclusions a e gi en in Sec ion 5. 2. Obse a ions We fi s desc ibe he obse a ional ma e ial acqui ed om he g ound, ollowed by a b ie desc ip ion o he HST obse a ions (Li e al. 2023). Table 1summa izes he echnical da a o he ins umen a ion used. Ape u e pho ome y o he bina y sys em was pe o med using he BOOTES-1 elescope. The Bu s Obse e and Op ical T ansien Explo ing Sys em (BOOTES)is a wo ldwide obo ic elescope ne wo k p ima ily designed o de ec and ollow gamma- ay bu s s (Cas o-Ti ado e al. 2012; Hu e al. 2023). The ape u e pho ome y measu emen s we e pe o med using BOOTES-1, which is a 0.3 m ape u e elescope loca ed in he Es ación de Sondeos A mos é icos in he Cen o de Expe imen ación, El A enosillo, Huel a, Spain. The ape u e size was selec ed au oma ically in he ange 6″–7″, depending on he seeing condi ions. The pho ome ic da a we e calib a ed using s anda d s a s in he Gaia G-band sys em (Weile 2018). The g ound-based images we e acqui ed om a p i a e obse a o y loca ed in he A acama dese (Chile)called San Ped o de A acama Celes ial Explo a ions (SPACEOBS), which is un by Alain Mau y. The A acama dese is an excellen place o as onomical obse a ions, wi h low humidi y and good anspa ency and seeing condi ions. All o he obse a- ions we e pe o med wi h a CCD came a moun ed on a 0.43 m ape u e elescope. The echnical in o ma ion on he ins u- men a ion used is displayed in Table 1. Images we e acqui ed om he impac da e (2022 Sep embe 26) o la e 2022 Decembe on 56 epochs in o al. The images we e acqui ed using a nonside eal acking mode, i.e., by acking on he bina y sys em, always using an exposu e ime o 300 s. The educ ion o he images was pe o med by s anda d echniques, including bias sub ac ion and fla -fielding. The sky back- g ound was es ima ed in each image by aking a median alue o field s a - ee egions in each ame. A median image was ob ained on each nigh by s acking up all o he a ailable educed images. The images we e calib a ed o magni udes a csec −2 using he pho ome ic da a om BOOTES-1 un il 2022 Oc obe 20. A la e epochs, we assumed o calib a ion o he images he V-band magni ude alues ob ained om he JPL-Ho izons web in e ace 24 o he Didymos sys em, as he ail con ibu ion is essen ially negligible on hose da es. This in ol es he assump ion ha he “naked”sys em has no expe ienced any b igh ness a ia ion om pos - o p eimpac condi ions, which is confi med by o he obse a ions. Thus, pho ome ic measu emen s by P. P a ec e al. (2023, p i a e communica ion) e eal a di e ence o jus –0.061 mag be ween he p e- and pos impac absolu e magni udes, which has been de ec ed only a he 1.9σle el ( o mal e o s), so i is only a ma ginal de ec ion o he bina y sys em’s b igh ening and no s a is ically significan . In line wi h his, B. Bu a i e al. (2023, pe sonal communica ion)did no epo any significan b igh - ness a ia ion in he sys em pos impac ei he , he di e ence being only –0.13 absolu e mag ela i e o he p eimpac da a. F om he la ge obse a ional da a se , we selec ed o modeling hose shown in Figu e 1, whose obse a ional pa ame e s a e gi en in Table 2. The ea lies images acqui ed only 1 o 2 days a e he impac al eady show a complex mo phology, whe e, in addi ion o o he smalle -scale ea u es, wo conspicuous ea u es di ec ed owa d he no h and sou heas ( he ejec a cone ea u es in Figu e 2, uppe panel) became appa en , as well as a well-de eloped ail in he an isola di ec ion. In addi ion, a seconda y ail appea ed no h o he main ail abou 6 days a e he impac , o ming a double ail ea u e ha is ba ely seen in he g ound-based images (see Figu es 8and 9)bu clea ly seen in he HST images (see Figu e 2, lowe panel; see also Li e al. 2023). The o igin o he no he n componen o he double ail is s ill unclea , bu i clea ly ollows he co esponding synch one a T 0 +6±1 day (whe e T 0 is he impac ime; Li e al. 2023). The HST images, al eady desc ibed in Li e al. (2023), we e acqui ed using he 2.4 m diame e HST wi h he Wide Field Came a 3 (WFC3). Addi ional echnical de ails o he ins umen a ion used a e p o ided in Table 1. We ha e selec ed o modeling a subse o he HST calib a ed images as shown in Table 3. Images coded as (l)and (o)a e depic ed in Figu e 2, showing he mos conspicuous ea u es obse ed in he images and p o iding a nomencla u e e e ence. In o de o e e all o he ape u e pho ome y da a and images o a common pho ome ic sys em, he sola spec um in combina ion wi h he eflec ance spec um o he bina y sys em should be aken in o accoun . The ou pu o ou Mon e Ca lo codes is gi en in sola disk in ensi y uni s (i/i e ) ha we con e ed o ¢ Sloan mag a csec −2 ,m, o compa e wi h he obse a ions acco ding o he equa ion  =W+-() ()mmii2.5 log 2.5 log , 1 10 10 whe e Ωis he solid angle sub ended by he Sun a 1 au exp essed in a csec 2 (Ω=2.893 ×10 6 a csec 2 ), and m e is he magni ude o he Sun in he ¢ Sloan fil e , m e =–26.95 (I ezić e al. 2001). I he eflec ed spec um we e pu ely sola , he ape u e pho ome y da a, gi en in he G-band sys em, could be con e ed o ¢ by ¢ =G+0.066 mag (Oszkiewicz e al. 2017). On he o he hand, he con e sion o magni udes in he Table 1 Technical Da a o he Ins umen a ion Used Telescope Loca ion CCD Came a Pla e Scale Fil e (La i ude; Longi ude)Field o View (a csec pixel −1 ) HST LMa coni 160″×160″0.04 F350LP SPACEOBS 22°57′09 8S; 68°10′48 7W ZWO ASI6200MM P o 49′×29′0.54 Clea BOOTES 37°05′58 2N; 06°44′14 9W Ando iXon EMCCD 16 8 ×16 8 1.97 Clea 24 h ps://ssd.jpl.nasa.go /ho izons/ 2 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. HST F350LP fil e o Johnson’sVhas been gi en by Nolan e al. (2019)as V=F350LP −0.12 mag. Then, using he ela ion ¢=- - +() VBV0.49 0.11 mag (Fukugi a e al. 1996), alid o s a s wi h (B−V)+1.5 mag, and he sola colo index (B−V)=0.629 mag (Willme 2018), we ge ¢ =F350LP −0.32 mag. Howe e , he eflec ance spec um o he unal e ed Didymos–Dimo phos sys em exhibi s empo al a ia ions in slope ha can be a ibu ed o a numbe o hings, including composi ional changes on Didymos’s su ace (Ie a e al. 2022), p e en ing us om pe o ming any p ecise pho ome ic co ec ion. In addi ion, he spec um o he eshly ejec ed ma e ial a e he DART collision migh be spec o- scopically di e en as well. Then, we decided o main ain all o he measu emen s in hei o iginal uni s. In any case, based on he gi en colo index con e sion assuming a sola -like spec um, we do no expec a ia ions highe han ≈0.3 mag among he di e en bands (Gaia Gand F350LP)and he ¢ Sloan magni udes. 3. Dus Tail Modeling Ou pu pose is o pe o m an in e p e a ion o he a ailable obse ed images wi h Mon e Ca lo echniques, i.e., by di ec calcula ion o he o bi s o he indi idual pa icles ejec ed a he ime o impac , and he compu a ion o hei posi ions in space a he ime o he obse a ion. To calcula e he o bi s o he ejec ed dus pa icles, we used wo di e en app oaches. The fi s one, which we call simple Mon e Ca lo modeling, assumes ha he pa icles a e ini ially placed ou o he Hill sphe e o he sys em, whe e he g a i y o he bina y componen s can be neglec ed, and hen he dus g ains a e influenced by sola g a i y and adia ion p essu e o ces only. In consequence, he pa icles unde go pu ely Keple ian o bi s a ound he Sun, and hei o bi s can be easily in eg a ed. This app oach is adequa e o desc ibe he escaping ejec a and sui ed o analyze he la ge-scale, low- esolu ion, g ound-based SPACEOBS obse a ions. The second app oach, which we call de ailed Mon e Ca lo modeling, is used o Figu e 1. Subse o SPACEOBS images used o modeling. Axes a e labeled in kilome e s o p ojec ed dis ance a he as e oid in all panels. Labels (a)–(i)indica e obse a ion ime as gi en in Table 2. Celes ial no h is up, and celes ial eas is le in all panels. Table 2 Log o SPACEOBS Obse a ions Time Time since h a Δ b PsAng c PlAng d α e Code (UT)Impac (days)(au)(au)(deg)(deg)(deg) 2022 Sep 30 07:41 3.32 1.037 8 0.072 292.45 48.67 58.24 (a) 2022 Oc 3 06:43 6.28 1.031 5 0.071 288.23 48.19 62.56 (b) 2022 Oc 16 07:12 19.30 1.014 5 0.081 280.87 34.34 75.20 (c) 2022 No 2 07:26 36.31 1.020 2 0.113 283.72 16.63 72.70 (d) 2022 No 18 08:24 52.35 1.053 3 0.147 284.81 6.80 60.31 (e) 2022 Dec 2 07:26 66.31 1.100 7 0.176 282.22 1.05 45.82 ( ) 2022 Dec 17 06:00 81.25 1.165 7 0.212 273.76 −3.30 28.33 (g) 2022 Dec 22 08:10 86.34 1.190 4 0.227 268.42 −4.38 22.32 (h) 2022 Dec 24 07:13 88.30 1.200 1 0.233 265.75 −4.73 20.06 (i) No es. a Heliocen ic dis ance. b Geocen ic dis ance. c Posi ion angle o he ex ended Sun- o-as e oid adius ec o . d Angle be ween obse e and as e oid o bi al plane. e Phase angle. 3 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. desc ibe he dynamics in he inne mos egion close o he bina y sys em and is based on he in eg a ion o he equa ions o mo ion o he pa icles, aking he g a i a ional fields o he wo bodies ully in o accoun . Owing o he supe b spa ial esolu ion o he HST images, his app oach is be e sui ed o analyze he de ails o he ea u es ha appea in hose images bu has he ob ious d awback o he la ge CPU ime needed o un he model in compa ison wi h he simple Mon e Ca lo model. A e se ing he pa icle sca e ing p ope ies, pa icula ly he geome ic albedo, he esul s o he models (i.e., he e olu ion o he ail b igh ness wi h ime)depend on h ee basic pa ame e s: he ejec ion eloci ies, he size dis ibu ion, and he o al dus mass ejec ed a e he DART impac . The combina ion o hose Figu e 2. The HST images ob ained on 2022 Sep embe 28 (code (l); uppe panel)and 2022 Oc obe 8 (code (o)) in Table 3, indica ing he mos ob ious ea u es encoun e ed in he images. Axes a e labeled in pixels, whe e 1 pixel ep esen s ≈2 km p ojec ed on he sky. Celes ial no h is up, and celes ial eas is le in bo h panels. Table 3 Log o he HST Obse a ions Time Time since h ΔPsAng PlAng αCode (UT)Impac (days)(au)(au)(deg)(deg)(deg) 2022 Sep 27 01:04 0.04 1.046 0.076 297.84 47.59 53.34 (j) 2022 Sep 27 07:25 0.31 1.045 0.075 297.39 47.73 53.74 (k) 2022 Sep 28 02:28 1.10 1.043 0.074 296.05 48.11 54.92 (l) 2022 Oc 1 16:12 4.67 1.035 0.072 290.42 48.63 60.25 (m) 2022 Oc 5 18:38 8.78 1.027 0.071 285.41 46.68 65.94 (n) 2022 Oc 8 19:40 11.82 1.022 0.073 282.95 43.76 69.56 (o) 2022 Oc 11 20:42 14.86 1.018 0.075 281.53 40.10 72.44 (p) 2022 Oc 15 10:26 18.43 1.015 0.078 280.90 35.46 74.80 (q) 4 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. pa ame e s a ec s he ail b igh ness in an in ica e manne . Thus, small dus pa icles a e highly a ec ed by adia ion p essu e and quickly popula e he a ail egions, while la ge pa icles need a much longe ime o lea e he nea -nucleus 25 egion, depending on ejec ion speed. In addi ion, in he de ailed model calcula ions, hose pa icles migh be apped o a long ime, o bi ing close o he bina y sys em and leaking ou om i e y slowly, especially i he ejec ion speeds a e close o he escape eloci y o Dimo phos. High ejec ion speeds sp ead ou he pa icles quickly, so ha he ail b igh ness will end o dec ease. The size dis ibu ion, which is commonly se o a powe -law unc ion, defines he ange o sizes ha domina e he o al mass. Thus, o powe exponen s lowe han –4, mos o he mass would be concen a ed in he smalles pa icles, while o exponen s highe han –3, mos o he mass would eside on he la ge ones. The fi ing p ocedu e is based on selec ing uppe and lowe limi s o he pa ame e inpu s and expe imen ing wi h hem un il a easonable ag eemen wi h all o he obse a ions is achie ed. Due o he many pa ame e s in ol ed, we canno ensu e ha he bes -fi ing pa ame e s cons i u e he only solu ion o he p oblem. The main weakness o he modeling esides in he di ficul y o cons aining he o al mass ejec ed, on one hand because o he p esence in he pa icle popula ion o me e -sized and la ge boulde s ha mos ly con ibu e o he mass bu no he b igh ness when compa ed wi h he much mo e abundan small pa icle popula ion, and, on he o he hand, because o he e y high-speed ejec a (see, e.g., Fi zsimmons e al. 2023)immedi- a ely a e impac , which lea es he field o iew o he came as in a e y sho ime in e al. We will come back o hese p oblems in he nex sec ion. 3.1. Simple Mon e Ca lo Modeling The in e p e a ion o he g ound-based dus ail b igh ness in e ms o he simple dynamical- adia i e models is made using ou Mon e Ca lo model as desc ibed in, e.g., Mo eno e al. (2022a, and e e ences he ein). In such an app oach, as s a ed abo e, he pa icles a e assumed o be a ec ed by he sola g a i y and adia ion p essu e only, igno ing he g a i y pe u ba ions o he wo componen s o he bina y sys em. Then, his model is alid ou o he Hill sphe e o he sys em; i.e., i is use ul o cha ac e ize he ma e ial ha has g a i a ionally escaped om he bina y sys em bu canno be used o desc ibe he complex dynamics in he icini y o he as e oid pai . In ac , we will see wi h he de ailed Mon e Ca lo model ha a significan ac ion o he ejec ed mass is los in collisions wi h ei he Didymos o Dimo phos, hus educing he dus mass ejec ed o in e plane a y space. In he simula ions, a la ge numbe (10 7 )o pa icles a e eleased wi h a ce ain eloci y dis ibu ion and pa icle size dis ibu ion. The o al ejec ed mass mus also be specified. Fo his applica ion o he code, all o he pa icles a e assumed o be ejec ed ins an ly, excep o a seconda y ejec ion e en occu ing a ew days a e he impac , which explains he de elopmen o an addi ional ail componen o ming a small angle owa d no h o he main ail, o be desc ibed a he end o his sec ion. The pa icles a e conside ed sphe ical, independen sca e e s, and hey do no expe ience collisions among hem o dis up ion o agmen a ion phenomena. Thei dynamics is go e ned by he so-called βpa ame e (no o be con used wi h he momen um ans e e ficiency due o he DART impac , usually also deno ed by β),defined as he a io o sola adia ion p essu e o ce o sola g a i y o ce, as β=F ad /F g a =C p Q p /(2ρ p ). In ha equa ion, is he pa icle adius, ρ p is i s densi y (assumed a 3500 kg m −3 ), C p =1.19 ×10 −3 kg m −2 is he adia ion p essu e cons an , and Q p is he sca e ing e ficiency o adia ion p essu e, which becomes Q p ≈1 o mode a ely abso bing pa icles wi h 1 μm(see, e.g., Mo eno e al. 2012, hei Figu e 5). The assumed densi y o ρ p =3500 kg m −3 co esponds o he densi y o o dina y chond i e me eo i es associa ed wi h he S- ype spec um exhibi ed by he Didymos–Dimo phos sys em (Dunn e al. 2013). All pa icles a e assumed o ha e he same densi y. The Keple ian ajec o ies o he pa icles can be de e mined om hei βpa ame e and he ejec ion eloci y ec o . A he end o he in eg a ion ime, hei posi ions on he sky plane a any ime a e ejec ion a e eco ded. The b igh ness con ib- u ion o each pa icle in a gi en pixel o he syn he ic image, m, exp essed in mag a csec −2 , is gi en by  pp a =´D - () () () p G 2.24 10 10 ,2 R hmm 2 22 220.4 whe e h is he as e oid heliocen ic dis ance in au, Δis he geocen ic dis ance o he as e oid, and m e is he appa en sola magni ude in he app op ia e passband. The pa icle’sgeome ic albedo a ze o phase angle is gi en by p R ,andG(α)=10 −0.4α is he phase co ec ion, whe e αis he phase angle, and is he linea phase coe ficien . Recen wo k by Lolachi e al. (2023), howe e , shows he calcula ed geome ic albedo dependence wi h phase angle, e ealing alues be ween 0.07 and 0.15 o p R o a ange o pa icle sizes, composi ions, and di e en po osi ies om se e al sou ces, including labo a o y da a by Muñoz e al. (2020)and emi ed pa icles om as e oid Bennu (He gen o he e al. 2020), o phase angles smalle han abou 60°,so ha weadop edp R =0.1 and G(α)=1. In any case, in he geome ic op ics app oxima ion, which holds o he de i ed size dis ibu ion unc ions, he ejec ed mass is di ec ly p opo - ional o he geome ic albedo, so ha o highe albedos, he dus mass ejec ed will be lowe acco dingly. In he icini y o he image op ocen e s, he con ibu ion o he nucleus eflec ed ligh (i.e., he sca e ed ligh o he sphe ical body ha ing an equi alen adius o he Didymos+Dimo phos sys em)is impo an , as i may be compa able o o highe han he dus cloud b igh ness. In ac , o images aken a ew weeks a e impac , he con ibu ion o he nucleus b igh ness o he o al b igh ness is dominan . The equi alen adius o he sys em can be app oxima ely compu ed as an a e age o he Didymos adius and he e ec i e adius o he Didymos+Dimo phos sys em, which u ns ou o be R n =395 m, i.e., only a bi highe han Didymos’s adius, as i has a much la ge su ace han Dimo phos. Then, o compu e he con ibu ion o he nucleus, we assume a sphe ical body wi h he same alue o geome ic albedo gi en abo e o he pa icles. Following he magni ude– phase ela ionship by She chenko (1997), o p R =0.1, we ge =- p0.013 0.0104 ln R =0.037 mag deg −1 . This alue is e y close o ha ob ained by B. Bu a i e al. (2023, pe sonal communica ion)o =0.035 ±0.001 mag deg −1 . 25 In he con ex o his pape , we always e e o he “nucleus”as o he wo as e oids o he bina y sys em ha a e seen as an e ec i e single body in he come -like appea ance in he Ea h-based elescope images o he ejec a. 5 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. The ejec ion o ma e ial is mainly modeled by wo ejec a componen s a eling a di e en speeds. This is jus ified below in o de o ep oduce bo h he an isola ail (slow-speed componen )and he conical ea u es (high-speed componen ). This componen , which con ibu es one- hi d o he o al ejec ed mass, is assumed o be cha ac e ized by a hollow conical shape whose axis is o ien ed o he equa o ial coo dina es R.A. =130°, decl. =17°, which is wi hin he ange o he cu en de e mina ions. The impac o di ec ion was R.A. =128°, decl. =18°(e.g., Hi abayashi e al. 2023). Recen de ailed calcula ions o he ejec a geome y by Hi abayashi e al. (2023) p edic an emission cone elonga ed along he no h–sou h di ec ion o Dimo phos wi h he cone axis o ien ed o R. A. =140°±4°,decl.=17°±7°( he unce ain ies a e 1σ alues). Howe e , he p ecise axis di ec ion does no ha e a significan impac on he esul s as long as i does no de ia e by mo e ha 10° om he assumed di ec ion. The cone ape u e is se o 140°, and he cone wall hickness is se o 10°. The second ejec a componen is desc ibed by a hemisphe ical ejec ion wi h he same axis as he conical emission and con ibu ing wo- hi ds o he ejec ed mass. The emaining model pa ame e s a e he size dis ibu ion and he ini ial speeds. The size dis ibu ion unc ion is ini ially se o a single di e en ial powe -law dis ibu ion unc ion wi h powe exponen κ, i.e., µk d n d , whe e dn is he numbe o pa icles be ween and +d . We assumed an ini ial alue o κo κ=–2.5, close o he alue ob ained by Li e al. (2023)on he ea lies HST images. The size dis ibu ion was assumed o be he same o all o he ejec a componen s. Conce ning ejec ion eloci ies, con en ional scaling laws o c a e ing ejec a gene ally e e o eloci y dis ibu ions as a unc ion o launch posi ion (e.g., Housen e al. 1983; Cin ala e al. 1999; Housen & Holsapple 2011)and do no include he e ec s o he di e en sizes o he pa icles popula ing he dis ibu ion. Only a ew expe imen al o obse a ional s udies p o ide in o ma ion on eloci y dis ibu ion as a unc ion o g ain size, bu in all cases, because o echnical limi a ions, hey e e o sizes in he millime e ange and la ge , up o boulde - sized deb is (e.g., Okawa e al. 2022). A e epea ed expe imen a ion wi h he model, i soon became appa en ha a double componen was needed o he ejec a speeds—one componen associa ed wi h as e pa icles gi ing ise o he wo ea u es associa ed wi h he conical ejec ion (high-speed componen )and ano he wi h ejec ion eloci ies close o Dimo phos’s escape eloci y (slow-speed componen )— o p ope ly model he leng h and hickness o he an isola ail ( he hemisphe ical ejec a componen ). The as e ejec a was modeled ollowing a powe -law unc ion o he pa icle size, as i has been se o model he ejec ion speeds o na u al impac s on as e oids (596)Scheila (Ishigu o e al. 2011)and 354P/LINEAR (Kleyna e al. 2013; Kim e al. 2017). On he o he hand, he eloci ies o he slow-speed componen a e modeled as =0.05(1+χ)ms −1 , whe e χis a andom numbe in he (0, 1)in e al. The andomiza ion in he speed dis ibu ion is imposed in an a emp o somehow mimic i s s ochas ic na u e. The high-speed ejec a was modeled by =0.375χ −0.5 ms −1 (wi h exp essed in me e s). Ejec a speed es ima es o ∼2ms −1 o millime e -sized pa icles ha e been epo ed by Ro h e al. (2023) om ALMA obse a ions o he DART impac . This is in line wi h ou a e age highe - speed ejec a es ima es o ∼6ms −1 o =1 mm pa icles. As s a ed abo e, he eason o a double ejec a componen is mo i a ed by he appea ance o he conical ea u e in combina ion wi h he an isola ail; his ail canno be modeled assuming he conical high-speed componen , as i would gene a e a ail a b oade and mo e di use han obse ed. The slow-speed componen , which encompasses mos o he ejec ed mass ( wo- hi ds o he o al dus mass), could be associa ed wi h he la ge amoun o ma e ial ha is ejec ed a a slow eloci y du ing he la e s ages o c a e o ma ion, as de e mined om con en ional scaling laws (e.g., Housen e al. 1983). In addi ion, he e is ano he mechanism ha migh be con ibu ing o his slow ejec a componen : he lo ing o pa icles owing o he p opaga ion o seismic wa es a e he impac (Tanc edi e al. 2023). In conjunc ion wi h he wo componen s o he ejec a jus desc ibed, a hi d dus ejec a emission e en ook place on 2022 Oc obe 2.5, leading o he seconda y no he n b anch o he ail. This e en is associa ed wi h he p esence o he no he n seconda y ail ha ollows he co esponding synch one a he gi en epoch. This ag ees wi h he iming ob ained by Li e al. (2023) om HST images (T 0 +6±1 days). The small ail hickness sugges s low ejec ion eloci ies, and i s ain ness compa ed o he main ail sugges s a much smalle ejec ed mass han he main ail slow componen o he ejec a. Fo simplici y, we adop he same pa ame e s o he slow ejec a componen men ioned abo e (i.e., =0.05(1+χ)ms −1 )and iso opic ejec ion. We will link his dus emission e en o he impac s o deb is pa icles on Didymos in he amewo k o he de ailed Mon e Ca lo app oach (see Sec ion 3.2). A his poin , i is in e es ing o no e ha a sligh inc ease in b igh ness has been obse ed a ound 6–9 days a e impac in bo h g ound-based and HST pho ome ic ligh cu es (Ka e a e al. 2023). This so- called “eigh h-day bump”could be associa ed wi h eimpac ing ma e ial on Didymos, as we will also show la e in Sec ion 3.2. The dus masses ejec ed o each componen ha be e fi s he ail p ofiles we e 2.2 ×10 7 (slow-speed), 7.4 ×10 6 (high- speed), and 3.7 ×10 6 kg (la e e en ), espec i ely, gi ing a o al mass ejec ed o 3.3 ×10 7 kg. The mass o he seconda y ejec a componen is jus a ough es ima e; he signal- o-noise a io o ha seconda y ail is oo low o allow o a be e cons ain . This es ima e will be imp o ed wi h he analysis o he much highe esolu ion HST images (Sec ion 3.2). The maximum pa icle size in he dis ibu ion was cons ained by he analysis o he la es images. Thus, he ini ial assumed adius max =1 cm had o be inc eased o la ge alues. The eason is ha o max =1 cm, he cen al condensa ion con aining he nuclei would be de ached om he ail a he la es obse a ion da es because he adia ion p essu e would be mo ing hose =1 cm pa icles away some mon hs a e impac . Then, a la ge size limi o max =5 cm was se ins ead, p o iding a be e fi o he nea -nucleus egion. Rega ding he minimum pa icle size, se ing min =1μm was ound o be adequa e o fi he ou e mos pa o he ail in he ea lies images. Also, as we will desc ibe la e in his sec ion, his lowe limi is e y well cons ained by he ea lies HST images, whe e he obse ed leng h o he ail is e y consis en wi h ha minimum size. Wi h all o he abo e model inpu s, he esul ing pho ome ic scans along he ails o he images in compa ison wi h he obse a ions a he da es shown in Table 2a e displayed in Figu e 3. Al hough he fi s o he ea ly images a e easonably good, he model does no pe o m well o images acqui ed la e han ≈10 days a e impac . Va ying he powe index κ does no p oduce any imp o emen ei he . As he obse ed 6 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. b igh ness in he nea -nucleus egion is clea ly o e es ima ed wi h his model, we imposed a b oken powe law wi h a “knee” in he millime e size ange o sea ch o an imp o emen in he fi s. We ound ha a b oken powe law wi h κ=–2.5 o pa icles smalle han 3 mm in adius and a highe slope o κ=–3.7 o pa icles ha ing adii la ge han 3 mm p oduces much be e fi s a all epochs, as can be seen in Figu e 4. The assump ion o a di e en size dis ibu ion wi h a highe slope on he la ges pa icles implies a ecalcula ion o he ejec ed masses ha now become a ac o o ≈8 smalle , i.e., 2.8 ×10 6 and 9.2 ×10 5 kg o he slow and as componen s, espec i ely, and 4.6 ×10 5 kg o he seconda y, la e, ejec a, gi ing a o al mass o 4.2 ×10 6 kg. I is impo an o ealize ha his dus mass cons i u es a s ingen lowe limi o he o al ejec ed mass om Dimo phos. On one hand, he high-speed (≈2kms −1 )ma e ial eleased igh a e he impac (see Shes ako a e al. 2023)is ou o he field o iew on ou images. On he o he hand, he p esence in he pa icle dis ibu ion o e y la ge pa icles, such as boulde s, migh con ibu e significan ly o he o al ejec ed mass bu e y li le o he b igh ness, becoming almos Figu e 3. Scans along he ails o he SPACEOBS images. The panels a e labeled (a)–(i), co esponding o he da es shown in Table 2(“Code”column). The black lines co espond o he obse a ions and he ed line o he model. The ho izon al axes a e labeled in kilome e s p ojec ed on he sky plane, and he e ical axes a e exp essed in mag a csec −2 . These scans we e ob ained using a single powe -law size dis ibu ion wi h κ=−2.5. The o al dus mass eleased is 3.2 ×10 7 kg. Figu e 4. Scans along he ails o he SPACEOBS images. The panels a e labeled (a)–(i), co esponding o he da es shown in Table 2(“Code”column). The black lines co espond o he obse a ions and he ed line o he model. The ho izon al axes a e labeled in kilome e s p ojec ed on he sky plane, and he e ical axes a e exp essed in mag a csec −2 . These scans we e ob ained using a b oken powe -law di e en ial size dis ibu ion unc ion wi h κ=−2.5 be ween 1 μm and 3 mm and κ=−3.7 be ween 3 mm and 5 cm. The o al dus mass sen o in e plane a y space is 4.2 ×10 6 kg. 7 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. unde ec able in he images. In ha espec , i is con enien o men ion he findings by Fa nham e al. (2023), who de ec ed a boulde popula ion a e DART impac by analyzing LICIACube LUKE images. Those au ho s ound a popula ion o some 100 me e -sized boulde s, so ha , assuming a densi y o 3500 kg m −3 , hey would gi e a o al mass o ≈1.5 ×10 6 kg. Those boulde s a e mo ing a speeds o 20–50 m s −1 ,so ha hey ca y a momen um ha migh be compa able o ha o he DART spacec a (Fa nham e al. 2023). Le us assume ha he ac ual boulde popula ion was a ac o o 100 highe , i.e., a o al mass o 10 8 kg, and ha his popula ion is dis ibu ed ollowing a powe law o index –3.7 (as in ou model)wi h ejec ion speeds o 20 m s −1 . This would esul in a un ealis ic momen um balance, bu we would like o ema k ha e en in his case, he boulde popula ion would add a negligible inc ease in he in eg a ed flux o only 0.06% ela i e o he co esponding model esul s on he da es shown in Table 2.E eni we educe he speed o hose boulde s o he much smalle speeds used in he modeling (see Table 4), ha will end o concen a e he boulde s much close o he op ocen e a all epochs, he con ibu ion o he o al flux coming om he boulde s would be o only 7% compa ed wi h he flux compu ed wi h he bes -fi model pa ame e s. The syn he ic images gene a ed a e con ol ed wi h a Gaussian unc ion o FWHM consis en wi h he a e age seeing poin -sp ead unc ion. The modeled images a e hen compa ed o he obse ed images in Figu es 5–7using he same g ay scale. As shown, he modeled images cap u e well many o he ea u es displayed in he obse ed images. The model image showing he double ail in compa ison wi h he SPACEOBS obse a ion is gi en in mo e de ail in Figu e 8. Fo pu poses o compa ison only, an addi ional image, aken a he LULIN Obse a o y 1 m ape u e elescope in Taiwan on Oc obe 12, i.e., 4 days be o e he SPACEOBS image on Oc obe 16, also displays he ea u e, wi h a sligh ly highe signal- o-noise a io han he SPACEOBS image in Figu e 8, see Lin e al. (2023; Figu e 9). By No embe 2, Table 4 Pa ame e s o he Bes -fi Models Ejec a Speed Ejec ed Ejec ion To al Unbound a Componen (ms −1 )Mass (kg)Mode Ejec ed Mass (kg) Simple Mon e Ca lo Model Slow 0.05(1+ξ)2.8 ×10 6 Hemisphe ical Fas 0.375χ −0.5 9.2 ×10 5 Conical 4.2 ×10 6 La e 0.05(1+ξ)4.6 ×10 5 Iso opic De ailed Dynamical Mon e Ca lo Model Slow 0.09 4.3 ×10 6 Hemisphe ical Fas 0.225χ −0.5 2.1 ×10 6 Conical 4.9 ×10 6 La e 0.09 3.0 ×10 6 Iso opic No e. a Deli e ed o in e plane a y space. No e ha in he case o he de ailed dynamical Mon e Ca lo model, his mass is no he sum o he o al masses ejec ed due o in e ening dynamical s i ing and collision o a sizable ac ion o he ejec a wi h Didymos and Dimo phos. Figu e 5. Panels (a),(b), and (c)display he SPACEOBS images a he co esponding da es in Table 2, and panels (a1),(b1), and (c1)display he co esponding syn he ic images gene a ed wi h he simple Mon e Ca lo model. All images a e s e ched be ween 28 and 22 mag a csec −2 . Axes a e labeled in kilome e s p ojec ed on he sky plane. No h is up and eas is o he le in all images. 8 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. Figu e 6. Panels (d),(e), and ( )display he SPACEOBS images a he co esponding da es in Table 2, and panels (d1),(e1), and ( 1)display he co esponding syn he ic images gene a ed wi h he simple Mon e Ca lo model. All images a e s e ched be ween 28 and 22 mag a csec −2 , excep syn he ic image (d1), which has been hea ily s e ched be ween 30 and 25 mag a csec −2 , ba ely showing he seconda y ail no h o he main ail. Axes a e labeled in kilome e s p ojec ed on he sky plane. No h is up and eas is o he le in all images. Figu e 7. Panels (g),(h), and (i)display he SPACEOBS images a he co esponding da es in Table 2, and panels (g1),(h1), and (i1)display he co esponding syn he ic images gene a ed wi h he simple Mon e Ca lo model. All images a e s e ched be ween 28 and 22 mag a csec −2 . Axes a e labeled in kilome e s p ojec ed on he sky plane. No h is up and eas is o he le in all images. 9 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. Figu e 16. Panels (m),(n), and (o)display HST images on he co esponding da es in Table 3, and panels (m1),(n1), and (o1)display he co esponding syn he ic images gene a ed wi h he de ailed Mon e Ca lo model desc ibed in Sec ion 3.2. All images a e s e ched be ween 22 and 17 mag a csec −2 . Axes a e labeled in kilome e s p ojec ed on he sky plane. No h is up, and eas is o he le in all images. Figu e 17. Panels (p)and (q)display HST images on he co esponding da es in Table 3, and panels (p1)and (q1)display he co esponding syn he ic images gene a ed wi h he de ailed Mon e Ca lo model desc ibed in Sec ion 3.2. All images a e s e ched be ween 22 and 17 mag a csec −2 . Axes a e labeled in kilome e s p ojec ed on he sky plane. No h is up, and eas is o he le in all images. 16 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. as e oids commonly occu s in a ime span o se e al weeks, while he DART ail is s ill obse able mo e han 9 mon hs a e impac (Li e al. 2023)Possibly, he bina y na u e o he impac ed objec is playing a ole in ha long su i abili y o he ail in keeping ela i ely la ge pa icles o bi ing he neighbo - hood o he bina y componen s o a long ime be o e being ejec ed o he in e plane a y medium. A hough ul analysis o he long-las ing ail is beyond he scope o his pape . 5. Conclusions The obse ed dus ejec a a e he collision o he DART spacec a wi h Dimo phos, he sa elli e o he (65803)Didymos sys em, has been modeled by Mon e Ca lo dus ail codes. The obse a ions— aken om he Ea h and HST, which has he ad an age o explo ing he ejec a beha io a wo di e en spa ial esolu ions and spa ial scales—a e analyzed by simple and de ailed Mon e Ca lo modeling. F om he g ound-based da a and using ou simple Mon e Ca lo model, we conclude ha he di e en ial size dis ibu ion o he pa icles could be ep esen ed by a b oken powe -law unc ion wi h index κ=–2.5 o pa icles be ween 1 μmand3mmandκ=–3.7 o pa icles o adii be ween 3 mm and 5 cm. The ejec a pa e n migh be explained, on one hand, by pa icles being ejec ed along he wall o a hollow cone wi h i s axis poin ing o R.A. =130°, decl. =17°wi h ela i ely la ge speeds, and on he o he hand, by pa icles emi ed hemisphe ically a Dimo phos’sescape speed, o ien ed in he same way as he emi ing cone. Wi h his configu a ion and an ejec ed mass o app oxima ely 6 ×10 6 kg, mos o he obse ed ea u es can be ep oduced bo h mo phologically and pho ome ically a all o he epochs included in he p esen analysis, keeping in mind ha his es ima e is always a lowe limi , as he p esence o la ge boulde s in he dis ibu ion, ha ing la ge mass bu con ibu ing negligibly o he b igh ness, canno be excluded. The de ailed Mon e Ca lo model akes in o accoun he igo ous mo ion o he pa icles in Figu e 18. Panel (l)displays he cen al po ion o image (l)(see Table 3), and panel (l1)displays he co esponding modeled image. The igh panel depic s he isopho e fields wi h con ou s a 20, 19, and 18 mag a csec −2 (black con ou s co espond o he obse a ion and ed con ou s o he model). All panels a e labeled in kilome e s p ojec ed on he sky and o ien ed no h up, eas o he le . Figu e 19. Ligh cu es om HST images wi h a 0 2 ape u e (Li e al. 2023; black filled ci cles)compa ed wi h BOOTES pho ome y (wi h 6"–7" ape u es depending on seeing condi ions; ed filled ci cles)and he model esul s (open ci cles connec ed by black and ed solid lines o HST and BOOTES, espec i ely). 17 The Plane a y Science Jou nal, 4:138 (18pp), 2023 Augus Mo eno e al. he neighbo hood o he bina y sys em. Wi h his model, a ious de ails obse ed in he ejec a on he HST images ha e been ep oduced, al hough he e emain some ha so a canno be cap u ed wi h such a model. In any case, he model pa ame e s used o explain he g ound-based images can also explain he de ailed s uc u es seen in he HST images, in pa icula , he no he n and sou heas e n s eams associa ed wi h he hollow cone emission, he ea ly e olu ion o he ejec a, he leng h o he an isunwa d ail, and he double ail pa e n. The no he n componen o he double ail could be associa ed wi h eimpac ing ma e ial on Didymos, as he momen um ca ied by he impac ing pa icles peaks a nea ly he same epoch as ha needed o gene a e he seconda y ail. Howe e , u he modeling is clea ly needed o es his conclusion. The e a e also many s uc u es eadily seen on he HST images, such as he no he n di use pa e n, un ep oducible wi h he model, ha needs u he modeling including addi ional p ocesses, such as pa icle collisions, agmen a ion, and dis up ion phenomena. Acknowledgmen s We a e e y g a e ul o he wo anonymous e e ees o hei ca e ul e iewing o he manusc ip and de ailed sugges ions, which ha e helped us o conside ably imp o e he manusc ip . Some o he da a p esen ed in his pape we e ob ained om he Mikulski A chi e o Space Telescopes (MAST)a he Space Telescope Science Ins i u e. The specific obse a ions analyzed can be accessed ia doi:10.17909/p c8- k24. F.M. acknowledges financial suppo om g an s PID2021- 123370OB-I00 and P18-RT-1854 om Jun a de Andalucia and CEX2021-001131-S unded by MCIN/AEI/10.13039/ 501100011033. A.C.B. acknowledges unding by he NEO-MAPP p ojec , g an ag eemen 870377, EC H2020-SPACE-2018-2020/ H2020-SPACE-2019. A.J.C.T. acknowledges suppo om Spanish MICINN p ojec PID2020-118491GB-I00 and he echnical s a a INTA-CEDEA, whe e he BOOTES-1 s a ion is loca ed. J.L.O. acknowledges suppo om con ac PID2020- 112789GB-I00. Facili ies: HST(STScI), SPACEOBS, BOOTES Global Ne wo k, LULIN. 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