The expansion of the GRB 221009A afterglow

A&A, 690, A74 (2024) h ps://doi.o g/10.1051/0004-6361/202348524 c The Au ho s 2024 As onomy & As ophysics The expansion o he GRB 221009A a e glow S. Gia a ana1,2,?, O. S. Sala ia3,4 , M. Gi ole i2, G. Ghi landa3,4, L. Rhodes5, P. A i6, B. Ma co e7, J. Yang8, T. An9,10 , G. Ande son11 , J. S. B igh 5, W. Fa ah12,21, R. Fende 5,13, J. K. Leung14,15,16 , S. E. Mo a3,5 , M. Pé ez-To es17,18,19 , and A. J. an de Ho s 20 1Dipa imen o di Fisica e As onomia, Uni e si à degli S udi di Bologna, Via Gobe i 93/2, 40129 Bologna, I aly 2INAF Is i u o di Radioas onomia, ia Gobe i 101, 40129 Bologna, I aly 3INAF Osse a o io As onomico di B e a, Via E. Bianchi 46, I-23807 Me a e, I aly 4INFN – sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, (MI), I aly 5As ophysics, Depa men o Physics, Uni e si y o Ox o d, Keble Road, Ox o d OX1 3RH, UK 6ASTRON, Ne he lands Ins i u e o Radio As onomy, Oude Hooge eensedijk 4, 7991 PD Dwingeloo The Ne he lands 7Join Ins i u e o VLBI ERIC, Oude Hooge eensedijk 4, 7991 PD Dwingeloo The Ne he lands 8Depa men o Space, Ea h and En i onmen , Chalme s Uni e si y o Technology, Onsala Space Obse a o y, SE-439 92 Onsala, Sweden 9Shanghai As onomical Obse a o y, Chinese Academy o Sciences, Nandan Road 80, Shanghai 200030, China 10 Key Labo a o y o Radio As onomy and Technology, Chinese Academy o Sciences, A20 Da un Road, Beijing 100101, PR China 11 In e na ional Cen e o Radio As onomy Resea ch, Cu in Uni e si y, GPO Box U1987, Pe h WA 6845, Aus alia 12 SETI Ins i u e, 339 Be na do A e, Sui e 200, Moun ain View, CA 94043, USA 13 Depa men o As onomy, Uni e si y o Cape Town, P i a e Bag X3, Rondebosch 7701, Sou h A ica 14 Da id A. Dunlap Depa men o As onomy and As ophysics, Uni e si y o To on o, 50 S . Geo ge S ee , To on o, On a io M5S 3H4, Canada 15 Dunlap Ins i u e o As onomy and As ophysics, Uni e si y o To on o, 50 S . Geo ge S ee , To on o, On a io M5S 3H4, Canada 16 Racah Ins i u e o Physics, The Heb ew Uni e si y o Je usalem, Je usalem 91904, Is ael 17 Ins i u o de As o ísica de Andalucía (IAA-CSIC), Glo ie a de la As onomía s/n, E-18008 G anada, Spain 18 Facul ad de Ciencias, Uni e sidad de Za agoza, Ped o Ce buna 12, E-50009 Za agoza, Spain 19 School o Sciences, Eu opean Uni e si y Cyp us, Diogenes s ee , Engomi 1516, Nicosia, Cyp us 20 Depa men o Physics, Geo ge Washing on Uni e si y, 725 21s S NW, Washing on, DC 20052, USA 21 Be keley SETI Resea ch Cen e, Uni e si y o Cali o nia, Be keley, CA 94720, USA Recei ed 9 No embe 2023 /Accep ed 3 July 2024 ABSTRACT We obse ed γ- ay bu s (GRB) 221009A using e y long baseline in e e ome y (VLBI) wi h he Eu opean VLBI Ne wo k (EVN) and he Ve y Long Baseline A ay (VLBA), o e a pe iod spanning om 40 o 262 days a e he ini ial GRB. The high angula esolu ion (mas) o ou obse a ions allowed us, o he second ime e e , a e GRB 030329, o measu e he p ojec ed size, s, o he ela i is ic shock caused by he expansion o he GRB ejec a in o he su ounding medium. Ou obse a ions suppo he expansion o he shock wi h a >4σ-equi alen signi icance, and con i m i s ela i is ic na u e by e ealing an appa en ly supe luminal expansion a e. Fi ing a powe law expansion model, s∝ a, o he obse ed size e olu ion, we ind a slope a=0.69+0.13 −0.14. Fi ing he da a a each equency sepa a ely, we ind di e en expansion a es, poin ing o a equency-dependen beha iou . We show ha he obse ed size e olu ion can be econciled wi h a e e se shock plus o wa d shock, p o ided ha he wo shocks domina e he emission a di e en equencies and, possibly, a di e en imes. Key wo ds. echniques: high angula esolu ion – echniques: in e e ome ic – gamma- ay bu s : gene al – adio con inuum: gene al – gamma- ay bu s : indi idual: GRB 221009A 1. In oduc ion On he 9 Oc obe 2022, all sa elli es equipped o an- sien de ec ion we e igge ed by he ex ao dina y γ- ay bu s (GRB) 221009A (Ve es e al. 2022;Bissaldi e al. 2022;U si e al. 2022;Piano e al. 2022;Go z e al. 2022; F ede iks e al. 2022;Tan e al. 2022;Mi chell e al. 2022; Liu e al. 2022;Lapsho e al. 2022;Xiao e al. 2022;Ripa e al. 2022;Dichia a e al. 2022;Kennea e al. 2022). A a edshi o z=0.151 (de Uga e Pos igo e al. 2022;Malesani e al. 2024), GRB 221009A holds he eco d o he highes e e measu ed iso opic equi alen ene gy (Eγ,iso &1055 e g – ?Co esponding au ho ; [email p o ec ed] Lesage e al. 2023). I is he b igh es GRB in he las 50 yea s and i is es ima ed o be a one in ∼10 000 yea s occu ence based on he obse ed lux dis ibu ion o o he known long GRBs (O’Conno e al. 2023;Bu ns e al. 2023;Malesani e al. 2024). Such a unique e en ini ia ed an unp eceden ed ollow- up campaign, cha ac e ised by ex ensi e empo al and spec al co e age. A he highes ene gies, he LHAASO Collabo a- ion epo ed he de ec ion o sus ained emission well abo e 1 TeV (LHAASO Collabo a ion 2023;Cao e al. 2023). A he lowe end o he elec omagne ic spec um, adio obse a- ions o GRB 221009A commenced jus h ee hou s pos -bu s and de ec ed he b igh es e e adio coun e pa , eaching a lux densi y o 60 mJy (B igh e al. 2023). Ini ial a emp s o model he mul i-wa eleng h a e glow emission conside ed 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. A74, page 1 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) con ibu ions om bo h he e e se shock (RS) and he o wa d shock (FS) esul ing om he decele a ion o he ul a- ela i is ic je by he su ounding ma e ial (Ren e al. 2023;Sa o e al. 2023; Laska e al. 2023;O’Conno e al. 2023;Gill & G ano 2023; Zheng e al. 2024). Howe e , unce ain ies pe sis in he inal in e p e a ion o he da a, despi e inco po a ing mos o he p esen ly known physical ing edien s go e ning he dynamics and emission o GRB je s. Unique measu emen s able o independen ly cons ain he a e glow e olu ion can be ob ained wi h millia scsecond eso- lu ion obse a ions. Ve y long baseline in e e ome y (VLBI) allows o di ec measu emen s o he size o he emis- sion egion, oge he wi h high-p ecision as ome y. As a esul , p ope mo ion and sou ce expansion can be measu ed (Taylo e al. 2004;Mooley e al. 2018;Ghi landa e al. 2019). I he iewing angle θ be ween he obse e line o sigh and he GRB je axis is smalle han he je hal -opening angle θj(‘on- axis’ GRB), he p ojec ed image du ing he a e glow is expec ed o expand, bu no o show app eciable p ope mo ion. Con- e sely, i he ou low is obse ed ‘o -axis’ (θ > θj), an appa - en supe luminal mo ion is expec ed. To da e, measu emen s o he size and expansion o he emi ing egion ha e only been possible o GRB 030329 (Taylo e al. 2004,2005), p o iding he i s di ec e idence o he ela i is ic expansion o GRB ou lows. O e he las wo decades, nume ous campaigns we e aimed o epea he success o GRB 030329 (e.g. Nappo e al. 2017;Sala ia e al. 2022;Gia a ana e al. 2022). Howe e , no e en shone b igh ly and long enough o allow o an expan- sion measu emen . On he o he hand, o he mul i-messenge e en GW 170817 (Abbo e al. 2017a,b;Ma gu i & Cho nock 2021), VLBI obse a ions we e undamen al o measu e he appa en supe luminal mo ion and o cons ain he size o he emi ing egion o he non- he mal elec omagne ic coun e pa (Mooley e al. 2018;Ghi landa e al. 2019), p o ing, o he i s ime e e , ha he me ge s o wo neu on s a s a e able o suc- cess ully launch ul a- ela i is ic je s. He e we p esen ou VLBI ollow-up campaign on GRB 221009A. The da a educ ion is de ailed in Sec. 2. The me hod implemen ed o measu e he sou ce p ope ies om adio obse a ions is desc ibed in Sec. 3. In Sec. 4we p esen he esul s o ou campaign and discuss he physi- cal implica ions in Sec. 5. Th oughou he wo k, we assume Planck Collabo a ion VI (2020) cosmological pa ame e s. Wi h hese pa ame e s, he angula diame e dis ance a z=0.151 is dA=560.3 Mpc. The e o e, 1 mas sepa a ion co esponds o 2.72 pc in p ojec ion a such a dis ance. 2. VLBI obse a ions and da a educ ion 2.1. Eu opean VLBI Ne wo k We obse ed he ield o GRB 221009A wi h he Eu opean VLBI Ne wo k (EVN) om 40 o 261 days pos -bu s (PI: Gia a ana, p ojec code: RG013). Gi en he a ge -o -oppo uni y na u e o he p oposal, no all an ennas we e a ailable a all epochs. Table A.1 lis s he an ennas joining each epoch. Table 1p esen s a summa y o he p ope ies o he obse a ions. The obse a- ions we e pe o med in wo di e en bands cen ed a 4.9 and 8.3 GHz. The da a we e eco ded a 4 Gbi s s−1. Dual pola i- sa ion p oduc s (RR, LL) we e co ela ed a he Join Ins i u e o VLBI in Eu ope (JIVE, Dwingeloo, Ne he lands) using he Supe FX Co ela o (SFXC; Keimpema e al. 2015) in o six een sub-bands wi h 32 MHz bandwid h and 64 channels each. Fo he las epoch, RG013 F, he da a we e co ela ed in o eigh sub- bands wi h 32 MHz bandwid h and 64 channels each. The i s , EVN epoch (RG013 A), ca ied ou 6 days pos bu s a a cen- al equency o 22.2 GHz, was no usable due o un a ou able obse ing condi ions. The obse a ions consis ed o phase- e e encing cycles wi h 4.5 and 2.5 minu es on he a ge a 4.9 and 8.3 GHz, espec i ely, and 1.5 minu es on he phase calib a o . Fu - he scans e e y app oxima ely 30 minu es on some ‘check’ sou ces we e also included. Th oughou he obse a ions, some scans on a inge inde we e pe o med. The adio sou ce J190536.4+194308 (J1905+1943 he ea e ) and he Ve y La ge A ay Sky Su ey (VLASS) compac adio sou ce J191142.50+195200 (J191142+1952 he ea e ) we e used as phase calib a o s in he i s wo (RG013 B and C) and in he las h ee obse a ions (RG013 D, E and F), espec i ely. The calib a ion was pe o med using AIPS1(G eisen 2003), ollowing he s anda d p ocedu e o EVN phase- e e enced obse a ions2. The ampli ude calib a ion, which accoun s o he bandpass esponse, he an enna gain cu es and he sys em empe a u es, was pe o med by applying he gains de i ed by he EVN pipeline. We pe o med a co ec ion o he dispe - si e delay using he IONEX iles om he In e na ional GNSS Se ice ( lba ec p ocedu e in AIPS), we calcula ed a manual single band delay on he inge inde ( lbampcl p ocedu e in AIPS) and we ca ied ou he global inge i ing on he phase calib a o ( ing ask in AIPS) using a model o he sou ce de i ed by a conca ena ion (in CASA,McMullin e al. 2007) and sel -calib a ion (in Di map,Shephe d e al. 1994) o all he is- ibili ies on he sou ce ob ained ac oss he a ious epochs. Solu- ions we e in e pola ed (clcal ask in AIPS) and applied o he phase calib a o i sel , he a ge and some check sou ces (see Appendix A). Fo he las h ee epochs, we co ec ed he isibil- i ies o J191142+1952 by ixing he phase cen e in CASA o he ac ual posi ion o he phase calib a o , as he ini ial posi ion o his phase calib a o was no cons ained wi h a sub-mas esolu- ion. Images o he sou ces we e p oduced using Di map. Fo he analysis p esen ed in his pape , we selec ed he image wi h he bes signal- o-noise a io (S/N) among he wo images p o- duced be o e and a e he sel -calib a ion o he phase calib a o , espec i ely. Fo RG013 C, he lux densi y o he GRB enabled a sel -calib a ion in phase in AIPS (solin =1 min). Fu he in o - ma ion on he s uc u e and he da a educ ion p ocess can be ound in Appendix A. 2.2. Ve y Long Baseline A ay The Ve y Long Baseline A ay (VLBA) da a we e acqui ed be ween 44 and 262 days pos -bu s (PI: A i, p ojec code: BA160). The cen al equency was 15.2 GHz, wi h a o al band- wid h o 512 MHz, di ided in o 4 spec al windows o 128 MHz and 256 channels each, in ull ci cula pola isa ion (RR, LL, RL, LR). The numbe o pa icipa ing s a ions con ibu ing use ul da a was 7, 8, 10 and 10 in expe imen s BA160 B, C, C1 and D espec i ely (see Table A.1). Each obse a ion included app ox- ima ely 30-minu e-long geode ic-s yle blocks a he beginning and a he end o he obse a ion, used o de e mine opo- sphe e modelling e o s. The cen al pa o he obse a ions included scans on inge inde b igh calib a o s and epe i ions 1The As onomical Image P ocessing Sys em (AIPS) is a so wa e package p oduced and main ained by he Na ional Radio As onomy Obse a o y (NRAO). 2h ps://www.e lbi.o g/e n-da a- educ ion-guide A74, page 2 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) Table 1. Log able o ou VLBI campaign and summa y esul s o ci cula Gaussian i s o sou ce isibili ies. Code Da e Time obs − 0A ay νobs bmaj (†)bmin (†)bp.a.(†)Phase Calib a o .m.s. (†)Fν(?)FWHM (?)hβappi(?) [hh:mm UT] [days] [GHz] [mas] [mas] [deg] [µJy/b] [mJy] [mas] c RG013 B 2022-11-18 09:30–13:30 40 EVN 8.1–8.6 1.4 0.45 11 J1905+1943 67 1.02+0.05 −0.05 0.12+0.05 −0.07 5.5+2.5 −3.2 RG013 C 2022-11-21 09:30–13:30 43 EVN 4.6–5.1 3.7 0.69 11 J1905+1943 16 1.42+0.03 −0.03 <0.20 <8.7 BA160 B 2022-11-22, 23 19:58–00:58 44 VLBA 14.9–15.3 1.4 0.40 −9 J1905+1943 130 0.80+0.08 −0.07 0.27+0.05 −0.06 11.3+2.1 −2.3 BA160 C 2023-01-31 15:15–20:14 114 VLBA 14.9–15.3 1.4 0.58 3.5 J1905+1943 66 0.27+0.06 −0.05 0.45+0.19 −0.19 7.4+3.1 −3.0 RG013 D 2023-02-03 05:30–11:30 117 EVN 4.6–5.1 7.1 0.9 7.8 J191142+1952 10 0.47+0.03 −0.03 0.35+0.09 −0.12 5.6+1.4 −1.9 RG013 E 2023-02-04 05:30–11:30 118 EVN 8.1–8.6 1.4 0.59 10 J191142+1952 21 0.59+0.07 −0.06 0.39+0.06 −0.07 6.2+0.9 −1.0 BA160 C1 2023-05-02 10:17–15:16 205 VLBA 14.9–15.3 1.5 0.51 −10 J1905+1943 35 0.33+0.09 −0.07 1.39+0.48 −0.42 12.6+4.3 −3.8 RG013 F 2023-06-27, 28 19:30–02:36 261 EVN 4.8–5.1 1.8 1.5 62 J191142+1952 10 0.16+0.01 −0.01 0.42+0.12 −0.17 3.0+0.9 −1.2 BA160 D 2023-06-28 04:08–09:19 262 VLBA 14.9–15.3 1.7 0.57 −3 J1905+1943 37 0.19+0.19 −0.07 <9.3<66 No es. (†)Beam majo axis, mino axis, posi ion angle, .m.s. noise le el wi h na u al weigh s. (?)Median and 68% con idence in e al o he lux densi y Fνand ull wid h a hal maximum FWHM om i ing a ci cula Gaussian o he sou ce isibili ies; and o he a e age appa en expansion speed hβappi, assuming ze o size a 0. I he lowe ex emum o he 68% c edible in e al is 0, we epo he 95% uppe limi ins ead. o a J1905+1943 – J1925+2106 – GRB 221009A sequence, whe e J1905+1943 and J1925+2106 a e known VLBA calib a- o s, wi h espec i e du a ions o 30s – 30s – 80s. The da a we e co ela ed a he NRAO in Soco o using he Dis ibu ed FX so wa e co ela o (DiFX; Delle e al. 2011). The da a educ ion was ca ied ou in AIPS, ollowing s anda d p ocedu es o con inuum phase- e e encing expe imen s. P o- cedu es lbaeops, lbacco , lbampcl, lbabpss, lbaamp we e ca ied ou in his o de o he ini ial bandpass and ampli- ude calib a ion. The ollowing s ep consis ed in he calib a ion o he oposphe e modeling e o s by unning he ask ing on he geode ic blocks, ollowed by mbdly and delzn. The inal phase, a e, and delay inge- i ing was ca ied ou sepa a ely on J1905+1943 and J1925+2106, yielding high S/N and well- beha ed solu ions o bo h sou ces. The solu ions om he close phase calib a o , J1905+1943, we e applied o he a ge ield. A e p epa ing a model o he phase calib a o using Di map, a cycle o ampli ude and phase solu ions we e de e mined o he calib a o i sel and applied o he a ge o u he e ine he cal- ib a ion. Finally, we p oduced single-sou ce equency-a e aged da ase s o he a ge , which we e imaged in AIPS wi h a na u al weigh ing scheme. Ou VLBA campaign included one mo e epoch, BA160 A, a app oxima ely 14 days pos -bu s . Howe e , as he an ennas we e poin ed a an inco ec posi ion in he sky, he GRB ell ou side he p ima y beam o he VLBA, which is app oxima ely 3 a cmin a 15 GHz. While he educed sensi i i y (app oxi- ma ely 25% o nominal) s ill allowed o he de ec ion o he bu s , a sa is ac o y calib a ion o he complex isibili ies was hampe ed. The e o e, we did no include his expe imen in ou analysis. 3. Me hods 3.1. Sou ce lux densi y, size and a e age appa en expansion eloci y es ima e In o de o ex ac in o ma ion abou he o al lux densi y, size and posi ion o he sou ce om each o ou epochs, we i ed a ci cula Gaussian sou ce model o he calib a ed isibili y da a adop ing a Ma ko Chain Mon e Ca lo (MCMC) app oach. The me hod, which is an ex ension o ha adop ed in Sala ia e al. (2022), is desc ibed in Appendix B. The p ojec ed angula diam- e e o he sou ce image is p opo ional o he ull wid h a hal maximum (FWHM) o he i ed ci cula Gaussian, wi h a p o- po ionali y cons an o o de uni y ha depends on he de ailed su ace b igh ness p o ile (G ano e al. 1999,2005;Taylo e al. 2004;Pihls öm e al. 2007;G ano 2008;Sala ia e al. 2022). In wha ollows, we se he p opo ionali y cons an equal o 1, and discuss i whene e ele an . Once his diame e is measu ed, he a e age appa en expansion eloci y can be calcula ed (assuming he size o be ze o a he ime 0o he explosion) as hβappi=(1 +z)dAs 2( obs − 0)c,(1) whe e sis he FWHM, obs is he ime o he obse a ion, and c is he speed o ligh . Table 1summa ises he esul o he ci cula Gaussian i - ing, along wi h he de i ed a e age appa en expansion eloc- i y. In Appendix Bwe p o ide mo e de ailed in o ma ion in he o m o co ne plo s ha illus a e he pos e io p obabili y den- si y o he lux densi y and sou ce size om he ci cula Gaussian i ing. Figu e 1addi ionally shows ‘ iolin plo s’ ha illus a e a ke nel densi y es ima e o he pos e io p obabili y densi y o he FWHM o each epoch. 3.2. Sou ce size e olu ion model i ing In o de o i a size e olu ion model sm( obs,θ) o he obse - a ions, whe e θis a ec o o ee pa ame e s, we adop ed a Bayesian app oach. By Bayes’ heo em, and gi en he ac ha he size es ima es om di e en obse a ions a e independen , he pos e io p obabili y on θis p opo ional o he p io π(θ) imes he p oduc o he likelihoods. This can be w i en as Pθ|{di}M i=1∝π(θ) M Y i=1 Psm( obs,i)|di π(s),(2) whe e M is he numbe o epochs included in he i , diis he da a (i.e. he isibili ies) o he i- h epoch, obs,iis he ime o he i- h obse a ion, π(s)= Θ(s) (whe e Θis he Hea iside s ep unc- ion) is he p io on he size adop ed in he ci cula Gaussian i s, P(s|di) is he pos e io om such i s (Eq. B.3) ma ginalised on all pa ame e s excep s. In o de o e alua e Eq. (2), we app ox- ima ed he ma ginalised pos e io on he size P(s|di) wi h a Gaussian ke nel densi y es ima e based on he pos e io samples de i ed om he MCMC desc ibed in Sec . 3.1. This allowed us o sample he pos e io on θagain h ough an MCMC app oach. A74, page 3 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) 101102103 Time a e explosion [days] 10 2 10 1 100 101 P ojec ed FWHM [mas] EVN (8.3 GHz) EVN (4.9 GHz) VLBA (15 GHz) 0.0 0.5 1.0 1.5 slope Fig. 1. Sou ce size as a unc ion o ime. The sou ce size cons ain s ob ained as desc ibed in Sec ion 3.1 a e shown in he o m o iolin plo s o di e en colou s, cen ed a he obse ing ime o each epoch and o p opo ional wid h o he pos e io p obabili y densi y o he FWHM. In addi ion, we show he median and 68% c edible in e al wi h an e o ba o he same colo , o he 95% c edible uppe limi wi h a iangle i he o me in e al ex ends o 0. The black dashed line and he wo g ey shaded a eas show espec i ely he median, 68% c edible in e al and 95% c edible in e al o he pos e io p edic i e dis ibu ion o he sou ce size e olu ion ob ained om i ing a powe law model s∝ a obs o he sizes om all he epochs. The inse shows he pos e io p obabili y densi y o he slope a om such i . 3.3. Fo wa d and e e se shock size e olu ion and p ope mo ion model In o de o in e p e ou obse a ions in he con ex o he s an- da d a e glow scena io, we de i ed a simple physical model o he size e olu ion and, in he case o a je no aligned wi h he obse e ’s line o sigh , he p ope mo ion o he sou ce expec ed i he emission is domina ed by ei he he FS o he RS p o- duced as a ela i is ic je expands in o an ex e nal medium wi h a powe law numbe densi y p o ile n(R)=A(R/R?)−k, whe e Ris he dis ance om he explosion si e (i.e. he p ogeni o es ige) and R?=5.5×1017 cm is a e e ence adius3. We assumed a uni o m je angula ene gy p o ile o simplici y, wi h an iso opic-equi alen kine ic ene gy E, a hal -opening angle θj, an ini ial Lo en z ac o Γ0and a du a ion T(which se s he je adial wid h ∆R∼cT). The iewing angle is assumed o be ei he θ =0 (on-axis, o he calcula ion o he p ojec ed size) o θ > θj(o -axis, o he calcula ion o he appa en p ope mo ion). The model is based on he s anda d ela i is ic- hyd odynamical heo y o a ela i is ic shock ha a ises om a ela i is ic explosion in o a s a ic, cold ex e nal medium (e.g. Mesza os & Rees 1993;Pi an e al. 1993;Sa i & Pi an 1995; Kobayashi e al. 1999;Kobayashi & Zhang 2003;Yi e al. 2013) and is desc ibed in de ail in Appendix D. We no e ha he model does no include he possible sideways expansion o he shock. The ee pa ame e s o he model a e he ene gy- o-densi y a io E/A, he du a ion T, he ini ial Lo en z ac o Γ0, he je 3Wi h his de ini ion, Ahas he same meaning as he usual A?= 1 ( ˙ Mw/10−5My −1)( w/1000 km s−1) cm−3pa ame e in he wind-like ex e nal medium case (k=2), whe e ˙ Mwand wa e he mass loss a e and he eloci y o he p ogeni o wind, assumed cons an (Panai escu & Kuma 2000). In he k=0 case, i is simply equal o he homogeneous ex e nal numbe densi y, A=n. hal -opening angle θj, he ex e nal medium densi y p o ile slope kand he iewing angle θ . He ea e we ix T=T90/(1 +z)= 251 s, whe e T90 e e s o he ime encompassing he 5% o 95% pe cen ile o he o al pho on coun s as seen in he obse e ’s e - e ence ame, and we assume Γ0=103based on he lowe limi s om Lesage e al. (2023). Mo eo e , we conside only wo al- ues o he ex e nal medium densi y p o ile slope, which a e k=0 (homogeneous ex e nal medium) and k=2 (wind-like ex e nal medium). The model p edic s he ime e olu ion o he p ojec ed angu- la diame e o he o wa d and e e se shock images (Eq. D.7). On he o he hand, ou es ima ed sou ce sizes a e ob ained by i ing a ci cula Gaussian model o he isibili y da a. The a io o he wo sizes ξ=sm/s(whe e smis he angula diame e p edic ed by he model, and sis he FWHM o he ci cula Gaussian) depends on he lowes -o de e ms o he MacLau- in expansion in UV adius o he Fou ie ans o m o he su ace b igh ness dis ibu ion o he sou ce (Thompson e al. 2017). Taking he su ace b igh ness om Bland o d & McKee (1976) as compu ed in G ano (2008) as e e ence, we expec 1.2.ξ.1.8. This ange accommoda es alues p e iously conside ed in he li e a u e, such as he alue ξ=1.4 used in Pihls öm e al. (2007) and ξ=1.3 used in Sala ia e al. (2022). Since ou model does no p edic he su ace b igh ness dis ibu- ion, we include ξin ou model as a nuisance pa ame e , wi h a uni o m p io in he ange [1.2,1.8]. 4. Resul s 4.1. Sou ce size expansion Figu e 1shows he sou ce size cons ain s om Table 1in he o m o a ‘ iolin plo ’, wi h he wid h o each shaded egion being p opo ional o he pos e io p obabili y densi y o he A74, page 4 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) FWHM, ho izon ally cen ed a he ime o he obse a ion. Addi ionally, we show he median and 68% symme ic c edi- ble in e al on he FWHM by means o an e o ba o each obse a ion, excep o cases whe e he pos e io p obabili y densi y does no show a clea peak, o which we show ins ead he 3σuppe limi wi h a downwa d-poin ing iangle. In o de o quan i y he sou ce size e olu ion om hese obse a ions, we i a simple phenomenological powe law e olu ion model, sm( obs)∝ a obs, o hese size measu emen s, h ough he me hod ou lined in Sec . 3.2. The esul ing pos e io p obabili y densi y o he powe law slope is shown in he inse o Figu e 1. The median and symme ic 68% c edible in e al is a=0.69+0.13 −0.14. We ound ha mo e han 99.99% o he pos e io p obabili y (>4σ-equi alen ) is loca ed a a>0. The e o e, ou obse a- ions s ongly suppo he expansion o he sou ce. In he main panel o Figu e 1, we show wi h a black dashed line he median o he pos e io p edic i e dis ibu ion, ha is, he p obabili y dis ibu ion o sm( obs) a each ixed obs, as de i ed om he i . The do ed lines encompass he 68% symme ic c edible in e al o he same dis ibu ion, illed wi h a g ey shade. A ligh e g ey shading shows he 95% symme ic c edible in e al. We no e ha he size measu emen s in ou 15 GHz VLBA epochs a 44 and 205 days a e in mild ension wi h he EVN measu emen s a sim- ila imes. To explo e he possibili y o a equency-dependen size, we epea ed he powe law size e olu ion model i con- side ing only obse a ions pe o med wi h he EVN o VLBA. Fig. 2shows he esul ing size e olu ion as i ed o EVN obse - a ions a 4.9 GHz and 8.3 GHz (uppe panel) o VLBA obse - a ions a 15 GHz (lowe panel). The plo s a e simila o Figu e 1, excep ha he epochs no conside ed in he i a e shown wi h a ligh g ey shading. The cons ain on a om hese i s esul s in medians and symme ic c edible in e als o a=0.79+0.19 −0.23 (4.9–8.3 GHz) and a=0.98+0.36 −0.38 (15 GHz), in ag eemen wi h each o he . On he o he hand, he no malisa ions o he EVN and VLBA powe laws di e a he ∼2σle el, as can be e inced om he wo-dimensional pos e io p obabili ies shown in Fig. 3. In o de o exclude he possibili y ha ou esul s wi h he EVN a e d i en by sys ema ic e ec s, we ca ied ou a se ies o es s including he check sou ce J1905+1943. We p esen he esul s o ou es s in Appendix C. The esul s o hese es s indi- ca e ha he obse ed e olu ion is no d i en by sys ema ic e o s in he calib a ions. 4.2. Appa en p ope mo ion VLBI obse a ions can cons ain he appa en p ope mo ion o he cen oid o he emission and, he e o e, he je iewing angle. The sou ce posi ion a each VLBA epoch is displayed in Fig. 4: ou esul s do no show any signi ican appa en p ope mo ion be ween 44 and 262 days pos -bu s , bu ou s a is ical e o s can accommoda e a displacemen o up o abou 0.6 mas (a he one-σle el) o e ha pe iod. As shown in Appendix D.2, such an uppe limi does no cons ain s ongly θ , which can s ill be se e al deg ees o he edge o he je , unless he ene gy- o-densi y a io o he explosion is e y la ge. S ill, a numbe o s udies including LHAASO Collabo a ion (2023) and O’Conno e al. (2023) ha e used hei da a o jus i y a e y small θ o GRB 221009A, indica ing ha we a e iewing he je close o on-axis. The lack o signi ican p ope mo ion obse ed du ing ou VLBI campaign is ully consis en wi h such on-axis scena io. We no e ha he EVN campaign was no used o such s udy because o he change in phase e e ence sou ce be ween he sec- ond and hi d epoch. While his change was mo i a ed by he 101102103 Time a e explosion [days] 10 2 10 1 100 101 P ojec ed FWHM [mas] EVN (8.3 GHz) EVN (4.9 GHz) 012 slope 101102103 Time a e explosion [days] 10 2 10 1 100 101 P ojec ed FWHM [mas] VLBA (15 GHz) 012 slope Fig. 2. Size e olu ion conside ing only obse a ions om a single a ay (uppe panel: EVN; lowe panel: VLBA). Each panel is simila o Figu e 1, excep ha he epochs no conside ed in he i a e shown wi h ligh g ey shading o cla i y. disco e y o a close phase calib a o (and hence a mo e e i- cien obse ing s a egy), he di e en sys ema ics and he lack o a eliable a p io i posi ion o he new calib a o p e en a eli- able as ome ic cha ac e isa ion. 5. Discussion 5.1. Slope o he size e olu ion wi h ime The size e olu ion powe law slope a=0.69+0.13 −0.14 we de i ed is compa ible wi h he expec ed slopes o a sphe ical Bland o d & McKee (1976) blas wa e expanding in o a homo- geneous medium, α=5/8=0.625, o a wind-like medium, α=3/4=0.75. Mo eo e , he e olu ion o he p ojec ed phys- ical size is qui e simila o ha o GRB 030329, he only o he bu s o da e wi h a measu ed expansion a e (Taylo e al. 2004, Fig. 5). A74, page 5 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) 0.0 0.2 0.4 0.6 0.8 Size a 100 days [mas] 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Slope EVN (4.9-8.3 GHz) VLBA (15 GHz) Fig. 3. Compa ison o he slope and no malisa ion o he size e olu ion as p obed by he EVN and he VLBA. The con ou s in he plo con ain 68% (da ke con ou s) and 95% (ligh e con ou s) o he pos e io p ob- abili y on he wo pa ame e s (slope and size a a e e ence ime o 100 days) o a single powe law i ed o he EVN (blue) o VLBA (o ange) size e olu ion. 0.40.20.00.20.4 RA [mas] 0.4 0.2 0.0 0.2 Dec [mas] Rela i e o 19h13m3.50116s +19d46m24.22970s BA160 B (44 d) BA160 C (114 d) BA160 C1 (205 d) BA160 D (262 d) Fig. 4. Sou ce posi ion in ou VLBA obse a ions. Fo each epoch, we show he s a is ical e o ba cen e ed on he median posi ion om he ci cula Gaussian sou ce i , wi h ba s spanning he symme ic 68% c edible in e al on he sou ce posi ion in each di ec ion. On he o he hand, a he ime o ou obse a ions, he aniso opy o he shock due o he ini e opening angle o he je should be obse able, bo h in e ms o a s eepening (a ‘je b eak’, Rhoads 1997) in he ligh cu es, and in e ms o a la ening in he e olu ion o he p ojec ed size (G ano e al. 2005). The p esence o a je b eak in he e y high ene gy a e glow ligh cu e a <1000 seconds pos - igge has been discussed by LHAASO Collabo a ion (2023). A je -b eak was also sugges ed by Le an e al. (2023) a .2600 seconds pos - igge , using op ical o mid-IR da a. The expec ed pos -je - b eak size e olu ion slope in he case o a homogeneous ex e - nal medium is a=1/4, in absence o an e icien sideways expansion o he shock (G ano e al. 2005). Such a shallow 101102 Time - T0/(1 + z ) [days] 1018 1019 1020 P ojec ed size [cm] GRB 030329 GRB 221009A Fig. 5. Compa ison o size measu emen s om VLBI obse a ions o GRB 030329 ( ed ci cles; Taylo e al. 2004;Pihls öm e al. 2007) and GRB 221009A (da k blue squa es, his wo k). T iangles ep esen uppe limi s. slope is in ension wi h he obse ed one a he ∼3σle el. Con- e sely, he expec ed a e age slope is s eepe and lies in he ange hai ∼ 0.6−0.8 i he shock expands sideways (G ano e al. 2005), which is compa ible wi h he obse ed one. The e o e, in he homogeneous ex e nal medium scena io, ou obse a- ions indica e ha ei he he je b eak has no happened ye , o ha he shock is expanding sideways. Gi en he e y la ge iso opic equi alen ene gy in he gamma- ays, he ‘la e je b eak’ scena io would pose e y demanding equi emen s on he o al ene gy (see e.g. O’Conno e al. 2023). On he o he hand, nume ical simula ions o ex e nal shocks a ising om ela i is ic je s and analy ical a gumen s seem o indica e ha he sideways expansion is ine icien , unless he ini ial open- ing angle is e y na ow ( an Ee en e al. 2010;De Colle e al. 2012;G ano & Pi an 2012). These di icul ies could be alle i- a ed i he je ea u es a s uc u e consis ing o a na ow ‘co e’ su ounded by ‘wings’ whe e he kine ic ene gy pe uni solid angle dec eases slowly, as sugges ed by O’Conno e al. (2023) and Gill & G ano (2023). This p o ile would s eepen he e olu- ion o he obse ed size, making i mo e simila o he sphe ical case, bu wi h a educed ene gy equi emen wi h espec o a wide je wi h a uni o m angula ene gy p o ile. In he wind medium case, he expec ed pos -je -b eak size e olu ion slope is a=1/2 in absence o sideways expansion. The e o e, in such a scena io, he obse ed e olu ion does no indica e he need o sideways expansion no o a non-uni o m s uc u e wi hin he opening angle. 5.2. Possible equency-dependen size As discussed in sec ion 4.1, ou da a sugges he p esence o a equency-dependen size e olu ion. We explo e he e a possi- ble a enue o in e p e his beha iou . The adio a e glow o his GRB canno be explained by a simple FS p opaga ing ei he in o a wind-like o a homogeneous en i onmen (Ren e al. 2023; A74, page 6 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) 101102 Time a e explosion [days] 10 2 10 1 100 101 P ojec ed FWHM [mas] k = 0 15 GHz: FS-domina ed 4.9-8.3 GHz: RS-domina ed RS FS 101102 Time a e explosion [days] 10 2 10 1 100 101 P ojec ed FWHM [mas] k = 2 15 GHz: FS-domina ed 4.9-8.3 GHz: RS-domina ed RS FS Fig. 6. Model size e olu ion in a FS plus RS scena io. The iolins and he e o ba s show he sou ce size e olu ion as in e ed by ou obse - a ions, in he same way as in Figu e 1. The g ay do ed line and o ange dashed lines show he medians o he pos e io p edic i e dis ibu ions o he FS and RS size, espec i ely, as ob ained by i ing he physi- cal model desc ibed in Appendix D o he sizes shown in he igu e, assuming a homogeneous ( op panel) o wind-like (bo om panel) ex e - nal medium, and assuming he FS o domina e a 15 GHz and he RS o domina e a 4.9 and 8.3 GHz. The shaded bands a ound hese lines show he 68% c edible in e al o he pos e io p edic i e dis ibu ion. Sa o e al. 2023;Ren e al. 2024;Zheng e al. 2024). Using a da a se encompassing obse a ions om he GeV o he adio domain, Laska e al. (2023) showed ha he s anda d a e glow model s uggles a explaining he adio emission bo h wi h a FS and a RS o a conical je p opaga ing h ough a wind-like en i onmen , leading hem o in oke an addi ional componen whose empo al e olu ion does no ollow he s anda d p esc ip- ions. Such a componen could be a RS, domina ing he emission a he lowe equencies (.10 GHz) up o obs .100 d (see he modelling o O’Conno e al. 2023;Gill & G ano 2023). S imula ed by hese s udies, we explo ed a scena io whe e he emission we obse ed is a supe posi ion o a FS and a RS. We i ed he model desc ibed in Sec . 3.3 o he obse ed size e olu- ion, lea ing E/Aand θjas ou ee pa ame e s, and addi ionally including he ξnuisance pa ame e (see sec ion 3.3). The ex e - nal medium powe law index was ixed o k=0 o k=2. Based 56.90+1.02 0.84 54 55 56 57 58 log(( E / A )/e gcm3) 8 16 24 je [deg] 54.0 54.5 55.0 55.5 56.0 56.5 57.0 6 12 18 24 30 je [deg] 21.18+8.82 8.23 (95%) 54.98+0.48 0.45 54.0 54.5 55.0 55.5 56.0 log(( E / A )/e gcm3) 6 12 18 24 30 je [deg] 53.0 53.5 54.0 54.5 55.0 6 12 18 24 30 je [deg] 23.14+6.86 7.12 (95%) Fig. 7. Co ne plo o he pos e io p obabili y densi y o he physical model pa ame e s in he homogeneous ( op panel) o wind-like (bo - om panel) ex e nal medium case and assuming he FS o domina e a 15 GHz and he RS o domina e a 4.9–8.3 GHz. In each panel, he ed his og ams show he ma ginalised pos e io p obabili y densi ies, wi h black solid e ical lines showing he median and dashed lines showing he 68% c edible in e al o , i he la e ex ends o he lowe (uppe ) ex emum o he p io ange, he 95% uppe (lowe ) limi ( al- ues epo ed on op o he panels). The illed con ou s show he small- es egions con aining 68% and 95% o he wo-dimensional pos e io p obabili y, wi h he black squa es showing he posi ion o he median. The g ey lines show con ou s o cons an o al je ene gy assuming A=1 cm−3. Each con ou is labeled wi h he base-10 loga i hm o he co esponding o al je ene gy. on he appa en possible equency-dependen beha iou , we assumed he highe - equency obse a ions (15 GHz) o be dom- ina ed by he FS, while he lowe - equency ones (4.9–8.3 GHz) o be domina ed by he RS. Fo each ex e nal densi y p o ile, he pos e io p obabili y densi y o (E/A, θj, ξ) was de i ed h ough he Bayesian app oach desc ibed in Sec . 3.2, wi h uni o m- in-log p io s on E/Ain he ange [1052,1058] e g cm3and on θjin he ange [0.5,30] deg, and a uni o m p io on ξin he ange [1.2,1.8], as explained in sec ion 3.3. Figu e 6shows he FS and RS model size e olu ion i ed o he obse a ions. Figu e 7shows co ne plo s o he pos e io p obabili y densi- ies, ma ginalized o e ξ. The model can easonably ep oduce he da a in bo h he homogeneous (k=0) and wind-like (k=2) ex e nal medium A74, page 7 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) cases. Due o he ela i ely s eep obse ed size e olu ion, and gi en ha ou model does no include sideways expansion, he equi ed je hal -opening angle is la ge (θj>13 deg a he 95% c edible le el o k=0; θj>16 deg o k=2). Assuming he s anda d e e ence ex e nal densi y A=1 cm−3, his in u n pushes he o al je ene gy Eje =E(1 −cos θj) o e y la ge al- ues, as demons a ed by he g ey con ou s in Fig. 7. The ene gy equi emen can be educed i he ex e nal densi y is much lowe han he e e ence alue, A1 cm−3, o i he je ea u es an angula s uc u e such ha he ene gy pe uni solid angle dec eases away om he je axis, as discussed in he p e ious sec ion. In his case, he dec ease mus be shallow (E∝θ−awi h a<2, whe e θis he angle om he je axis), o he wise bo h he emission and he size e olu ion would simply e lec hose o a uni o m je (O’Conno e al. 2023), leading again o he same di icul ies wi h he je opening angle and o al ene gy. 6. Conclusions In his pape , we p esen ed VLBI obse a ions o he b igh es γ- ay bu s e e obse ed, GRB 221009A. The high angula esolu- ion p o ided by he EVN and he VLBA allowed us o cons ain he size and he expansion o he blas wa e p oduced by he GRB ejec a o he second ime e e . The expansion a e is con- sis en wi h he expec a ion o a sphe ical Bland o d & McKee (1976) blas wa e (i.e., an ul a- ela i is ic FS) p opaga ing in o a homogeneous o wind-like medium. This could be aken as an indica ion ha he shock is aniso opic only on angula scales la ge han hose p obed by ou obse a ions (i.e. he je b eak has no happened ye ). This in u n poin s o an ex emely la ge o al (collima ion-co ec ed) ene gy in he je , especially i he ex e nal medium ea u es a homogeneous densi y. The demand- ing ene gy equi emen could be alle ia ed i he ex e nal den- si y we e much lowe han usually assumed, o i he je ene gy pe uni solid angle dec eased slowly wi h he angle om he je axis, as p oposed by O’Conno e al. (2023) and Gill & G ano (2023). Al e na i ely, he shock could be unde going sideways expansion du ing he ime co e ed by ou obse a ions. Addi ionally, ou obse a ions sugges a equency- dependen size e olu ion, wi h he VLBA obse a ions a 15 GHz showing a somewha la ge size a 40 days a e he explosion, and a as e inc ease a e wa ds, wi h espec o he EVN obse a ions a 5 and 8 GHz. This could be due o he emission being domina ed by he e e se shock a he lowe equencies, and by he o wa d shock a he highe equencies. Ou wo k highligh s he c ucial ole played by mul i-wa eleng h VLBI moni o ing o ansien e en s bo h a ea ly and la e imes, and in p o iding a i al insigh in o he physics o such e en s. Da a a ailabili y A copy o he educed isibili ies is a ailable a he CDS ia anonymous p o cdsa c.cds.unis a. (130.79.128.5) o ia h ps://cdsa c.cds.unis a. / iz-bin/ca /J/ A+A/690/A74 Acknowledgemen s. The Eu opean VLBI Ne wo k is a join acili y o indepen- den Eu opean, A ican, Asian, and No h Ame ican adio as onomy ins i u es. Scien i ic esul s om da a p esen ed in his publica ion a e de i ed om he ollowing EVN p ojec code: RG013. The Na ional Radio As onomy Obse a- o y is a acili y o he Na ional Science Founda ion ope a ed unde coope a i e ag eemen by Associa ed Uni e si ies, Inc. This wo k made use o he Swin- bu ne Uni e si y o Technology so wa e co ela o , de eloped as pa o he Aus alian Majo Na ional Resea ch Facili ies P og amme and ope a ed unde licence. SG would like o hank Z. Pa agi and he s a o JIVE o hei help and suppo du ing his isi ing pe iod in Dwingeloo. We would like o hank he di ec o s and s a o all he EVN elescopes o app o ing, execu ing, and p ocessing ou ou -o -session ToO obse a ions. The esea ch leading o hese esul s has ecei ed unding om he Eu opean Union’s Ho izon 2020 Resea ch and Inno a ion P og amme unde g an ag eemen No. 101004719 (OPTICON RadioNe Pilo ). 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C., Wu, X. F., & Dai, Z. G. 2005, MNRAS, 363, 93 A74, page 9 o 16 Gia a ana, S. e al.: A&A, 690, A74 (2024) 54 55 56 57 log( E / A ) [e g cm 3 ] 1 2 3 4 5 6 7 8 9 10 11 12 13 j [deg] j= 0.5 deg 54 55 56 57 log( E / A ) [e g cm 3 ] j= 1.0 deg 54 55 56 57 log( E / A ) [e g cm 3 ] j= 2.0 deg 54 55 56 57 log( E / A ) [e g cm 3 ] Wind ex . medium ( k = 2 ) j= 4.0 deg 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 (262d) (44d) [mas] 56 57 58 59 log( E / A ) [e g cm 3 ] 1 2 3 4 5 6 7 8 9 10 11 12 13 j [deg] 56 57 58 59 log( E / A ) [e g cm 3 ] 56 57 58 59 log( E / A ) [e g cm 3 ] 56 57 58 59 log( E / A ) [e g cm 3 ] Homog. ex . medium ( k = 0 ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (262d) (44d) [mas] Fig. D1. Cons ain on he iewing angle om he absence o an obse ed sou ce appa en displacemen in ou VLBA obse a ions. In each panel, illed con ou s show he displacemen o he cen e o he i ed Gaussian expec ed be ween 44 d and 262 d, colo coded as shown in he colo ba on he igh , as a unc ion o he E/A a io and o he o -edge iewing angle θ −θj. The ed con ou shows ∆(262 d) −∆(44 d) =0.6 mas, which ep esen s he la ges displacemen compa ible a 1 σwi h ou obse a ions. The ed con ou hence con ains he excluded pa ame e egion. The uppe panel ow e e s o a wind-like ex e nal medium, while he lowe ow e e s o a homogeneous ex e nal medium. Each column assumes a di e en je hal -opening angle, gi en a he op o he column. A74, page 16 o 16