ARENBERG DOCTORAL SCHOOL
Facul y o Science
De ailed s uc u e and e olu ion modeling o he
igh es massi e bina y s a s
Ma hias FABRY
Examina ion commi ee:
p o . d . J. Magdaleni´c Zhuko , chai
p o . d . H. Sana, supe iso
d . P. Ma chan , co-supe iso
p o . d . J. O. Sundq is
p o . d . C. Ae s
p o . d . N. Lange
(Uni e si ä Bonn, Ge many)
d . E. Laplace
(Heidelbe g Ins i u e o Theo e ical S udies,
Ge many)
p o . d . O. R. Pols
(Radboud Uni e si ei Nijmegen, he
Ne he lands)
Disse a ion p esen ed in pa ial
ul illmen o he equi emen s
o he deg ee o Doc o o Sci-
ence (PhD): As onomy and As o-
physics
May 2024
Co e A : Inez Fab y
Acknowledgmen s: M.F. ecei ed unding om he Flemish esea ch ounda ion (FWO, Fonds oo We enshappelijk
Onde zoek) unde PhD ellowship No. 11H2421N.
The esea ch leading o hese esul s has ecei ed unding om he Eu opean Resea ch Council (ERC) unde he
Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og am (g an ag eemen numbe s 772225: MULTIPLES).
This wo k has made use o da a om he Eu opean Space Agency (ESA) mission
Gaia
(h ps://www.cosmos.esa.
in /gaia), p ocessed by he
Gaia
Da a P ocessing and Analysis Conso ium (DPAC, h ps://www.cosmos.esa.in /
web/gaia/dpac/conso ium). Funding o he DPAC has been p o ided by na ional ins i u ions, in pa icula he
ins i u ions pa icipa ing in he Gaia Mul ila e al Ag eemen .
Based on obse a ions made wi h he Me ca o Telescope, ope a ed on he island o La Palma by he Flemish
Communi y, a he Spanish Obse a o io del Roque de los Muchachos o he Ins i u o de As o ísica de Cana ias.
Figu e 1.12 ep oduced unde he ai use policy o A . XI.191/1 o Belgian Law. No copy igh in ingemen is
in ended.
©2024 KU Leu en – Facul y o Science
Ui gege en in eigen behee , Ma hias Fab y, Celes ijnenlaan 200D box 2401, B-3001 Leu en (Belgium)
Alle ech en oo behouden. Nie s ui deze ui ga e mag wo den e menig uldigd en/o openbaa gemaak wo den
doo middel an d uk, o okopie, mic o ilm, elek onisch o op welke ande e wijze ook zonde oo a gaande
sch i elijke oes emming an de ui ge e .
All igh s ese ed. No pa o he publica ion may be ep oduced in any o m by p in , pho op in , mic o ilm,
elec onic o any o he means wi hou w i en pe mission om he publishe .
Acknowledgmen s
Wha a jou ney.
In he i s hal o my PhD, he pandemic equi ed all in e ac ions wi h s uden s
and colleagues o mo e o Zoom (2/10; would no ecommend). Despi e he
unbelie able si ua ion, once i eceded, I had he pleasu e o mee many cool
people ab oad, bo h a con e ences and du ing isi s. Special hanks go o Sil ia,
Tome and he API membe s o hos ing me in Ams e dam, and o No be and
he Bonn s ella g oup o le ing me s ay o ou mon hs! Also big hanks o
he
MESA
de s, TAs and s uden s a he San a Ba ba a summe school o you
inexhaus ible en husiasm o s a s.
Back home, I had a mos wonde ul ime a he Ins i u e o As onomy he e in
Leu en. I eally is a ib an place, wi h lo s o spo s, ecep ions, and an especially
igh -kni g oup o PhD s uden s and pos -docs. Special hanks go o:
The olleyball g oup (Robin, Ka an, Nico, Oli ie , Luka, and many, many
mo e...) o he many F iday nigh s on he sand (and a Spuye o Me a oo
a e wa ds...).
The 2023 Ui je eam ( oo many o lis he e, THANKS e e yone!) o o ganizing
h ee days in he A dennes ull o ac i i ies.
The UK/I eland olks (Abi, Ga e h, Calum, Emily) who in oduced me o Six
Na ions Rugby a he I ish Pub.
The SkI S ad en u e s (Nich, Luc, JP, Silke, ...) whom I was able o join wice,
aking in he sigh s, showing o he d ip and sh edding he gna on he F ench
alps.
My ellow San F ancisco a ele s (Ma hias, Joey, Vincen , Annachia a) o
si ing h ough my e y i s li e baseball game wi h me, walking h ough he
Cali o nian Redwoods, d i ing down highway 1, and o he wise making my
i s ip o he US one o ne e o ge .
Clio, Ani a, Monique and he Sys em eam who all play in aluable suppo
oles in he ins i u e.
Jesus and Saskia o he suppo on my h ee ips o La Palma. Obse ing on
he moun ain is uly an un o ge able expe ience!
i
ii ACKNOWLEDGMENTS
And inally, massi e hanks o he massi e-s a g oup. I conside mysel blessed
o ha e been able o mee e e y single one o you. Thanks Abi, Calum, Ga e h,
Lau en , Annachia a, Kunal, Soe kin, Tinne, Emma, Alan, F ank, Jaime, Julia,
Michael, Cyp ien, Philippe, Maddalena, Tome , Reinhold, Dominic and Ka an o
he many ui ul discussions, in e es ing science and he un on he eam Ui jes! I
delibe a ely le Hugues and Pablo ou o his lis . Tha ’s because, o cou se, you
wo dese e sepa a e pa ag aphs.
Me ci mille ois, Hugues, d’ê e un supe men o . I conside s a ing a PhD wi h
you one o he bes decisions I ha e e e made in my li e (9/10; would ecommend).
E en be o e comple ing my wo k on you p ojec , hell, e en be o e inishing
he i s h ee mon hs, you immedia ely le me w i e my own p ojec , which
esul ed in he bulk o his hesis. You kick-s a ed my academic ca ee (which
I hope o con inue o many yea s o come) and suppo ed me in my pe sonal
and p o essional de elopmen . I especially like you no-nonsense a i ude, and
making i ex emely clea wha o do and also (maybe mo e impo an ly) wha no
o do. I e e I need an opinion on any hing e en emo ely ela ed o as ophysics,
I know who o u n o.
Muchas g acias, Pablo, po mos a me las es ellas. While I lea ned wha s a s
a e in my cou ses, i was you who augh me he ine de ails and showed me he
inne wo kings o
MESA
, no o men ion he my iad e ec s o bina y p ocesses. You
ha e p o ided he majo i y o he scien i ic mo i a ion o his hesis, which I will
ne e ake o g an ed. Alongside Hugues, you ha e suppo ed me h oughou
he pas yea s, and hanks o you, I ha e become he scien is I am oday. You a e
a con inuous inspi a ion o me and I will always look o you and you wo k as he
gold s anda d o esea ch in heo e ical s ella as ophysics. I hope we can wo k
oge he o many mo e yea s. Thanks also o being a cool guy and a good iend,
and 3D p in ing some o he designs I made o e he yea s.
I’d be emiss no hanking all my o he iends. Thanks o my pals a Gol Pa k
Te u en (and o me ly DMGC), o all he un ounds on he many Sunday
mo nings. I also ha e good memo ies om ou In e club compe i ion campaigns!
Y’all made i a eally, eally un club I eel a home a . The li le nine-hole cou se I
lea ned o play on will always ha e a special place in my hea . I’ll de ini ely isi
in he u u e!
Thanks o he people o Guikui (who came up wi h his name anyway?), o
dealing wi h me du ing ou bachelo deg ee. You a e a bi c azy some imes, bu
ha ’s OK, and exac ly why I like you. I was maybe no he mos ou going in his
g oup du ing my PhD, bu I s ill wan o hank you since you helped me ge ing
h ough some o he oughes imes o my li e.
Big hanks o my h ee pa ne s o he Lucky Bas e ds, Ba , Jo di and Lenna .
By my es ima ion we played boa d games oge he o o e a housand hou s,
ad en u ing in Gloomha en, F os ha en, he 7 h con inen , Aeon T espass, Scy he,
ACKNOWLEDGMENTS iii
and elsewhe e. I akes a special kind o geek o be willing o do his, and I’m
happy o ha e ound hem in you guys. I hink i ’s all wo h i hough, igh ?
Remembe ick ock, he (glo ious) deus ex machina d10 c i oll in Aeon, me
ying despe a ely o make expedi ions wo k in Odin, and, o cou se, he e ible
semi-coope a i e shenanigans in Gloom/F os ha en. On o 10000?
Fo some mo e andom acknowledgmen s: Thanks o deadmau5 o slapping
some cho ds oge he ha I enjoy,
1
F ank He be o w i ing Dune, he Mille
b o he s and Cyan Wo lds o c ea ing Mys and he D’ni uni e se, James SA
Co ey o w i ing he Expanse, Mike Pondsmi h o c ea ing Cybe punk upon
which bo h CDPR’s 2077 and Ne unne a e based (no andom: huge hanks o
he Belgian Ne unne communi y, bu especially Ruben o playing many games
wi h me Tuesday nigh s in Demo-Spel!), Asobo s udios o c ea ing he bes ligh
simula o o da e, wi h which I can eel jus a bi close o my la e g and a he ,
and Jon an Caneghem o de eloping He oes o Migh and Magic III, a simply
awesome and imeless game. These people will p obably ne e ead his, bu s ill,
hanks, you kep me busy du ing mos o my o - ime in he pas yea s, and ha ’s
equally impo an I hink.
Ik heb de belang ijks en als laa s gehouden. Dank u me e , oma en opa om
in e esse e blij en onen in mijn s e enkundige a on u en. Wa ik bes udee lig
misschien wa e an jullie bed, maa och la en jullie mij maa doen, me mijn
hoo d in de uim e. Dank u pe e om mijn science-nonkel e zijn. Klink als ie s
ui een Ame ikaanse si com, maa ik heb he geluk een éch e e hebben! Dankzij
jou wee ik wa mogelijk is als academicus, en ook bij wie ik e ech kan oo
pe soonlijk ad ies. Ten laa s e, dank u mama, papa en zus oo de ongeziene
s eun. Dank u om en mins e also e doen geïn e essee d e zijn in wa ik doe, en
e blij en luis e en naa mijn gepala e o e s e en, pindano en en zwa e ga en.
De oo bije ja en wa en nie al ijd de gemakkelijks e, maa desondanks heb ik di
boekje kunnen sch ij en, en di was nie mogelijk zonde jullie hulp gedu ende
mijn hele le en. Ik ben hie ex eem ie op, en ik hoop jullie ook.
Ma hias
1like “some cho ds” – deadmau5
Abs ac
Massi e s a s a e esponsible o p oducing hea y elemen s in he uni e se, and
sca e hem in o hei cosmic neighbo hoods as hey explode as supe no ae.
These p ocesses pa e he way o he o ma ion o a new gene a ion o s a s ha
ha e he possibili y o o m plane s ou o he o med dus , po en ially ha bo ing
li e.
Wi h masses o e eigh imes ha o ou Sun, massi e s a s lead b igh and as
li es, aking only millions o yea s om bi h o dea h, compa ed o billions o
sola - ype s a s. Fu he mo e, much mo e so han he low-mass s a s, massi e s a s
come p edominan ly in pai s, called bina y sys ems. Th oughou hei e olu ion,
s a s in bina y sys ems in e ac , ans e ing mass be ween each o he , which has
majo implica ions o hei ul ima e a e. This is one o he easons ha makes
s udying massi e bina y s a s so in e es ing.
Fo igh bina y sys ems in pa icula , a whole lis o ex a e ec s come in o play.
Some bina ies o bi so close ha hey de o m h ough idal in e ac ion ( he same
p ocess ha sloshes he oceans on Ea h), and in he ex eme case, he s a s come
in o con ac , o ming “peanu s a s.” Massi e con ac bina ies do no necessa ily
immedia ely me ge in o one, as hey a e o a ing a ound each o he e y quickly,
and hey can be s able o millions o yea s. Wha is mo e is ha , by obse ing
samples o con ac bina ies, hey seem o be s able while ha ing unequal masses.
The mass- a io dis ibu ion is ai ly uni o m be ween mass a ios o a ound one
hal and uni y. Un o una ely, no bina y models ha simula e he li e o hese
s a s ha e been able o ep oduce his cons ain , and p edic ha con ac sys ems
should o e whelmingly appea as equal-mass bina ies.
When modeling he li e o s a s, oday, we use compu e s o sol e he se o
di e en ial equa ions ha go e n hei e olu ion. We include se e al e ec s ha
modi y he s uc u e o he s a s, such as o a ion, in e nal mixing, s ella winds, as
well as he bina y-s a p ocess o mass ans e . Up un il oday, idal de o ma ion
and he p ocess o ene gy ans e in con ac bina ies ha e no ye been accoun ed
o in de ailed models o bina y e olu ion. Ins ead, he s a e-o - he-a models a e
sphe ically symme ic, o a mos de o med cylind ically due o o a ion.
Vulga ise ende samen a ing
S e en zijn g o e ballen gas en plasma da ons aan doo de samen ekking an
s o - en gaswolken in he heelal. Hun massa’s kunnen a ië en an enkele
honde ds en o e elijke ien allen maal de massa an onze Zon, dewelke op
zijn beu miljoenen ke en zwaa de is dan de hele Aa de. Als een s e mee
dan onge ee ach kee zwaa de is dan de Zon, sp eken we an een zwa e, o
massie e s e . Massie e s e en zijn we kelijke kosmische machines, omda zij
zwa e elemen en, zoals ijze , in hun ke nen maken en deze dan la e doo heen hun
melkwegs elsel slinge en wannee zij explode en in een supe no a. Laagmassie e
s e en zoals de Zon hebben deze eigenschappen nie . Hie doo is he bes ude en
an massie e s e en heel belang ijk, omda zij e an woo delijk zijn oo he
c eë en an elemen en da le en mogelijk maak in he heelal. IJze is bij oo beeld
een essen ieel ing ediën in he menselijk bloed.
Aan de hand an gea ancee de (compu e )modellen kunnen as o ysici ui sp aken
doen o e he le en an een s e . Zo kunnen we bij oo beeld de empe a uu o de
le ensduu an een s e be ekenen, maa we kunnen ook hun binnens e s uc uu
bes ude en. In de laa s e honde d jaa zijn we zo e we en gekomen da s e en
hun ene gie ui nucleai e usie eac ies halen. De ene gie die in he cen um an
de s e gep oducee d wo d , wo d dan naa de bui ens e lagen ge anspo ee d
o ewel doo s aling, o ewel doo “ e bubbelen,” ne zoals wa e kook in een po
op een gas uu . Al deze p ocessen hebben als einddoel de s uc uu an de s e
s abiel e houden. Moes e imme s geen ene gie gep oducee d wo den, zouden
s e en als een kaa enhuis in elkaa zakken en ge olge an hun immens gewich !
Massie e s e en komen heel aak oo me een pa ne , éél mee dan hun
laagmassie e soo geno en. Doo hun wede zijdse aan ekkingsk ach d aaien zij
ond elkaa , ne als een koppel op de dans loe . Zulke o ma ie noemen we een
binai e s e , o ook een binai sys eem. Soms o men s e en zó dich bij elkaa da
doo heen hun le ens hun bui ens e lagen in con ac komen, een si ua ie waa an
u op de co e een a is ieke illus a ie kan beschouwen. Hun o m lijk wel op een
pindanoo , en de naam “pindanoo -s e ” blij ook bij we enschappe s plakken.
Di is een zee ex eme ase in dewelke binai e s e en kunnen oo komen, maa
xiii
xi VULGARISERENDE SAMENVATTING
och gebeu di ij equen . Gede aillee de modellen le en ons da onge ee 40%
an alle binai e pa en an massie e s e en een con ac ase meemaken. Om goede
oo spellingen e doen o e he le en en de eigenschappen an binai e s e en,
moe de con ac ase dus nauw onde de loep genomen wo den.
In een compu e model an een (binai e) s e is he belang ijk da alle ele an e
ysische p ocessen mee wo den be ekend. In he bijzonde zijn e in een
con ac sys eem namelijk ex a e ec en die oege oegd moe en wo den, die o nog
oe enkel benade d o zel s helemaal e waa loosd we den. Bij oo beeld wo den
s e en aak oo ges eld als pe ec onde bollen, maa he moe duidelijk zijn doo
de illus a ie op de co e da di nie klop oo de s e en in een con ac sys eem.
De e o ming an de s e en is he ge olg an ge ijdenwe king, dezel de k ach
da de zeeën op Aa de op en nee doe gaan. Deze hesis bouw een me hode om
zulke e o ming oo e s ellen in onze compu e modellen. Een weede e ec in
con ac sys emen da e waa loosd we d is ene gie ans e . Omda de bui ens e
lagen in zulke sys emen o e lappen, kan e ene gie loeien an de ene s e naa
de ande e. Hoe de wa m es oom in een con ac sys eem e p ecies ui zie , is nog
onduidelijk. De e wach ing is ech e wel da ene gie loei an de mee helde e
s e naa de minde helde e s e , wa e geleken kan wo den me hoe wa m e
loei an binnenin een he e adia o naa een koele kame e bui en. Di e wach e
esul aa modelle en we in deze hesis, om zo he e ec an ene gie ans e op he
le en an een con ac sys eem e bes ude en.
Ui de nieuwe compu e modellen an con ac sys emen, waa in we de e ec en an
ge ijden e o ming en ene gie ans e in ekening b ach en, moe en we beslui en
da zij nog s eeds de eigenschappen an waa genomen con ac sys emen nie
helemaal kunnen e kla en. Ookal e be e en zij enkele aspec en, moe en e
dus nog e schillende ande e e ec en o p ocessen zijn die we in de modellen
o e he hoo d zien. He laa s e woo d o e con ac sys emen is dan ook nog
nie gesch e en. Desondanks zijn binai e sys emen in con ac unieke labo a o ia
waa mee as o ysici de heo ieën an s e e olu ie kunnen es en, om zo s eeds
be e e en be e e s e modellen e bouwen.
AGB Asymp o ic gian b anch (s a )
BC Bounda y condi ion
BH Black hole
CDF Cumula i e dis ibu ion unc ion
CE Common en elope
CHE Chemically homogeneous e olu ion
ET Ene gy ans e
GRB Gamma- ay bu s
GW G a i a ional wa e
HMXB High-mass X- ay bina y
HRD He zsp ung-Russell diag am
MB Magne ic b aking
MC Mon e Ca lo
MCMC Ma ko chain Mon e Ca lo (me hod)
MESA Modules o Expe imen s in S ella As ophysics (code)
MT Mass ans e
NIR Nea in a ed
PMS P e-main sequence
RGB Red gian b anch (s a )
RL Roche lobe
RLOF Roche-lobe o e low
RV Radial eloci y
S/N Signal- o-noise a io
SB Spec oscopic bina y
SN Supe no a
WR Wol -Raye (s a )
ZAMS Ze o-age main sequence
ZKL on Zeipel-Kozai-Lido (cycles)
Lis o ac onyms
x
Con en s
Abs ac
Beknop e samen a ing ix
Vulga ise ende samen a ing xiii
x
Lis o ac onyms x
Con en s x ii
1 In oduc ion 1
1.1 Single s a s ................................. 4
1.1.1 The equa ions ........................... 10
1.1.2 S a o ma ion ........................... 19
1.1.3 Main-sequence e olu ion .................... 19
1.1.4 Pos -main-sequence e olu ion ................. 22
1.1.5 Open ques ions in massi e-s a s uc u e and e olu ion . . . 25
1.2 Bina y s a s ................................ 30
1.2.1 O bi al cha ac e iza ion and dynamical masses ........ 34
1.2.2 S ella mul iplici y ........................ 37
1.2.3 The Roche po en ial and mass ans e ............. 39
1.3 Con ac bina ies .............................. 44
1.3.1 Obse ed sys ems ........................ 45
1.3.2 Modeling e o s ......................... 46
1.3.3 E olu iona y pa hways – Me ge s ............... 48
1.4 G a i a ional wa e as onomy ..................... 50
1.4.1 G a i a ional-wa e p ogeni o s ................. 53
1.5 This hesis ................................. 57
2 The dynamical mass o 9 Sagi a ii 61
x ii
x iii CONTENTS
2.1 In oduc ion ................................ 62
2.2 Obse a ions ................................ 64
2.2.1 Op ical spec a .......................... 64
2.2.2 Nea -in a ed In e e ome y .................. 65
2.3 O bi al analysis .............................. 67
2.3.1 As ome ic o bi ......................... 67
2.3.2 Spec al disen angling ...................... 68
2.3.3 Dis ance .............................. 72
2.4 A mosphe e modeling .......................... 74
2.4.1 Se up ................................ 74
2.4.2 Resul s and discussion ...................... 75
2.5 E olu iona y modeling .......................... 76
2.6 Conclusions ................................ 78
3 Modeling idal dis o ion in 1D s ella s uc u e 81
3.1 In oduc ion ................................ 82
3.2 Me hods .................................. 84
3.2.1 Modi ica ions o he sphe ical s ella s uc u e equa ions . . 84
3.2.2 Single s a o a ion ........................ 86
3.2.3 Synch onized Roche bina ies .................. 87
3.2.4 Con ac shells ........................... 88
3.2.5 Nume ical calcula ions ...................... 89
3.2.6 In eg a ion esul s ........................ 90
3.2.7 Compa ison o li e a u e ..................... 91
3.3 The 𝛺𝛤 limi ................................ 92
3.4 A mosphe ic bounda y condi ions ................... 96
3.5 S ella model compa ison ........................ 98
3.5.1 Physical ing edien s ....................... 99
3.5.2 Main sequence e olu ion o a single o a ing s a . . . . . . . 100
3.5.3 E olu ion o a de ached B- ype s a ..............102
3.5.4 E olu ion o a win con ac bina y ...............102
3.6 Discussion and conclusions .......................104
4 Modeling ene gy ans e in con ac bina ies 107
4.1 In oduc ion ................................108
4.2 Theo y o ene gy ans e ........................109
4.2.1 Simple conside a ions ......................109
4.2.2 Models o ene gy ans e ....................111
4.2.3 Ene gy ans e in he Roche geome y ............112
4.3 Me hods ..................................112
4.3.1 Physical assump ions in MESA ..................112
4.3.2 Shellula i y and Roche lobe geome y .............113
4.3.3 Mass ans e ...........................115
4.3.4 Ene gy ans e ..........................115
CONTENTS xix
4.4 S ella models ...............................117
4.4.1 De ailed example o ET e olu ion ...............117
4.4.2 Mass e sus luminosi y a ios ..................122
4.5 Conclusions ................................124
5 Ene gy ans e in a popula ion o massi e con ac bina ies 125
5.1 In oduc ion ................................125
5.2 Me hodology ...............................126
5.2.1 Physical assump ions in MESA ..................126
5.2.2 Bina y ini ializa ion .......................127
5.2.3 Ou comes and e mina ion ...................128
5.2.4 Popula ion syn hesis compu a ions ..............128
5.3 Resul s and discussion ..........................130
5.3.1 Roche-lobe o e low a he ze o-age main sequence . . . . . 130
5.3.2 In e ac ion on he main sequence ................137
5.3.3 Luminosi y a ios .........................141
5.3.4 Reju ena ion ...........................142
5.3.5 Unce ain popula ion dis ibu ions ...............145
5.4 Conclusions ................................148
6 Conclusions and ou look 151
6.1 Fu u e wo k and pe spec i es ......................153
A Addi ional igu es and ables o Chap e 2 161
B spinOS 169
C Dis o ion in he single o a ing s a po en ial 173
C.1 Polynomial i s in he o a ing shell po en ial .............173
C.2 Ma ching he idal and single o a ing s a dis o ion models . . . . 175
D Addi ional conside a ions o ene gy ans e 177
D.1 Ene gy ans e loca ion .........................177
D.2 Con e gence s udy ............................178
D.2.1 Explici e alua ion ........................178
D.2.2 Implici e alua ion ........................178
E Addi ional igu es and ables o Chap e 5 181
Bibliog aphy 185
Lis o Publica ions 209
Chap e 1
In oduc ion
“Space, he inal on ie ...”
— Cap ain James T. Ki k
When you look up on a clea nigh in o a ound Leu en, you can see a housand o
so objec s on he sky. I abo e he ho izon, he moon will be mos p ominen , and
i p esen s i sel as a c escen o a b igh disk depending on i s phase. The nex
b igh es objec s a e he plane s, such as Jupi e , Venus, Ma s and Sa u n. E en
wi h a modes elescope o good binocula s, you can see Sa u n’s ings, o he big
ed spo o Jupi e . Me cu y can only be seen when i is e lec ing a good amoun
o sunligh back o he Ea h, while o U anus you will p obably need o squin
you eyes in o de o see i . Nep une is in isible o he naked eye in any bu he
e y da kes skies, o which he Belgian ones anno 2024 de ini ely do no quali y.
Nex in line a e objec s ou side ou own sola sys em, like s a s, galaxies and
nebulae. In con as o he plane s, hese objec s a e up o housands o ligh yea s
away in he case o s a s o nebulae in ou own Galaxy o millions o e en billions
o ligh yea s o o he galaxies. The b igh es s a in he sky is Si ius, he ‘Dog
s a ’, and can be seen owa d he sou h in win e ime. I is so b igh i bea s
signi ican impo ance in many socie ies and cul u es h oughou his o y. I was
used by he Egyp ians o p edic he looding o he Nile (Clage ,1989), while
many Polynesian and o he sea a ing na ions used i (and o he b igh s a s) o
na iga ion pu poses (Ba eson,1959). In mode n day Leu en some o he b igh
s a s you will be able o spo easily include Pola is, he pole s a , Be elgeuse and
Rigel in he cons ella ion o O ion, Cas o and Pollux in Gemini ( wins), and o he s
like Vega, Deneb and Al ai . When i comes o seeing galaxies, nebulae and s a
1
8 INTRODUCTION
This numbe is a ound en imes bigge han he es ima ed age o he uni e se!
I he Sun has an age o se e al billion yea s, only a ew pe cen o he mass o
he Sun needs o be ac ually a ailable as usible hyd ogen. The s a s hus ha e
plen y o nuclea uel in hei anks o explain hei b igh ness and age. This also
means ha almos all ene gy sou ces on Ea h a e nuclea in o igin.
4
Wi h his
inal ealiza ion, i was Edding on who comple ed he se o equa ions ha go e n
he li e o a s a , he gene al o m o which has emained unchanged since.
Fu he ad ances we e made hanks o new quan um-mechanical insigh s. Mo e
accu a e heo ies on he opaci y o s ella ma e ial (i.e., he deg ee wi h which
pho ons in e ac wi h a oms) we e de eloped (K ame s,1923;Rosseland,1924).
Addi ionally, he p ecise a es a which nuclea eac ions ake place and he
amoun o ene gy eleased as a esul o hem, depending on he empe a u e and
densi y, could be compu ed om new nuclea heo ies (A kinson & Hou e mans,
1929;A kinson,1936;Be he & C i ch ield,1938;Be he,1939). As a consequence
o he exclusion p inciple pu o wa d by Wol gang Pauli (Pauli,1925), which
s a es ha pa icles wi h hal -in ege spin (such as elec ons) canno occupy he
same quan um s a e, elec on degene acy p essu e was disco e ed and applied
o as ophysics by Fowle (1926) in he o m o an equa ion o s a e independen
o empe a u e. Elec on degene acy would be ins umen al in unde s anding
he la e s ages o s ella e olu ion. Mos no ably, Sub ahmanyan Chand asekha
compu ed, using he ela i is ic e sion o elec on degene acy, ha he e is a
maximal mass o an objec ha ing his equa ion o s a e (Chand asekha ,1931a,b).
Despi e a a he public dispu e be ween he well es ablished Edding on, who
ejec ed he heo y o he up-and-coming s uden Chand asekha (Mena & Pe es,
2018), his idea u ned ou o be co ec , and he limi ing mass o a whi e dwa
(see Sec . 1.1.4) is now known as he Chand asekha limi .
Un il he in en ion o he mul ipu pose, digi al compu e , all calcula ions
in ol ing s ella s uc u e and e olu ion had o be done by hand, which
necessi a ed ha he equa ions we e sol able analy ically. This o en equi ed
making mul iple simpli ica ions and app oxima ions o he physical condi ions,
o example by s a ing ha he gas o he whole s a could be desc ibed by a single
poly ope. A poly ope is he solu ion o he Lane-Emden equa ion (Lane,1870;
Ri e ,1898;Emden,1907), which was he hen-p e ailing equa ion desc ibing
s ella s uc u e. I assumed a single mode o ene gy anspo inside he s a ,
and could no accoun o ene gy gene a ion in he co e. Fu he mo e, Edding on
4
Excep ions o his a e idal powe , which o igina es om he g a i a ional ene gy o he Ea h,
Moon and Sun, and geo he mal ene gy, coming om he hea o Ea h’s in e io . The la e , in u n, is
spli in wo sou ces: g a i a ional om i s o ma ion and se ling o he co e, and nuclea , his ime
om he adioac i e decay o na u ally occu ing adioiso opes wi hin he Ea h. The ac ion ha he
nuclea componen con ibu es o geo he mal ene gy is highly deba ed, and anges om 50% (Gando
e al.,2011) o 80% (Tu co e & Schube ,2014), depending on he model.
SINGLE STARS 9
had o ini ially conside ully homogeneous s a s (wi h a cons an gas- o-pho on
p essu e, a sou ce o deba e) o keep he equa ions manageable, meaning he could
only compu e models o ully-mixed, uni o m s a s.
The homogeneous s ella models o Edding on we e no able o simul aneously
explain he wo popula ions in he HRD. You ha e he gian s a s who we e o de s
o magni ude mo e luminous han he main-sequence s a s (also called dwa
s a s) a oughly he same su ace empe a u e. In 1938, E ns Öpik i s conside ed
s ella models ha we e no homogeneous by en e aining he idea ha he in e io
o s a s is gene ally no well mixed (Öpik,1938). This mean ha he co e o a s a
could un ou o hyd ogen o bu n and become ine . He p edic ed he co e would
apidly con ac and elease g a i a ional ene gy ins ead, expanding he en elope
ha is s ill hyd ogen ich. This p ocess, i.e., he deple ion o hyd ogen in he co e
and he accompanying s uc u al changes o he s a , would na u ally p oduce he
gian s a s in he HRD.
Once compu e s en e ed he scene in he 1950s, s ella models could be compu ed
using nume ical echniques. In hose days, machines wi h he compu a ional
powe o oday’s pocke calcula o s had he size o whole class ooms, bu he
physical p ocesses occu ing in s a s could be p og ammed (wi h ape o punch
ca ds!
5
), and de ailed s ella -e olu ion models we e p oduced as ea ly as 1952.
As compu e s became mo e and mo e pa o he s anda d equipmen o he
as ophysicis , inno a i e echniques in he implemen a ion o he di e en ial
equa ion sol e we e de eloped, mos no ably wi h he con ibu ion by Louis
Henyey (Henyey e al.,1959). The so-called Henyey me hod is s ill used oday o
calcula ing s ella -e olu ion models, and uses a ial solu ion and a gene alized
New on-Raphson oo - inding algo i hm o i e a i ely app oach a be e solu ion
o he di e en ial equa ions.
As a omic and nuclea da a kep imp o ing, in pa d i en by esea ch on nuclea
weapons du ing Wo ld Wa II and he Cold Wa , and compu e s kep ge ing as e
exponen ially, decade a e decade saw subs an ial p og ess in he calcula ion o
s ella models. Ra he han compu ing single models o one pa icula mass and
composi ion, whole se s, o popula ions, o s ella models could be explo ed. This
enabled o s udy he e olu ion o clus e s o s a s, whe e he assump ion can be
made ha all s a s a e bo n mo e o less a he same ime and mo e o less wi h
he same composi ion. Al e na i ely, ha ing a lo o compu ing powe allows o
s udy he impac o a ying ce ain pa ame e s ha desc ibe physical p ocesses,
such as he e iciency o in e nal mixing, he a e a which ce ain nuclea eac ions
occu , o he s eng h o mass loss o he s a du ing i s li e (see Sec . 1.1.5).
5
Anecdo ally, Rudol Kippenhahn ecalls ha , om he pa e ns in he lashing ligh s and noises he
compu e made, he and his collabo a o s could in e which speci ic pa o he s a i was calcula ing
(Ginge ich,1978).
10 INTRODUCTION
Today, al hough supe compu e s a e used o calcula e g ids o ens o e en hun-
d eds o housands o s ella models, as ophysicis s do no need supe compu e s
hos ed a academic o go e nmen al ins i u ions o p oduce de ailed models. Mos
s ella -e olu ion codes a e designed o un on basically any compu e , and i will
ake only on he o de o hou s o comple e he in eg a ion o , o example, he li e
o he Sun, aking a ound en housand in eg a ion s eps. This is in s a k con as
wi h he si ua ion o yes e yea , as Ma in Schwa zschild no ed in 1958:
A pe son can pe o m mo e han wen y in eg a ion s eps pe day ...
so ha o a ypical single in eg a ion o , say, o y s eps, less ha wo
days is needed. — Schwa zschild (1958).
I one would a emp o compu e he s ella models con ained in his hesis
wi hou he help o mode n compu e s, ins ead elying on manual in eg a ion
like in he imes o Schwa zschild, his hesis would equi e an es ima ed 8000
yea s o comple e, and no unding agency would wan o back a g an o ha
long. This goes o show how powe ul compu e s ha e become in jus se en y
yea s o so. P og ess in compu e science does no s and s ill howe e , and,
wi h machine lea ning echnologies inding applica ions in he na u al sciences,
including as ophysics, maybe he ield o s ella s uc u e and e olu ion, in
pa icula he compu a ion o s ella models, will again be e olu ionized soon.
1.1.1 The equa ions
The s udy o s ella s uc u e and e olu ion e ol es a ound he solu ion o a se o
pa ial di e en ial equa ions, each one encoding a undamen al physical p inciple.
They a e:
(i) Conse a ion o mass,
(ii) Conse a ion o momen um,
(iii) Conse a ion o ene gy,
(i ) T anspo o ene gy, and
( ) Changes in composi ion.
Unde gene al condi ions, he se o equa ions associa ed o hese p inciples is
no analy ically sol able since we a e dealing wi h coupled, non-linea , pa ial
di e en ial equa ions in h ee spa ial dimensions plus a ime dimension. Howe e ,
by obse ing ha s a s a e, gene ally speaking, ound balls, we can make use o
SINGLE STARS 11
sphe ical symme y o cas he equa ions in he ollowing o m:
𝜕𝑀
𝜕𝑟 =4𝜋𝑟2𝜌, (1.5a)
𝜕𝑃
𝜕𝑟 =−𝜌𝜕𝛷
𝜕𝑟 =−𝜌𝐺𝑀
𝑟2,(1.5b)
𝜕𝐿
𝜕𝑟 =4𝜋𝑟2𝜀, (1.5c)
𝜕𝑇
𝜕𝑟 =−𝜌𝐺𝑀
𝑟2
𝑇
𝑃∇,(1.5d)
𝜕𝑋𝑖
𝜕𝑡 =∑︁
𝑗
𝑟𝑗𝑖 −∑︁
𝑘
𝑟𝑖𝑘 −1
𝜌𝑟2
𝜕
𝜕𝑟 𝜌𝑟2𝐷𝜕𝑋𝑖
𝜕𝑟 .(1.5e)
We now ha e only one spa ial dimension,
𝑟
, and he ime,
𝑡
, le as independen
a iables. In hese equa ions, se e al unc ions a e de ined. We ha e he se
consis ing o
𝑀=𝑀(𝑟, 𝑡)
, which is he mass a ime
𝑡
enclosed in a sphe e o
adius
𝑟
, and
𝑃=𝑃(𝑟, 𝑡), 𝑇 =𝑇(𝑟, 𝑡)
and
𝐿=𝐿(𝑟, 𝑡)
, which a e, espec i ely, he
p essu e, empe a u e and luminosi y o he s ella ma e ial a he adius
𝑟
and
ime
𝑡
. These ou unc ions speci y he s uc u e o he s a a each poin in ime.
We ha e also
𝜌=𝜌(𝑃, 𝑇, 𝑋𝑖)
, he densi y o he ma e ial, which is compu ed ia
an equa ion o s a e, and he po en ial
𝛷=𝛷(𝑀, 𝑟)
, which is he g a i a ional
po en ial associa ed wi h he mass con igu a ion o he s a , and can be sol ed
om Poisson’s equa ion,
∇2𝛷=
4
𝜋𝐺𝜌
. The unc ion
𝜀=𝜀(𝜌, 𝑇, 𝑋𝑖)
encodes ene gy
gene a ion by nuclea eac ions as well as he modynamic hea ing o cooling by
comp ession o expansion, and
∇
is he loga i hmic de i a i e o empe a u e
o p essu e equi ed o anspo he luminosi y
𝐿(𝑟)
h ough he s a . The las
equa ion wi h
𝜕𝑋𝑖
𝜕𝑡
is ac ually a se o mul iple equa ions ha desc ibes, o each
elemen
𝑖
, he ansmu a ion o
𝑖
in o di e en elemen s
𝑘
a a a e o
𝑟𝑖𝑘
, while
o he elemen s
𝑗
a e changed in o
𝑖
a he a e
𝑟𝑗𝑖
. The hi d e m ep esen s
in e nal mixing o he s ella ma e ial, occu ing wi h di usion coe icien
𝐷
. A
ull solu ion o all equa ions, gi ing he unc ions
𝑀(𝑟, 𝑡), 𝑃(𝑟, 𝑡), 𝑇 (𝑟, 𝑡), 𝐿(𝑟, 𝑡)
and
he elemen al species
𝑋𝑖(𝑟, 𝑡)
, wi h app op ia e ini ial and bounda y condi ions is
hen called a s ella model.
Equa ions o con inui y and hyd os a ic equilib ium
The i s wo equa ions a e esul s o classical physics and we e well unde s ood
al eady a he s a o he wen ie h cen u y. The i s equa ion, Eq.
(1.5a)
, called he
con inui y equa ion, simply s a es ha he mass,
𝑑𝑀
, con ained in a hin shell o
adius
𝑟
is he p oduc o he densi y,
𝜌
, o he shell, i s a ea, 4
𝜋𝑟2
, and i s hickness,
12 INTRODUCTION
Table 1.1: Final assembly eac ions in he pp chain o hyd ogen usion in o helium.
pp-I pp-II pp-III
3He + 3He −−−→ 4He + 2 1H3He + 4He −−−→ 7Be + 𝛾
7Be + e–−−−→ 7Li + 𝜈e7Be + 1H−−−→ 8B + 𝛾
7Li + 1H−−−→ 24He 8B−−−→ 8Be + e++𝜈e
8Be −−−→ 24He
𝑑𝑟
. Equa ion
(1.5b)
asse s ha o ces a ising om he g a i a ional po en ial,
𝛷
,
a e balanced by he in e nal p essu e g adien coming om he gas making up he
s a and he adia ion i p oduces. We call his si ua ion hyd os a ic equilib ium.
In mos o he li e o a s a , hyd os a ic equilib ium is sa is ied, o e y nea ly so.
Howe e , o example, when a massi e s a collapses a he end o i s li e, we
speak o hyd odynamic e olu ion, and an accele a ion e m
−𝜌 𝜕2𝑟𝜕𝑡2
mus be
added o he igh hand side o Eq.
(1.5b)
. Addi ionally, Eq.
(1.5b)
igno es he ac
ha , a he su ace o he s a , adia i e o ces can launch a s ella wind, bu his
e ec is usually aken in o accoun by inco po a ing mass loss in he models (see
Sec . 1.1.5).
Equa ions o ene gy conse a ion & ansmu a ion
The hi d equa ion, Eq.
(1.5c)
, is he equa ion o conse a ion o ene gy and is
in ima ely connec ed wi h Eq.
(1.5e)
speci ying he ansmu a ion o elemen s in o
o he s. As discussed in he in oduc o y pa ag aphs, i was ound in he ea ly
wen ie h cen u y ha he main sou ce o s ella ene gy is nuclea . On he main
sequence, s a s con e hyd ogen in o helium. In low-mass s a s (
𝑀≲
2
𝑀⊙
), his
p ocess is achie ed by di ec assembly, also called he p o on-p o on o pp chain.
Two hyd ogen nuclei,
1H
, which consis o jus a single p o on, combine while one
o hem changes in o a neu on, o ming deu e ium, and emi a posi on
e+
and a
neu ino 𝜈e:1H+1H−−−→ 2H+e++𝜈e(1.6)
Adding hen ano he p o on o ms 3He and emi s a pho on, 𝛾:
2H+1H−−−→ 3He +𝛾(1.7)
Final assembly o
4He
hen occu s by ei he combining wo
3He
nuclei (pp-I) o ,
wi h
4He
al eady p esen , going ia be yllium and li hium (pp-II) o bo on (pp-III),
as p esen ed in Table 1.1. Each o hese eac ions eleases an amoun o ene gy,
𝜀
,
o be used o hold up he s a agains i s own g a i y.
SINGLE STARS 13
Figu e 1.4: CNO cycle o hyd ogen bu ning in o helium. In massi e main-sequence
s a s his is he dominan p ocess om which nuclea ene gy is libe a ed. Adap ed
om Ma chan (2018).
The second way o use hyd ogen is h ough he CNO cycle, which is he dominan
p ocess in in e media e-mass (2
𝑀⊙≲𝑀≲
8
𝑀⊙)
and massi e s a s (
𝑀≳
8
𝑀⊙
).
Ins ead o assembling a helium nucleus om sc a ch by s a ing om single
p o ons, in his p ocess we s a wi h al eady p esen ca bon and ni ogen nuclei,
and add p o ons one a a ime un il a helium nucleus spli s o . This happens in
wo dis inc cycles, going h ough bo h s able and uns able iso opes o ca bon,
ni ogen, oxygen and luo ine as shown in Fig. 1.4. An impo an p ope y o his
cycle is ha he p o on cap u e o ni ogen,
14N+1H−−−→ 15O+𝛾(1.8)
occu s he slowes , meaning ha he CNO cycle, while bu ning hyd ogen o
helium, con e s a ac ion o he ca bon and oxygen in o ni ogen. Obse ing an
enhanced ni ogen abundance on he su ace o a s a is he e o e a good indica ion
ha nuclea ly p ocessed ma e ial o he co e has been b ough o he su ace by
in e nal mixing p ocesses (see Sec . 1.1.5).
When a s a lea es he main sequence as i s hyd ogen uns ou in he co e, he
nex nuclea usion eac ion a ailable is he combina ion o h ee helium nuclei o
a ca bon nucleus, which is called he iple alpha p ocess:
34He −−−→ 12C+𝛾. (1.9)
14 INTRODUCTION
This eac ion is c ucially dependen on he s abili y o he in e media y p oduc
be yllium: 4He +4He −−−→
←−−− 8Be.(1.10)
Be yllium-8 is uns able, and decays back o wo
4He
nuclei wi h a hal -li e o only
abou 8
.
2
×
10
−17
s. Luckily, he densi y and empe a u e in a s ella co e whe e
hese eac ions ake place is high enough o he hi d helium nucleus o encoun e
he 8Be pa icle be o e i decays again, and use o 12C. The inal eac ion:
8Be +4He −−−→ 12C∗−−−→ 12C+𝛾, (1.11)
howe e passes h ough an exci ed esonance s a e o ca bon,
12C*
. P edic ed by
F ed Hoyle in he ea ly i ies (Hoyle,1954), i is now called he Hoyle s a e. The
s a e was expe imen ally ound no much la e a Cal ech (Dunba e al.,1953;
Cook e al.,1957), bu i ook un il 2011 o heo e ically calcula e i s exis ence
(Epelbaum e al.,2011). The ac ha s ella nuclea bu ning has o go h ough
12C*
and ha his s a e exis s almos exac ly a he sum o ene gies o
8Be
and
4He
is o conside able in e es o philosophe s in con ex o he an h opic p inciple
and he ine- uning o he uni e se (K agh,2010). I his s a e did no exis , o
had i s p ope ies al e ed by sligh ly di e en undamen al physical cons an s, he
p oduc ion o elemen s hea ie han helium could ha e been adically di e en ,
o s alled al oge he , and li e as we know i migh no ha e eme ged.
In massi e s a s hea ie han abou 8
𝑀⊙
, he monuclea bu ning p oceeds om
12C
o hea ie elemen s like
16O
,
20Ne
,
28Si
, and inally
56Fe
. All hese s eps use
mul iple in e media y nuclei and in e ac ion channels, much oo la ge o lis in
his hesis. Ins ead we e e o wo ks on s ella nucleosyn hesis, such as Clay on
(1983). We do howe e no e he eac ion
12C+4He −−−→ 16O+𝛾, (1.12)
which has a pa icula ly ha d o de e mine eac ion a e (deBoe e al.,2017).
Because his a e es ablishes he p ecise ca bon o oxygen a io a e helium
bu ning, i has impo an implica ions o he inal a e o he massi e s a (e.g., on
he masses o black holes, Fa ag e al.,2022).
Figu e 1.5 shows he binding ene gy pe nucleon o he mos s able iso ope o each
mass numbe
𝐴
. I shows ha he i s s ep in he usion chain, hyd ogen bu ning
o helium, is he mos ene ge ically a o able eac ion since he jump in binding
ene gy is by a he la ges . This ansla es in he ac ha a s a will spend he as
majo i y o i s li e bu ning hyd ogen in i s co e. Each p og essi e bu ning s age
eleases less ene gy, which means hese s ages will ake less ime o comple e. As
soon as
56Fe
is o med in he co e o a massi e s a , nuclea usion s ops as i is
no longe ene ge ically a o able o he s a . The whole machine y o p oducing
ene gy o main ain a p essu e g adien which coun e ac s g a i y alls apa like
a house o ca ds. This announces he d ama ic end o he li e o he massi e s a
wi h a supe no a (see Sec . 1.1.4).
SINGLE STARS 15
Figu e 1.5: Binding ene gy pe nucleon o he mos s able iso ope pe mass numbe
𝐴
. Indica ed a e se e al impo an iso opes in s ella nucleosyn hesis, as well as
sil e -108, gold-198 and u anium-235, he la e being he p ime issile ma e ial
in mode n nuclea powe plan s. The e ical line indica es i on-56, which is he
mos e icien ly bound nucleus. Nuclea da a om Huang e al. (2021); Wang e al.
(2021).
In many o he nuclea eac ions o he pp chain and he CNO cycle e e ed o
abo e, neu inos a e p oduced as a esul o in e ac ion wi h he weak nuclea
o ce. Fu he mo e, in bu ning s ages o ca bon and la e , he condi ions in he
s a a e such ha pho ons a e ene ge ic enough o c ea e an elec on-posi on
pai , which occasionally u n in o a neu ino-an ineu ino pai . Howe e , once
p oduced, neu inos a e no o iously non- eac i e wi h p o ons and elec ons, and
he e o e do no deposi hei ene gy in o he gas o he s a . S a s hus lose ene gy
o neu ino p oduc ion, which can be subs an ial in he la e phases o massi e-s a
e olu ion. Addi ionally, gas could be hea ing, cooling, con ac ing o expanding,
all o which a ec s he ene gy balance in he s ella laye s. Ra he han hiding
all possible ene gy e ms in he gene al unc ion
𝜀
, he ene gy equa ion is mo e
explici ly w i en as:
𝜕𝐿
𝜕𝑟 =4𝜋𝑟2𝜀nuc −𝜀neu −𝑐𝑃
𝜕𝑇
𝜕𝑡 +𝛿
𝜌
𝜕𝑃
𝜕𝑡 ,(1.13)
whe e he ou e ms on he igh hand side a e, in o de , he nuclea ene gy
eleased om usion, ene gy losses by neu inos, he modynamic hea ing o
cooling wi h speci ic hea capaci y
𝑐𝑃
and olume ic expansion o comp ession
wi h 𝛿=−𝜕ln 𝜌/𝜕ln𝑇.
16 INTRODUCTION
Equa ion o ene gy anspo
The anspo o ene gy wi hin he s a , speci ied by he ou h equa ion, Eq.
(1.5d)
,
was i s hough o be media ed solely by con ec i e blobs o gas. Con ec ion
happens all a ound us, ake o example boiling wa e on a s o e. The in ense hea
nea he bo om o he po c ea es bubbles ha ise o he su ace. The same e ec
also powe s ocean cu en s, wea he pa e ns in Ea h’s a mosphe e, and causes
ec onic ac i i y o Ea h’s c us as he man le below con ec s hea om he co e
o he su ace. Howe e , i was pu o wa d by Ka l Schwa zschild ha adia ion
can also play a signi ican ole in he anspo a ion o ene gy om he co e o he
ou e laye s o a s a (Schwa zschild,1906). This idea was pu on solid heo e ical
g ounds some ime la e by Edding on (1916), whe e he also discussed he sou ces
o opaci y inside s ella ma e ial.
The ype o ene gy anspo , adia i e o con ec i e, is an impo an ac o in
de e mining he s uc u e o a s a . Conside ing i s adia i e anspo , which is a
di usi e p ocess, Fick’s law applies (Fick,1855), which, when applied o adia ion,
gi es he adia i e lux:
F ad =−4𝑎𝑐
3
𝑇3
𝜅𝜌 ∇𝑻.(1.14)
He e he opaci y,
𝜅
, o he ma e ial is used, and
𝑎=
8
𝜋5𝑘4
B/(
15
𝑐3ℎ3)
is he adia ion
densi y cons an , de ined using he undamen al cons an s o Bol zmann,
𝑘B
, and
Planck,
ℎ
. Using hen ha he luminosi y is
𝐿(𝑟)=
4
𝜋𝑟2F ad
, om Eq.
(1.14)
we
ha e o a sphe ically symme ic con igu a ion:
𝜕𝑇
𝜕𝑟 ad
=−3
16𝜋𝑎𝑐
𝜅𝜌𝐿
𝑟2𝑇3,(1.15)
which, di iding by he equa ion o hyd os a ic equilib ium, de ines a adia i e
empe a u e g adien ∇ ad:
∇ ad ≡dln𝑇
dln 𝑃 ad
=3
16𝜋𝑎𝑐𝐺
𝜅𝐿𝑃
𝑀𝑇4.(1.16)
I ene gy is anspo ed ully by adia ion, he empe a u e wi hin he s a will
hus a y wi h p essu e as gi en by Eq. (1.16).
Con ec i e ene gy anspo will occu i gas mo ion is uns able agains adiaba ic
displacemen . Pu simply, i a blob o gas ises (o sinks) in he s a and inds i sel
in an en i onmen ha makes i ise (o sink) e en mo e, la ge-scale con ec i e
bubbles will be c ea ed. De ining he adiaba ic empe a u e g adien a any poin
in he s a :
∇ad ≡dln𝑇
dln 𝑃𝑠
,(1.17)
SINGLE STARS 17
he subsc ip meaning o ake he de i a i e wi h cons an speci ic en opy,
𝑠
, a
s ella laye will be uns able o con ec ion i :
∇ad <∇ ad.(1.18)
This is called he Schwa zschild c i e ion o con ec ion (Schwa zschild,1906). The
Belgian as ophysicis om Liège, Paul Ledoux, modi ied his c i e ion by also
conside ing s abilizing molecula g adien s (Ledoux,1947). He ealized ha when
a con ec i e blob mo es o a egion whe e he chemical composi ion is di e en ,
when accoun ing o his in he s abili y c i e ion, an ex a e m needs o be added
ha depends on he composi ion g adien . The Ledoux c i e ion o con ec ion is:
∇ad <∇ ad −𝜑
𝛿∇𝜇,(1.19)
wi h he de ini ions
𝜑≡𝜕ln 𝜌/𝜕ln 𝜇
,
𝛿≡ − 𝜕ln 𝜌/𝜕ln𝑇
and
∇𝜇≡𝜕ln 𝜇/𝜕ln 𝑃
o
w i e he s abili y condi ion concisely. The mean molecula weigh ,
6𝜇
, is de ined
as he amoun o mass he a e age ee pa icle has in uni s o 𝑚u:
𝜇=𝜌
𝑛𝑚u,(1.20)
wi h
𝑛
he numbe o ee pa icles pe uni olume in he mix u e. A highe
𝜇
hus means ha pa icles ha e mo e nucleons on a e age. Con e sely, a lowe
𝜇
means ha pa icles a e ligh e , o ha he mix u e is mo e ionized, which
inc eases he numbe o ee elec ons. Looking a Eqs.
(1.16)
and
(1.19)
, we see
ha con ec ion will se in i ei he he luminosi y o mass a io,
𝐿/𝑀
, o he opaci y,
𝜅
, is pa icula ly high, while s eep composi ion g adien s,
∇𝜇
(e.g., le behind by
nuclea bu ning), ha e s abilizing e ec s.
Compu ing he ac ual empe a u e g adien ,
∇
, o he s a in a con ec i e zone
equi es a heo y o u bulen mo ion. To emphasize,
∇
o a con ec i e egion
is no a esul o he s ella -s uc u e equa ions, bu is a he an inpu equi ed o
desc ibe he empe a u e g adien o a con ec i e egion. Desc ibing con ec ion is
ha d, as i is a u bulen low ha is chao ic. In p inciple, he ull Na ie -S okes
equa ions need o be sol ed in h ee dimensions, a e which pa ame e iza ions o
one dimension can be a emp ed. We can howe e make he app oxima ion ha
con ec ion will occu on a ypical leng h scale, and conside ha all con ec i e
bubbles a el ha leng h and hen p omp ly deposi hei excess hea in o he
en i onmen . This se o assump ions de ines he mixing leng h heo y (MLT)
o con ec ion, and was pionee ed by E ika Böhm-Vi ense as she i s desc ibed
con ec ion in he Sun (Vi ense,1953;Böhm-Vi ense,1958). MLT allows o compu e
a one-dimensional e iciency o con ec ion,
𝜁
, by compa ing he con ec i e hea
6
This is, in my opinion, ano he con using misnome , as, in he con ex o s ella s uc u e, his has
no hing o do wi h weigh no wi h molecules. I p e e mean pa icle mass.
24 INTRODUCTION
Figu e 1.10
: C ab Nebula as
imaged by he James Webb
Space Telescope. I is he
emnan o a supe no a ha
e up ed in 1054. Image
c edi : NASA, ESA, CSA,
STScI, T. Temim.
collapses unde i s own weigh , causing a iolen supe no a accompanied by a
neu ino ou bu s (howe e , a supe no a can be a oided al oge he , see below).
In a supe no a, pa o he mass o he massi e s a p ogeni o (including i s
nuclea ly p ocessed ma e ial as well as hea y elemen s p oduced in he supe no a
i sel ) is ejec ed in o he in e s ella medium, making massi e s a s he p ima y
engine o chemical en ichmen in galaxies (Bu bidge e al.,1957). Some o he
bes s udied supe no ae a e he ones ha occu ed in ou galac ic neighbo hood.
In 1054, Chinese and Japanese as onome s de ec ed a “gues s a ” (S ephenson
& G een,2003), which e ol ed in o he C ab Nebula we see oday (Fig. 1.10).
Mo e ecen ly, SN 1987A e up ed in Feb ua y 1987 in he La ge Magellanic Cloud
(LMC), one o he sa elli e galaxies o ou Milky Way. I was he i s nea by
supe no a s udied wi h mode n elescopes and ins umen s, including neu ino
de ec o s, and hus i p o ided excellen obse a ional da a abou he physics o
co e-collapse supe no ae.
The emnan o a co e-collapse supe no a is ei he a neu on s a o black hole (BH),
he o me being a dense ball o nuclea ma e ial held up by neu on degene acy
p essu e and s ong nuclea in e ac ions, he la e being a singula i y ha can
cu en ly only be desc ibed by Eins ein’s heo y o gene al ela i i y (Eins ein,
1915;Schwa zschild,1916). The main pa ame e s ha de e mine wha emnan
(i any) will be le behind a e a supe no a is he mass o he s ella co e, and
he me allici y o he ma e ial, i.e., he ac ion o mass in me als. Me allici y is
impo an as i d i es mass loss (see Sec . 1.1.5) and so i in luences he mass
o he ca bon-oxygen co e by he ime helium is deple ed. Fu he mo e, he
explodabili y o he co e depends on he complex in e ac ion be ween neu inos
and he co e s uc u e i sel ha is beyond he scope o his hesis (see Hege
SINGLE STARS 25
e al. 2003,2023, o summa ies). Finally, i he co e is oo ha d o explode, a
supe no a may no occu a all, and he s a collapses di ec ly in o a BH wi hou
launching much ma e ial in o in e s ella space. The main esul om all hese
conside a ions is ha i is ha de o o m BHs a sola me allici y han a low
me allici y, whe e ca bon-oxygen co es a e hea ie . Those ha do o m a e due o
supe no a allback, whe e ma e ial ha is launched in he supe no a shock d ops
back on o he p o o-neu on s a and ips i o e he mass limi o become a BH.
Howe e , he o bi al cha ac e is ics o Cygnus X-1 and VFTS 243 indica e ha BH
o ma ion h ough di ec collapse is likely wha occu ed in hese sys ems, and
bo h a e loca ed in he high-me allici y en i onmen s o he LMC and he Milky
Way (Mi abel & Rod igues,2003;Shena e al.,2022). We d aw his conclusion
because he eccen ici y o he o bi is e y low, which sugges s ha he e was
e y li le mass loss du ing he o ma ion o he BH, hus a o ing a di ec collapse
o e a supe no a.
1.1.5 Open ques ions in massi e-s a s uc u e and e olu ion
The physical p ocesses ou lined abo e gi e, gene ally, a good pic u e o massi e-
s a s uc u e and e olu ion. We can explain he gene al ends in he li e o a
massi e s a , and unde s and he chemical en ichmen by hea y elemen s esul ing
om hei winds and supe no ae. We ha e de ec ed bo h neu on s a s and black
holes le behind by massi e s a s. The i s neu on s a was disco e ed by Jocelyn
Bell using adio obse a ions (Hewish e al.,1968), while he i s con incing black
hole candida e was Cygnus X-1, iden i ied independen ly by Bol on (1972) and
Webs e & Mu din (1972) using a combina ion o X- ay obse a ions om he
UHURU space elescope and measu emen s om adio an ennas.
Despi e hese successes, he e a e s ill mul iple unsol ed disc epancies be ween
heo y and obse a ion ega ding s ella e olu ion, pa icula ly so in he massi e-
s a egime, some o which we ou line below.
Mass disc epancy
A majo p oblem p esen in he li e a u e is he so-called mass
disc epancy p oblem. Fo single s a s, he e a e wo main ways o calcula ing i s
mass. One in ol es measu ing he luminosi y and e ec i e empe a u e o he
s a , and ma ching hem o he posi ion o e olu iona y models in he HRD; he
bes i gi ing he e olu iona y mass. The o he equi es a measu emen o he
su ace g a i y
𝑔=𝐺𝑀/𝑅2
h ough spec oscopic da a, which, i he adius can be
es ima ed om he luminosi y and e ec i e empe a u e, gi es he spec oscopic
mass o he s a . Many s udies howe e ind a sys ema ic misma ch o bo h
mass calcula ions (He e o e al.,1992;Mokiem e al.,2007;Ma ins e al.,2012;
Tkachenko e al.,2020;Mahy e al.,2020). A his momen i is unclea whe he
26 INTRODUCTION
he p oblem lies in he accu acy o de e mining su ace g a i ies, which can be
challenging o O s a s, o i he e olu iona y acks su e om unce ain physical
modeling.
Mass loss
Massi e s a s ha e a s ong s ella wind on he main sequence (o
o de 10
−6𝑀⊙y −1
o mo e). This is in con as wi h low-mass s a s, which lose
negligible mass on he main sequence (e.g., he Sun has a mass loss a e o me ely
10−14 𝑀⊙y −1).
Since s ella -s uc u e and e olu ion heo y does no (a p io i) p edic he mass-loss
a es o s a s, o he heo ies o expe imen s need o be used. The i s e idence o
s ella winds in massi e s a s came om Mo on (1967), who used measu emen s
in he ul a iole o in e ou lows om h ee s a s.
9
La e , i was p oposed by
Lucy & Solomon (1970) and Cas o e al. (1975) ha such ou lows a e d i en by
he abso p ion and sca e ing o adia ion by elemen s (in pa icula me als) in he
ou e laye s o he s a . The adia ion coming om he in e io he e o e deposi s i s
momen um in hese a oms and d i es hem ou . Ac ually quan i ying how s ong a
adia ion ield can accele a e a oms his way is a di icul p oblem. Vink e al. (2001)
calcula ed he in e ac ions o a la ge numbe o pho ons wi h o e 10
5
di e en
a omic lines o elemen s anging om hyd ogen o zinc. Mo e ecen calcula ions
howe e p edic lowe mass-loss a es (Sundq is e al.,2019;Bjö klund e al.,
2021,2022). Fo example, a 15
𝑀⊙
s a wi h luminosi y o
log(𝐿/𝐿⊙)=
4
.
5 and a
su ace empe a u e o 30000K has a p edic ed mass-loss a e by Vink e al. (2001)
o
¤
𝑀≈
1
×
10
−8𝑀⊙y −1
, while Bjö klund e al. (2022) p edic
¤
𝑀≈
2
×
10
−9𝑀⊙y −1
,
almos an o de o magni ude lowe .
I has become appa en ha mass loss, bo h on he main sequence and beyond,
can signi ican ly change he e olu ion o a massi e s a . Fo example, i mass
loss du ing he ed supe gian phase is pa icula ly s ong and blows away he
whole hyd ogen ich en elope, he ho helium co e can become exposed o ou side
obse e s. I i does, such objec s a e called Wol -Raye (WR) s a s, a e Cha les
Wol and Geo ges Raye disco e ed hey had s ong and b oad emission lines
in hei spec a, which a e di ec consequences o an op ically hick wind. I is
howe e unclea i WR s a s can be p oduced om a supe gian wind alone (e.g.,
Lange e al.,1994;Meyne & Maede ,2005), o i bina y in e ac ion is needed o
ake away he hyd ogen laye s (e.g.,Pauli e al.,2022). The e o e, unde s anding
he e olu iona y pa hway and obse able quan i ies o a s a depends on knowing
he p ecise mass-loss a es ha massi e s a s ha e du ing he a ious phases o
hei li es.
9
Since ul a iole ligh is la gely abso bed by he a mosphe e, he needed a ocke -moun ed
spec og aph o ge o al i udes o 200km o ge su icien signal.
SINGLE STARS 27
Ro a ion and in e nal mixing
In s a ing he equa ions o s ella s uc u e in
Eqs.
(1.5)
, we assumed no o ces o he han sel -g a i y and in e nal p essu e
g adien s ac on he s ella ma e ial. Howe e , as soon as a ball o gas is spinning
abou an axis, ano he e m should be conside ed in he balance o o ces o
Eq.
(1.5b)
. To sus ain o a ional mo ion, a cen ipe al accele a ion is needed, which
will be supplied by he g a i a ional a ac ion. Con e sely, when iewed in he
ame o e e ence o he o a ing s a , a cen i ugal o ce appea s ha poin s away
om he o a ion axis. In he equa ion o hyd os a ic equilib ium, we w i e i as:
𝜕𝑃
𝜕𝑟 =−𝜌𝜕𝛷
𝜕𝑟 −𝜌𝑟 𝛺2,(1.24)
whe e
𝛺
is he angula eloci y abou he axis o o a ion. A majo issue wi h
Eq.
(1.24)
is ha i is only alid o in he equa o ial plane o a s a . By in oducing
o a ion, we necessa ily added an independen spa ial coo dina e, ypically chosen
o he he pola angle,
𝜃
. In 2D hen, he equa ion o hyd os a ic equilib ium,
including o a ion, eads:
∇𝑃=−𝜌∇𝛷−1
2𝜌𝛺2∇(𝑟sin 𝜃)2.(1.25)
No e ha he g a i a ional po en ial
𝛷
is also no longe sphe ically symme ic,
since he mass dis ibu ion is de o med. In p inciple his means all equa ions o
s ella s uc u e need o be sol ed in wo dimensions,
𝑟
and
𝜃
. While possible, his
usually means sac i icing sol ing he ime dependen p oblem, so no e olu ion is
simula ed, as is done by Roxbu gh (2004) and Espinosa La a & Rieu o d (2013).
Deup ee (1990,1995) and ecen ly, Momba g e al. (2023) succeeded in compu ing
wo-dimensional s ella -s uc u e models while also compu ing he e olu ion
along he main sequence. An al e na i e ea men o o a ion exis howe e , as
Kippenhahn & Thomas (1970) p o ided a me hod o include o a ion in o 1D
s ella -s uc u e calcula ions, which we will in oduce in Sec . 3.2.
Ano he unce ain physical p ocess associa ed o o a ion is o a ionally induced
mixing. Ve y ea ly on, i was iden i ied by Ed a d Hugo on Zeipel ha a o a ing
s a canno be in hyd os a ic and adia i e he mal equilib ium a he same ime
( on Zeipel,1924). This a ises since, in hyd os a ic equilib ium, he amoun o
ene gy ha adia ion can anspo is p opo ional o he magni ude o he local
e ec i e g a i y,
F ad ∼𝒈e ≡ −∇𝛷−1/2𝛺2∇(𝑟sin𝜃)2
. This ela ion is called on
Zeipel’s heo em. In adia i e he mal equilib ium, he equa ion o conse a ion
o ene gy can be w i en as:
∇·F ad =𝜀nuc (1.26)
I a s a is in hyd os a ic equilib ium, laye s o cons an densi y, empe a u e,
p essu e and composi ion align wi h su aces o cons an e ec i e po en ial
𝛹=
𝛷+1
2𝑟2𝛺2
( on Zeipel,1924). Von Zeipel hen showed ha , on an equipo en ial
su ace, he le -hand side o Eq.
(1.26)
is a unc ion o he pola angle
𝜃
, while he
28 INTRODUCTION
Figu e 1.11: Ci cula ion cu en s in a slowly, uni o mly o a ing s a . These cu en s
balance ou he non-uni o m adia i e ene gy anspo as a esul o on Zeipel’s
heo em. Figu e om Swee (1950).
igh -hand side is cons an , since
𝜀nuc =𝜀nuc(𝜌, 𝑇, 𝑋𝑖)
is a unc ion o mic ophysical
p ope ies. In hyd os a ic equilib ium hen, unde gene al condi ions, adia i e
equilib ium canno be sa is ied.
I is ha d o imagine ha he nuclea usion p ocesses conspi e wi h s ella o a ion
o keep isoba s and su aces o cons an empe a u e aligned wi h he e ec i e
po en ial. Ins ead, Edding on (1925) and Vog (1925) independen ly p oposed a
solu ion o his p oblem by conside ing me idional ci cula ion cu en s. These
cu en s ope a e o ca y away he excess ene gy gene a ed in one egion and
deposi hem in ano he so as o main ain adia i e he mal equilib ium and
hyd os a ic equilib ium. La e , Swee (1950) calcula ed he ci cula ion eloci ies in
he case o a slowly, uni o mly o a ing s a , and p oduced he pic u e in Fig. 1.11.
Such lows a e now called Edding on-Swee ci cula ions.
While he o igin o me idional ci cula ion is easonably well unde s ood, he
mixing o s ella ma e ial associa ed wi h hese cu en s is no . F om Fig. 1.11, i
is clea ha ma e ial nea he co e, which is al e ed by nuclea p ocesses, can be
anspo ed up o he su ace. Howe e , his pic u e is a a he simpli ied iew o
he de ailed physics ha happens. Zahn (1992) shows ha u bulen iscosi y is a
he oo o he mixing p ocess, a he han he me idional ci cula ion, and de i es
i can only be e icien o e y as o a o s. Fu he mo e, since used elemen s
nea he co e (like helium, ca bon, e c.) a e hea ie han un ouched hyd ogen in
he en elope o he s a , he e is a molecula weigh g adien ha inhibi s hese
SINGLE STARS 29
ci cula ion cu en s. Di e en g oups implemen his in e ac ion in di e en ways
(compa e, e.g.,Hege e al.,2000, and Meyne & Maede ,2000), and p oduce
di e en esul s. While bo h models ind elemen s like ni ogen a e enhanced
nea he su ace, only he models o Meyne & Maede (2000) p oduce signi ican
helium en ichmen o as - o a ing, massi e main-sequence s a s (unless he s a s
o a e so as hey e ol e chemically homogeneously, see Sec . 1.4.1). Using he
2D ea men , Momba g e al. (2023) implemen ed he heo y o Zahn (1992) and
showed in hei models ha me idional lows a e indeed ine ec i e a anspo ing
chemical elemen s om he co e o he su ace.
Con ec i e-co e o e shoo ing
In massi e s a s, he co e is uns able agains
con ec i e mo ion o ma e ial, while in he en elope, adia i e anspo o ene gy
occu s (see Fig. 1.6). A he edge o he co e is hus a con ec i e bounda y, beyond
which con ec i e blobs should no exis anymo e. I is howe e no un easonable
o hink ha he ine ia o such blobs ca y hem pas he bounda y. This p ocess is
called con ec i e-co e o e shoo ing. The ques ion is how a con ec i e elemen s
can pene a e in o he adia i e en elope (i a all), and in so doing mix he ma e ial
ha would o he wise no pa icipa e in he nuclea bu ning o he co e. The ne
e ec o o e shoo ing is he leng hening o he li e o he s a , since he mo e
nuclea uel a s a has a ailable, he longe i can main ain equilib ium agains
g a i y. Fu he mo e, s onge o e shoo ing inc eases he mass o he helium
co e, which has impo an ami ica ions o he u he e olu ion (and a e) o he
massi e s a .
Jus like desc ibing con ec ion i sel (see Sec . 1.1.1), heo e ically de i ing he
p ope ies o co e o e shoo ing is e y ha d. We no e he e he wo k o Canu o
(1992), who de i ed a heo y o con ec ion ha na u ally includes a desc ip ion o
o e shoo ing, bu i has no ound wide-sp ead use in s ella -s uc u e calcula ions.
Ins ead, o e shoo ing is commonly pa ame e ized by a dimensionless numbe ,
𝛼o
, which signi ies how many p essu e scale heigh s,
𝐻𝑃
, con ec ion ex ends
beyond he con ec i e bounda y se by he Ledoux c i e ion, wi h
𝐻𝑃=−dln 𝑃
d𝑟−1
.(1.27)
Using obse a ions o clus e s o s a s, B o e al. (2011) calib a ed he o e shoo
pa ame e o
𝛼o =
0
.
335 o massi e s a s in he LMC, while Sch ode e al. (1997)
cons ained i o
𝛼o ∈ [
0
.
24
,
0
.
32
]
o in e media e-mass s a s. Mo e ecen ly,
Tkachenko e al. (2020) ound alues consis en wi h
𝛼o =
0
.
4 o mos s a s in a
sample o galac ic eclipsing bina ies.
Ano he way o cons ain o e shoo ing is by employing he heo y o as e oseis-
mology, he s udy o s ella oscilla ions (Ae s e al.,2010). Jus as wi h Ea h’s
seismology, he su ace oscilla ions o s a s allow us o pee inside hem and gauge
30 INTRODUCTION
he mass o he con ec i e co e (see e.g.,Johns on,2021 o a lis o such con ec i e
co e measu emen s), and hus cons ain he s eng h o he o e shoo (e.g.,Ae s
e al.,2003;Mo a eji e al.,2016;Bu ssens e al.,2023).
Finally, wi h inc easing e iciency o ha dwa e and inno a ion in compu a ional
algo i hms, in pa icula he use o CPU pa alleliza ion and GPU compu ing,
mo e and mo e h ee-dimensional simula ions o con ec ion in s ella co es a e
pe o med (e.g.,B owning e al.,2004;C is ini e al.,2017;Edelmann e al.,2019;
Vanon e al.,2023;And assy e al.,2024). Such simula ions could p o ide a di ec
mass and luminosi y dependen measu e o he o e shoo pa ame e , as well as
he empe a u e g adien in he co e.
1.2 Bina y s a s
The his o ical in oduc ion in his sec ion is mainly based on Ai ken (1935).
While ou own Sun is a lonely s a , i is no impossible o s a s o li e and e ol e
nea each o he . One amous example ea u ed in pop cul u e is he s a s hos ing
he plane Ta ooine in he S a Wa s anchise. As Luke looks o e he dese
a sunse (suns-se ? O jus sunse s?), wo eddish s a s a e sinking owa d he
ho izon, see Fig. 1.12. Thanks o New on’s syn hesis o mo ion and g a i a ion
(New on,1687,1999), we know ha he o ces ha keep such s a s e ol ing
a ound each o he a e he same ones ha make he Ea h mo e a ound he Sun, o
ha make apples all om ees.
The s udy o bina y s a s, like egula single s a s, is no a ecen a ai . As ea ly
as he G eek and Roman pe iod, P olemy o Alexand ia used he e m double
s a o desc ibe
𝜈
Sagi a ii, which a e wo s a s sepa a ed by only abou hal
o he Moon’s appa en diame e (which is a ound hi y a cminu es o hal a
deg ee). The i s mode n obse a ions o double s a s a e usually a ibu ed o he
I alian Jean Bap is e Riccioli in 1650, who ound he s a Miza in U sa Majo had
wo b igh componen s. Some his o ians challenge his c edi , and le e s om
Galileo o Benede o Cas elli indica e i was Cas elli who i s esol ed Miza in o
wo componen s as ea ly as 1617 (Ond a,1999). Ch is iaan Huygens and Robe
Hooke, among many o he s, made simila such obse a ions in he se en een h
cen u y. Ini ially, he e was no speci ic in e es in hese objec s, as i was hough i
was me e coincidence ha wo s a s happen o align when iewed om Ea h.
I was in he mid-eigh een h cen u y ha his pe spec i e changed. Based upon
gene al p obabili y a gumen s, John Mi chell a gued in 1767 ha s a s ha appea
close o each o he a e indeed e y likely o physically be nea each o he oo (as
opposed o sca e ed andomly h oughou he galaxy and aligning by chance),
and hus compose a s ella sys em. This ealiza ion spa ked William He schel and
BINARY STARS 31
Figu e 1.12: Bina y s a sunse o e he plane Ta ooine in he mo ie S a Wa s
Episode IV: A New Hope. © Lucas ilm L d., Wal Disney S udios. Rep oduced
unde he Fai Use Ac .
Ch is ian Maye o sys ema ically look o bina y s a s, and by 1782, hund eds
o double s a s we e ca aloged (He schel & Wa son,1782). A he s a o he
nine een h cen u y, i was He schel who made he o mal dis inc ion be ween
double s a s and p ope bina y sys ems, by equi ing o he la e ha he wo
s a s a e “uni ed by he bond o hei own mu ual g a i a ion owa d each o he ”
(He schel,1802). One yea la e , he published he i s se o measu emen s ha
i e ocably p o ed ha New on’s laws de e mine he mo ion o some o he
obse ed double s a s (He schel,1803).
Up o his poin , only isual bina ies we e disco e ed. These a e bina y sys ems
ha a e esol able by di ec imaging wi h elescopes, whe e he size o he main
mi o (and he quali y o he a mosphe e) de e mines how close s a s can be
dis inguished. Howe e , wi h he ad en o spec oscopy, in 1887, An onia Mau y
a Ha a d no iced ha he da k lines in he spec a om Miza A appea ed
single some imes, and double o he imes, see Fig. 1.13. Toge he wi h Picke ing,
hey concluded ha one componen o he isual bina y is in i sel ano he
bina y sys em (Picke ing,1890). Based upon he Dopple -Fizeau p inciple, in
aspec oscopic bina y (SB) hen, as a s a al e na ely mo es owa d and away
om he obse e , a spec al line ge s blueshi ed o sho e wa eleng hs as he s a
is app oaching, o edshi ed o longe wa eleng hs when he s a is eceding. The
e ec we expe ience in daily li e when hea ing an ambulance, ca o ai plane pass
by is analogous. The posi ion o he line co ela es wi h he adial eloci y (RV)
o he s a , i.e., he speed wi h which i app oaches o ecedes om he obse e .
When he spec al line doubled, he Ha a d eam hus saw one s a eceding
and he o he s a app oaching he Ea h, while when he line was single, bo h
32 INTRODUCTION
Figu e 1.13: Two pho og aphic pla es showing he line spli ing o Miza A’s
spec a. On he le panel, a double line can be dis inguished, while on he igh ,
he line appea s single. Images ep oduced om Ond a (1999).
s a s we e mo ing angen ially o he plane o he sky ( ela i e o he mo ion o
he bina y sys em as a whole). Only mon hs la e , He mann Ca l Vogel saw he
spec a o Algol mo e in co ela ion wi h he pe iodic dimming o he s a (Vogel,
1890). The heo y by John Good icke ha a da k(e ) objec passed in on o he
s a (Good icke,1783) hus u ned ou o be co ec , and he class o eclipsing
bina ies had i s i s disco e y.
In he la e nine een h cen u y, me hods o cha ac e izing o bi s we e de eloped.
Tho ald N. Thiele published a me hod o compu ing he o bi al elemen s o a
isual bina y by desc ibing he appa en ellipse he o bi d aws on he sky (Thiele,
1883). This was la e de eloped on by Robe T.A. Innes ( an den Bos,1926),
esul ing in wha is now called he se o Thiele-Innes cons an s desc ibing isual
bina ies. While he equa ion desc ibing he RV o spec oscopic bina ies can be
de i ed om elemen a y geome y, ac ually i ing obse ed RVs was a challenge
in he pas . Today, we can use nume ical me hods and compu ing powe o ind a
well i ing o bi , bu in he ea ly wen ie h cen u y se e al algo i hms in ol ing
i e a ion we e composed, e.g., om Lehmann-Filhés (1894), Schwa zschild (1900)
and Russell (1902), among o he s.
F om he mid- wen ie h cen u y, obse a ions wi h X- ay elescopes unco e ed
ano he way o de ec bina ies. When a s a is dona ing ma e ial o a nea by
companion, an acc e ion disc can o m a ound he acc e o whe e he in- alling
ma e hea s up emendously. This ho gas hen emi s high-ene gy adia ion in
he o m o X- ays, o which a speci ic ype o elescope is needed o de ec hem.
The e y i s such ins umen s we e ocke moun ed o ge abo e he Ea h’s
a mosphe e, bu subsequen designs we e ull space elescopes like UHURU
(Giacconi e al.,1971), Chand a (Weisskop e al.,2000), XMM-New on (Jansen
e al.,2001), and NuSTAR (Ha ison e al.,2013). In o de o each he empe a u es
equi ed o gene a e X- ays, he acc e o needs o be e y compac so he in- alling
ma e gains signi ican ene gy. Ele a ed X- ay emission om an objec is hus a
ell ale sign o a bina y sys em ha is exchanging mass, and i also p o ided he
i s indi ec e idence o black holes exis ing in na u e.
BINARY STARS 33
One o he main bene i s o s udying bina y sys ems is ha hey p o ide a di ec
measu emen o he masses o he s a s h ough Keple ’s laws. Such dynamical
mass es ima es usually su e om much less unce ain y han gauging he mass
om e olu iona y models o ha ing o ely on a mosphe ic modeling as is done
o single s a s. I is s ill a om a i ial ask, as, o isual bina ies, compu ing
he masses equi es knowing he dis ance o he sys em as well as he mo ion o
bo h componen s, a he han hei ela i e mo ion. Fo spec oscopic bina ies, a
majo unknown is he inclina ion o he o bi (i.e., he angle o he o bi al plane
o ha o he sky), and allows only o es ima e a lowe limi on he mass o he
s a s. Gi en hese p oblems, eclipsing bina ies a e pa icula ly use ul, as, in o de
o be eclipsing, he inclina ion mus be e y close o 90 deg ees. Fu he mo e, he
du a ion o he ansi s ela es o he ela i e sizes o he s a s o he o bi and so
mass and adius can be measu ed simul aneously.
Fas - o wa ding o oday, we ha e ca alogs o hund eds o housands o bina y
sys ems, coming om la ge all-sky su eys such as OGLE
10
(G aczyk e al.,2011),
ASAS
11
(Paczynski e al.,2006) and space missions like Keple (Bo ucki e al.,
2010), TESS12 (Ricke e al.,2015) and Gaia (Gaia Collabo a ion e al.,2023). Wi h
such ca alogs, s a is ical s udies on he pe iod, eccen ici y and mass dis ibu ions
can be unde aken o gain a be e unde s anding o s ella and bina y e olu ion.
I would be nai e o assume ha a s a ha is pa o a bina y would e ol e in
p ecisely he same way as i i was single. Se e al key di e ences a e immedia ely
clea :
Ι
n close bina y sys ems, idal in e ac ions can di ec ly in luence he o a ion
a e o ei he s a . The e is a pa allel wi h he Ea h-Moon sys em, as idal
in e ac ions be ween he Ea h and Moon ha e made he o a ion pe iod o he
Moon equal i s o bi al pe iod a ound he Ea h (so ha i always shows he same
ace o us, we call his p ocess idal locking). In a single s a howe e , o a ion
is una ec ed by ex e nal o ces. Bu mo e impo an is he p oximi y o he
companion i sel . As s ella e olu ion d i es s a s o become bigge , a some poin
he mo e massi e s a o he wo, which e ol es as e , g ows o he poin whe e
he g a i a ional pull o he companion on i s ou e a mosphe ic laye s supe sedes
i s own. A ha momen , a phase o mass ans e will be ini ia ed, whe e ma e ial
o he dono s a alls owa d he acc e o s a . Cons uc ing de ailed models o
his o m o bina y in e ac ion, and he impac on he e olu ion o he s a s, is he
subjec o many cu en e o s in bina y-s a as ophysics.
10Op ical G a i a ional Lensing Expe imen
11All Sky Au oma ed Su ey
12T ansi ing Exoplane Su ey Sa elli e
40 INTRODUCTION
gian . I we hen make he app oxima ion ha he dense co es o he s a s a e
poin masses, and ha he o al g a i a ional po en ial is una ec ed by he enuous
ou e laye s o he s a s, we ge :
𝑉(𝒙)=−𝐺𝑀1
|𝒙|−𝐺𝑀2
|𝒙−𝒂|,(1.34)
whe e
𝒂
is he ec o sepa a ing he cen e s o mass o he s a s. F om his po en ial
i ollows ha he cen e s o mass o he s a s mo e a ound one ano he in ellipses,
jus as Johannes Keple ound o he plane s a ound he Sun. I we make one inal
simpli ica ion, namely ha he o bi s o he s a s a e ci cula , and hen mo e ou
e e ence ame in o he o a ing bina y, we a i e a he Roche po en ial o he
bina y s a :
𝑉(𝒙)=−𝐺𝑀1
|𝒙|−𝐺𝑀2
|𝒙−𝒂|−1
2(𝜴×𝒙)2,(1.35)
whe e
𝜴
is he angula eloci y ec o poin ing along he axis o o a ion. I s
magni ude,
𝛺2=𝐺(𝑀1+𝑀2)
𝑎3,(1.36)
ollows om Keple ’s hi d law. Equa ion
(1.35)
is named a e nine een h-cen u y
F ench ma hema ician Édoua d Roche, who s udied he es ic ed h ee-body
p oblem using his po en ial. Figu e 1.16 shows an imp ession o he Roche
po en ial in he equa o ial plane. We can see he wo wells o he poin masses
ha ep esen he hea y co es o he s a s, as well as he ou e egions ha
a e domina ed by he cen i ugal o ce which appea ed when changing in o an
accele a ing e e ence ame. Wi hou any o he o ce, he slopes in Fig. 1.16
de e mine how a ma ble would s a o oll i placed in his po en ial.
I we now conside su aces o cons an po en ial, i.e., equipo en ials, in he
equa o ial plane, we a i e a he con ou s pic u ed in Fig. 1.17. Ma ked he e also
a e he Lag angian poin s, L
1
h ough L
5
. These a e special loca ions whe e he
g adien (i.e., he slope) o he po en ial is ze o. A pa icle ha is exac ly a any
o he Lag angian poin s would hus s ay he e inde ini ely i no dis u bed by
an ou side o ce. Howe e , poin s L
1
, L
2
and L
3
a e uns able, meaning ha he
inies pe u ba ion om hem will ampli y, and he pa icle will lea e ha poin .
In con as , L
4
and L
5
can be s able hanks o he Co iolis o ce ac ing on a mo ing
pa icle (which canno be ep esen ed in a po en ial g aph).14
The exis ence o he Lag angian poin s has impo an ami ica ions o he li e
o s a s ha ha e a nea by companion. Gene ally, a s a will g ow in adius
h oughou i s li e. This is mos appa en when a s a exhaus s i s hyd ogen in he
14
This is no a i ial calcula ion. Only o mass a ios bigge han
𝑀1/𝑀2=25+3√69
2≈
25 a e L
4
and
L
5
s able. Fu he mo e, hey a e ne e asymp o ically s able, meaning a pa icle will ne e exac ly
e u n o L
4
o L
5
, bu only s ay in hei icini y. This nice esul can be ob ained om he es ic ed
h ee-body p oblem by sol ing he linea ized equa ions o mo ion a ound he Lag angian poin s
(G eenspan,2014).
BINARY STARS 41
Figu e 1.16
: Rep esen a ion
o he Roche po en ial in he
equa o ial plane o
𝑞=𝑀2
𝑀1=
0
.
3. The deep g a i a ional
wells a e c ea ed by he
massi e co es o he s a s,
while he ou e egions a e
domina ed by he cen i ugal
o ce.
Figu e 1.17
: Roche equipo-
en ial con ou s in he equa-
o ial plane ( o
𝑞=𝑀2
𝑀1=
0
.
3), along wi h he i e
Lag angian poin s ma ked
as ed do s.
42 INTRODUCTION
co e and becomes a gian , whe e he s ella size changes ypically by wo o de s o
magni ude. Bu , e en du ing he main sequence i sel , as he chemical mix u e in
he co e is slowly changing om hyd ogen o helium, he s a needs o adap and
g ow in size o accommoda e o he inc easing luminosi y gene a ed (see Fig. 1.8).
The i s Lag angian poin , L
1
, is a shows oppe howe e , as, when a s a g ows so
i s su ace c osses L
1
, mass will s a unneling owa d he companion. Figu e 1.18
shows ano he iew o he Roche po en ial, his ime along he axis connec ing he
cen e s o he s a s. In he le panel, bo h s a s si nicely inside hei own wells,
he so-called Roche lobes (RLs), while in he igh panel, he mo e massi e s a has
g own o ill i s RL. We gi e hese bina y con igu a ions he names ‘de ached’ and
‘semi-de ached’, espec i ely. The size o he RL depends on he sepa a ion,
𝑎
, and
he mass a io,
𝑞
, o he bina y, and can be compu ed by pe o ming a nume ical
in eg a ion o he Roche po en ial o Eq.
(1.35)
. A widely used analy ical i o he
olume-equi alen adius, i.e., he adius o a sphe e ha ing he same olume as
he ea d op-shaped RL, exis s (Eggle on,1983):
𝑅RL,1=0.49𝑞−2/3
0.6𝑞−2/3+ln 1+𝑞−1/3𝑎, (1.37)
whe e we de ined
𝑞=𝑀2/𝑀1
. The main ea u e o his equa ion is ha he RL size
inc eases as he sepa a ion (o o bi al pe iod) inc eases, and ice e sa.
The p ocess o mass spilling o e he L
1
poin is called mass ans e (MT), o
Roche-lobe o e low (RLOF), and ma e ial ha was once pa o he dono s a can
end up on he acc e o s a . Fo main-sequence dono s a s, we expec i s adius
o ne e signi ican ly inc ease beyond he RL, because he mass- ans e a e, i.e.,
he amoun o mass ans e ed pe second, inc eases wi h he deg ee o o e low
(Kolb & Ri e ,1990). Fo gian dono s howe e , whose ou e laye s a e a e ied
and enuous, he o e low can become qui e la ge e en o mode a e mass- ans e
a es. Mass- ans e phases a e ca ego ized depending on he e olu iona y s a us
o he dono s a , coined by Kippenhahn & Weige (1967). Fo main-sequence
dono s, we use he case A designa ion. Fo pos -main-sequence dono s, we use
case B and case C o s a s ha ha e exhaus ed hyd ogen and helium in hei
co es, espec i ely.15
I is impo an o conside also he e ec s o mass ans e on he o bi al pa ame e s
hemsel es, and he e we ocus on he case whe e mass ans e is ully conse a i e,
i.e., all mass los om he dono is acc e ed on o he companion. When mass is
ans e ed om a mo e massi e s a o a less massi e s a , conse a ion o (o bi al)
angula momen um says he sepa a ion be ween he s a s mus sh ink. Con e sely,
mass ans e om a lowe -mass s a o a highe -mass s a will widen he o bi .
This means ha a mass- ans e phase o a main-sequence dono (case A) can be
15
Al hough Lau e bo n (1970) s a es Kippenhahn & Weige (1967) ca ego ized all h ee cases, o ou
knowledge, Kippenhahn & Weige (1967) made no men ion o case C. Ins ead, we ind Lau e bo n
(1970) makes he i s men ion o case C mass ans e .
BINARY STARS 43
Figu e 1.18: Con igu a ions in he Roche po en ial. On he le , bo h s a s eside
wi hin hei own Roche lobes, while on he igh , mass ans e h ough Roche-lobe
o e low is occu ing.
spli in o wo sub-phases. Fi s , as he mo e massi e s a eaches i s RL, i will s a
mass ans e o a less massi e companion, meaning he o bi will sh ink, esul ing
in a dec easing RL size. The dono s a canno he mally adjus quick enough o
his new RL size, and will o e low i e en mo e, inc easing he mass- ans e a e,
which exace ba es he p oblem. This scena io is called as o he mal imescale
mass- ans e , as i akes ( oughly) one he mal imescale o he dono o adjus i s
adius o i s new mass (Pols,1994). In mos bina y sys ems, he acc e o s a will be
gaining so much mass du ing as case A i becomes he mo e massi e s a in he
sys em. F om ha momen on, he bina y sepa a ion widens as a esul o u he
mass ans e , which inc eases he RL size o bo h componen s. Once he dono
has egained he mal equilib ium, and he s a can immedia ely adjus i s adius
o i s new mass, he mass- ans e a es emain a he benign as i is he nuclea
expansion o he dono ha d i es mass ans e . This scena io is called slow case
A mass- ans e , and occu s on he nuclea imescale o he dono . In e y close
bina y e olu ion, bo h phases occu one a e he o he (see, e.g., Sec . 4.4.1).
The ans e o mass be ween componen s in a bina y is he main in e ac ion
p ocess by which bina y-s a e olu ion di e s signi ican ly om single-s a
e olu ion. Bina y in e ac ion has been e y success ul in explaining a ious
as ophysical phenomena ha could no be unde s ood in e ms o single-s a
s uc u e and e olu ion. One amous such case is he so-called Algol pa adox.
Algol is bo h a spec oscopic and eclipsing bina y,
16
and om i s analysis he
masses and adii o he wo s a s could be de e mined wi h good p ecision. The
p ima y s a , Algol A, has
𝑀𝐴=
3
.
17
±
0
.
21
𝑀⊙
and
𝑅𝐴=
2
.
73
±
0
.
20
𝑅⊙
, while he
seconda y, Algol B, has
𝑀𝐵=
0
.
70
±
0
.
08
𝑀⊙
and
𝑅𝐵=
3
.
48
±
0
.
28
𝑅⊙
(Ba on e al.,
2012). I we make he assump ion ha hese wo s a s we e o med a he same
ime, hen con en ional, single-s a e olu ion a gumen s should make he mo e
16I is ac ually a iple s a sys em, bu we ocus on he inne bina y he e.
44 INTRODUCTION
Figu e 1.19: Con ac con igu a ion o a bina y s a . Bo h s a s a e o e lowing
hei espec i e RLs, and sha e common laye s abo e L
1
. The su ace laye o he
con ac bina y is a Roche equipo en ial.
massi e s a e ol e as e and become la ge be o e he less massi e s a . In Algol,
his is clea ly no he case as
𝑀𝐴> 𝑀𝐵
, bu
𝑅𝐵> 𝑅𝐴
, so i seems he less massi e
s a has g own as e han he mo e massi e one! In p inciple i could be ha bo h
s a s we e no bo n simul aneously, and la e pai ed up o become he bina y s a
we see oday, bu ha is impossible as a 0
.
7
𝑀⊙
s a has no had enough ime ye o
expand o 3
.
5
𝑅⊙
, he uni e se is oo young. Ins ead, mass ans e can ai ly easily
esol e he pa adox: In he pas , Algol B was mo e massi e han Algol A, and
e ol ed mo e apidly, un il mass was ans e ed om Algol B o Algol A, which
has lipped he mass a io. Algol A and B a e hus a bina y in e ac ion p oduc s.
Algol A appea s younge han i s age sugges s because o much o i s li e i was
much less massi e. Algol B on he o he hand appea s much olde o i s mass, o
he opposi e eason.
1.3 Con ac bina ies
When a s a g ows o ill i s RL, i will s a a phase o mass ans e o i s
companion in a semi-de ached con igu a ion. Howe e , i is possible ha bo h
s a s simul aneously wan o o e ill hei espec i e RLs. In his case, bo h s a s will
g ow o ha e hei su ace laye s on he same equipo en ial somewhe e be ween
L
1
and L
2
, see Fig. 1.19. As a esul , he s a s en e in o a con ac con igu a ion, and
hei shape me i s he colloquialism “peanu s a s.” In he li e a u e, bo h he e ms
o e con ac and con ac a e used o he same con igu a ion. “O e con ac ” mos
likely o igina es om he po man eau o “o e low” and “con ac ,” and some
au ho s dis inguish “o e con ac ” om “con ac ” om he amoun o o e low.
The e is howe e no clea , physical bounda y sepa a ing con ac bina ies om
deep “o e con ac ” bina ies. The e o e, in his hesis, we use he e m “con ac ”
o any con igu a ion whe e bo h componen s o e low hei RLs.
CONTACT BINARIES 45
Figu e 1.20: Su ace model o VFTS 352, a massi e con ac bina y in he LMC.
The colo indica es he local e ec i e empe a u e, which a ies ac oss he su ace
because o he loss o a sphe ically symme ic g a i a ional ield. Rep oduced
om Abdul-Masih e al. (2021).
1.3.1 Obse ed sys ems
The cha ac e iza ion o massi e bina ies is no an easy ask. I equi es using
se e al obse a ional echniques o accu a ely de e mine he o bi al and s ella
pa ame e s. In Sec . 1.2.1, we in oduced spec oscopic and as ome ic cons ain s,
bu a combined spec oscopic and pho ome ic analysis can also ully de e mine a
sys em. Fo con ac bina ies, his is he mos widely used app oach (e.g.,Lo enzo
e al.,2014;Abdul-Masih e al.,2021;Kobulnicky e al.,2022).
One o he bes cha ac e ized massi e con ac bina ies is VFTS 352, in he Ta an ula
s a - o ming egion o he LMC. By simul aneously i ing RV and ligh cu e da a,
Almeida e al. (2015) ound i is a 28
.
6
±
0
.
3
𝑀⊙+
28
.
8
±
0
.
3
𝑀⊙
bina y, i.e., a nea ly
equal-mass sys em. They also ound bo h s a s a e o e luminous and ho e han
wha single-s a heo y would p edic . Abdul-Masih e al. (2021) cha ac e ized
wo mo e con ac bina ies, and simila ly ound o e luminous componen s in bo h
o hem. A po en ial sou ce o his disc epancy is ha 1D, sphe ical a mosphe e
models we e i ed o (clea ly) non-sphe ical, highly dis o ed s a s. Abdul-Masih
e al. (2020) de eloped a spec oscopic pa ch model ha mo e accu a ely i s he
spec a han he 1D models. This me hod assigns a “local” spec um o pa ches
ac oss he su ace, hus aking in o accoun empe a u e a ia ions due o idal
dis o ion (see Fig. 1.20), and in eg a es he con ibu ions o all pa ches o come o
a composi e spec um.
I is qui e challenging o con iden ly say an obse ed con ac bina y s a is indeed
expe iencing a con ac phase, o whe he i is nea ly so in a (semi-)de ached
con igu a ion. Usually, con ac bina y candida es a e iden i ied h ough hei
46 INTRODUCTION
pho ome ic ligh cu es ob ained om s ella a iabili y su eys like OGLE,
ASAS, MACHO, Keple , Gaia and TESS (Szymanski e al.,2001;Paczynski e al.,
2006;Rucinski e al.,2007;P ša e al.,2011a;Ricke e al.,2015;Gaia Collabo a ion
e al.,2016,2023). Such su eys p o ide hund eds o housands o (p ima ily low-
mass) candida es. Un o una ely, a ligh cu e o a con ac bina y looks e y much
like a “nea ly-in-con ac -bu -s ill-de ached” bina y. To con i m he s a us o he
bina y, in pa icula o ha e an independen cons ain on mass a io, spec oscopic
measu emen s need o be aken, whose analysis has i s own challenges alluded o
in he p e ious pa ag aph. E en wi h homogeneous da ase s, Mahy e al. (2020)
could only con i m he con ac con igu a ion o VFTS 352. Th ee o he massi e
con ac bina y candida es (ou o a sample o hi een a ge s) ha e measu ed
pa ame e s whose unce ain y anges make hem consis en wi h a de ached
con igu a ion.
Despi e he obse a ional challenges, o e he pas decade, he ollowing pic u e o
obse ed, massi e con ac bina ies eme ged. A good ac ion o con ac bina ies a e
obse ed wi h unequal-mass componen s, meaning hei mass a io,
𝑞=𝑀2/𝑀1
, is
signi ican ly away om uni y. In Fig. 1.21, we plo he cumula i e dis ibu ion o
obse ed, massi e (nea -)con ac bina ies (Tables E.1 and E.2), and deduce, unde
he assump ion his is a ep esen a i e sample, ha he p obabili y o inding
con ac bina ies wi h mass a ios mo e ex eme han, say,
𝑞=
0
.
85, is a ound
50%. Fu he mo e, om s udying he e olu ion o he o bi al pe iod o e he pas
hi y yea s, Abdul-Masih e al. (2022) de e mined ha he pe iod de i a i e in
six con ac bina ies is a he low, on he o de o he nuclea imescale, meaning
hei mass- a io e olu ion also occu s on he nuclea imescale. I hus seems ha
con ac bina ies wi h unequal-mass componen s a e s able in ha con igu a ion.
This is an impo an obse a ional cons ain ha models o con ac bina ies would
ideally wan o ep oduce.
1.3.2 Modeling e o s
Modeling o con ac bina ies go well unde way in he la e 1960s, hanks o
g owing su eys o a iable s a s. Many o hese a iable s a s u ned ou o
be pa ially eclipsing, low-mass con ac bina ies, called W UMa s a s (Eggen,1967;
Binnendijk,1970). The su eys e ealed many sys ems con ained unequal-mass
componen s. Theo e ically, his was no expec ed, as Kuipe (1941) iden i ied
ha he mass- adius ela ionship o single s a s in equilib ium, combined wi h
he geome ic cons ain om he Roche geome y, did no allow o con ac
bina ies wi h mass a ios di e en om uni y. This disc epancy be ween heo y
and obse a ion is called Kuipe ’s pa adox. To esol e i , Lucy (1967b) i s
conside ed he p ocess o ene gy ans e in he common laye s o he con ac
bina y. Also Bie mann & Thomas (1972), Vilhu (1973), Flanne y (1976), Shu e al.
(1976) and Webbink (1977) all wo ked on ad ancing he models o W UMa s a s.
CONTACT BINARIES 47
Figu e 1.21: Cumula i e p obabili y dis ibu ion unc ion o he obse ed sample
o massi e (nea -)con ac bina ies (see Tables E.1 and E.2). I ollows nea ly a
𝑞3
dis ibu ion. Abou a hal o he sys ems a e obse ed a mass a ios mo e ex eme
han 𝑞=0.85.
Un o una ely, all o hese models con ained inconsis encies, o we e no applicable
o he massi e con ac bina ies. Fo example, Hazlehu s (1993) showed ha he
model o Shu e al. (1976) iola ed he second law o he modynamics, while he
ene gy- ans e model o Lucy (1967b) is no applicable o he adia i e en elopes
o massi e s a s.
Since he ini ial e o s o model W UMa s a s, li le p og ess has been made a
ob aining a comple e model o con ac bina y s a s, especially he massi e ones.
As a esul , much o he heo e ical wo k om i y yea s ago is igno ed in ecen
modeling. Now ha i is clea massi e s a s unde go con ac phases equen ly,
many s udies ocus on close bina y e olu ion wi h speci ic conside a ions o he
con ac phase, e.g. Ma chan e al. (2016), Menon e al. (2021), Sen e al. (2022),
and Henneco e al. (2024). Howe e , hey ea con ac phases in a udimen a y
way only. Mass ans e is compu ed consis en ly, by equi ing ha he mass-
ans e a e is adjus ed so ha he su aces o bo h s a s lie on a common
Roche equipo en ial. Un o una ely, o he e ec s a e igno ed in all models and
calcula ions. Fo example, he sphe ically symme ic s ella -s uc u e equa ions,
Eqs.
(1.5)
, o hei equi alen s o single o a ing s a s, a e sol ed, so ha any
e ec o idal de o ma ion canno be aken in o accoun . Fu he mo e, Has ings
e al. (2020) calcula ed ha in e nal mixing cu en s in idally de o med s a s a e
s onge han he o iginal p esc ip ion o Edding on (1925) and Swee (1950) by
48 INTRODUCTION
abou a ac o o wo. Tidally dis o ed s a s could hus li e longe and ha e la ge
co es as hey mix mo e esh uel in o hei s ella co es. Finally, he p ocess o
ene gy ans e , which was hea ily discussed o W UMa s a s in he se en ies, and
aken up again by a ew pape s in he 2000s (e.g.,Kähle ,2004;Yaku & Eggle on,
2005), has ne e been conside ed in massi e bina y e olu ion.
1.3.3 E olu iona y pa hways – Me ge s
Con ac bina ies ep esen a unique, bu no -uncommon phase in bina y e olu ion.
Recen calcula ions es ima e ha a leas 40% o all massi e s a s unde go a leas
one con ac phase h oughou hei li es (Henneco e al.,2024). Gene ally speaking,
he e a e wo ways a con ac con igu a ion migh o m wo wi h main-sequence
componen s (see Henneco e al.,2024, o a comple e iew o con ac phases in
all s a s). Fi s , when he acc e o in a semi-de ached con igu a ion swells as a
consequence o i gaining mass, i can o e low i s Roche lobe oo and engage in
con ac wi h i s dono (case AR as in oduced in Nelson & Eggle on,2001). This
happens in igo ous, he mal- imescale mass- ans e phases. Second, in cases o
slow mass ans e , he acc e o can “o e ake” he e olu ion o he dono (cases
AS/AE o Nelson & Eggle on,2001). In his case, he acc e o , g owing on i s
nuclea imescale, en e s con ac wi h he dono , and he mass- ans e a e will
soon e e se. Bo h hese scena ios all unde he “acc e o expansion” ou come o
Henneco e al. (2024).
A ound hal o all case A bina y sys ems ha en e a con ac phase a e expec ed o
me ge (Henneco e al.,2024), and his numbe ises o essen ially all sys ems i one
conside s nuclea - imescale con ac phases only (Menon e al.,2021). The me ge
p ocess o wo s a s is a iolen and dynamical occu ence, esul ing in ma e
being lung ou wa d, and a empo a y b igh ening o he sys em (e.g.,Pejcha e al.,
2016). Because he p obabili y o obse ing a phenomenon is p opo ional o
i s du a ion, s ella me ge s a e ha d o ca ch, as hey happen a he quickly on
as onomical imescales. Ne e heless, hanks o long- e m moni o ing p og ams
and ansien su eys, which use dedica ed elescopes ha scan he whole sky
e e y ew days, we ha e caugh se e al e en s called Luminous Red No ae (LRNe;
see Kasliwal e al.,2017;Pas o ello e al.,2019). Such e en s a e dimme han
supe no ae, bu mo e luminous han classical no ae coming om whi e dwa s.
The e up ions o V838 Monoce o is and V1309 Sco pii a e he canonical examples
o LRNe. Bo h we e a he un ema kable s a s ha qui e suddenly b igh ened
by se e al magni udes. V838 Mon e up ed in Janua y 2002, and was i s no iced
by B own e al. (2002) as a “peculia a iable.” In he ollowing yea s, a ious
models o he e en we e pu o wa d, like an expanding ed gian swallowing
i s close plane s (Re e & Ma om,2003), o he me ge o a low-mass s a wi h
an in e media e-mass one (Soke & Tylenda,2003). Because he dis ance o his
CONTACT BINARIES 49
sou ce was adjus ed by analysis o he ligh echo (i.e., he delayed illumina ion o
he su ounding gas and dus ; Tylenda,2004), Tylenda e al. (2005) modeled he
p ogeni o as a massi e,
𝑀∼
8
𝑀⊙
, s a . Mo e ecen ly, mass loss om he ou e
Lag angian poin is also conside ed as he o igin o he e up ion o V838 Mon
(Pejcha e al.,2016). Apa om i s simila i y o V1309 Sco, he p ecise na u e o
he e up ion o V838 Mon has no been unco e ed ye . By compa ing he a ious
p oposed models, Tylenda & Soke (2006) conclude a me ge o a 8
𝑀⊙
s a wi h
a 0
.
4
𝑀⊙
p e-main-sequence s a is he mos p omising (al hough i is unsu e
whe he a con ac bina y o med).
V1309 Sco is pe haps e en be e s udied, because he e is well- esol ed OGLE da a
o he p ogeni o , which was undoub edly a con ac bina y. I s ou bu s in 2008
was disco e ed by Nakano e al. (2008), a e which Mason e al. (2010) es ablished
ha i was no a classical no a. Tylenda e al. (2011) analyzed bo h he ou bu s and
he p ogeni o da a, and conclusi ely showed ha his objec was indeed a con ac
bina y ha expe ienced a dynamical me ge e en . The OGLE da a e ealed ha ,
in he yea s p eceding he me ge , he o bi al pe iod was dec easing exponen ially,
which hin s a a unaway, sel -s eng hening p ocess. Du ing he ou bu s , he
sys em b igh ened by abou se en magni udes in a ound six mon hs, inishing
wi h a sha p, en-day ise o h ee mo e magni udes. Soon a e , S
e¸
pie´n (2011)
compu ed e olu iona y models o he p ogeni o , and es ima ed ini ial masses o
𝑀1≈
1
.
2
𝑀⊙
and
𝑀2≈
0
.
6
𝑀⊙
o he componen s, espec i ely. These s a s i s
e e se hei mass a io by case A mass ans e , a e which he second phase o
mass ans e becomes uns able, esul ing in he obse ed inspi al and me ge .
Pejcha (2014) ocused on he exponen ial ise o he ou bu s , and connec ed i o
apid mass loss om he ou e Lag angian poin L
2
. Se e al mo e LRNe ha e been
de ec ed and connec ed o me ge s o common-en elope e en s (Smi h e al.,2016;
Pas o ello e al.,2019;Blago odno a e al.,2017,2020,2021). In common-en elope
e olu ion, like con ac bina ies, bo h componen s sha e hei ou e laye s, bu a
c ucial di e ence is ha he e is loss o co- o a ion (I ano a e al.,2013;Röpke &
De Ma co,2023). While a con ac bina y can be desc ibed as a (quasi-)hyd os a ic
objec , a common-en elope phase is always a hyd odynamical a ai , whe e he
co e o a s a , o a compac objec , is mo ing h ough he sha ed en elope o
he sys em. Fu he mo e, in common-en elope e en s, he Roche po en ial is no
longe a sui able app oxima ion o he geome y o he sys em.
The esul o a me ge , once i has he mally elaxed, is o en a a he peculia
s a . I i is pa o a s ella clus e , i will appea younge han i s ellow clus e
membe s. I will also appea ho e (“blue ”) on he HRD, gi ing hese s a s
he name blue s aggle s, i s iden i ied in he globula clus e M3 by Sandage
(1953). While he e is ag eemen ha s ella me ge s con ibu e o he numbe o
blue s aggle s (Hills & Day,1976), he p ecise amoun is unclea (Leona d,1989).
O he possibili ies a e ha hey a e s a s wi h pa icula ly well mixed ma e ial (see
Sec . 1.1.5;Wheele ,1979), ha hey ha e acc e ed a signi ican amoun o mass
56 INTRODUCTION
a black hole as i s nuclea uel is expended. A his poin in he e olu ion,
he o bi al pe iod is s ill on he o de o hund eds o days, meaning no
g a i a ional wa e me ge is expec ed i he second s a also became a black
hole a his pe iod. We hus need a mechanism ha sh inks he o bi by a
leas wo o de s o magni ude. As ime mo es on, he second s a s a s
e ol ing signi ican ly, causing i o ill i s Roche lobe. I he mass a io is
low,
𝑀BH ≪𝑀2
, he second phase o mass ans e is uns able, o ming
a common en elope (CE). In common-en elope e olu ion, he black hole
spi als in h ough he en elope o he gian seconda y, and expe iences d ag.
I is hen pos ula ed ha he g a i a ional ene gy libe a ed om he inspi al
is con e ed o unbind he ou e laye s o he gian . In his p ocess, he o bi
o he black hole can sh ink by o de s o magni ude, enabling a g a i a ional-
wa e me ge e en o happen in a Hubble ime (e.g.,Tu uko & Yungelson,
1993;S e enson e al.,2017;Giacobbo & Mapelli,2018). While we unde s and
he quali a i e pic u e o a common-en elope inspi al, he e a e s ill many
unce ain ies in i s quan i a i e e olu ion, in pa icula on he e iciency o
ene gy con e sion, and on he endpoin o he common-en elope phase (see,
e.g.,I ano a e al.,2013, o a e iew). One o he possible ou comes is ha
he en elope is no ejec ed, and ha he black hole me ges wi h he co e o
he gian s a , p eemp ing any g a i a ional-wa e me ge happening om
he sys em (e.g.,Klencki e al.,2021).
3.
S able mass ans e . A sis e o he common-en elope channel, his scena io
equi es ha he second phase o mass ans e is s able. Unde ce ain
condi ions, i mass los om he dono is no acc e ed by he companion,
he pe iod o he bina y can sh ink wi hou he mass- ans e a e inc easing
uncon ollably ( an den Heu el e al.,2017). In essence, he non-conse a i e
mass ans e ac s as an ene gy and an angula momen um sink, wi h
which he bina y can also sh ink in pe iod by se e al o de s o magni ude.
Cha ac e izing exac ly when mass ans e is s able is c ucial o unde s and
wha pa ame e space o bina ies migh go h ough his channel (Ma chan
e al.,2021;Klencki e al.,2021;Olejak e al.,2021;Gallegos-Ga cia e al.,
2021).
4.
Popula ion III s a s As ea ly as 1926 by Jan Hend ik Oo , and la e in
1944 by Wal e Baade, s a s we e ca ego ized in wo classes depending on
hei measu ed eloci y. Popula ion I s a s a e he slowly mo ing, me al-
ich s a s o he galac ic disk, while popula ion II s a s we e as e mo ing,
me al-poo e s a s loca ed in he galac ic bulk and globula clus e s in he
halo. Since globula clus e s a e e y old, popula ion II s a s a e olde han
popula ion I s a s. La e , he hi d class, popula ion III, was added o he
s a s ha we e o med e y soon a e he big bang (in he i s billion yea s
o so, Abel e al.,2002). Because hey a e he e y i s gene a ion, and one
equi es s a s o o m hea y elemen s, popula ion III s a s a e hough o be
essen ially me al- ee (B omm e al.,2002). Popula ion III s a s should be
THIS THESIS 57
mo e compac o he same mass as popula ion I s a s, so hey mo e easily
a oid in e ac ion du ing hei li es. Kinugawa e al. (2014) modeled a la ge
se o popula ion III s a s, and compu ed he compac objec me ge a es
om i . They ind ha , e en hough he me ge a e om popula ion III
s a s is lowe han o popula ion I s a s, hey a e mo e easily obse able
because he a e age masses o he me ging black holes a e much highe o
he popula ion III s a s. This conclusion howe e depends on cosmological
pa ame e s and he s a - o ma ion- a e his o y.
The communi y has no ye eached consensus on which o he channels p oduces
he obse ed g a i a ional-wa e me ge s. Mos p obably, all o hem will
con ibu e o a ce ain deg ee, bu unce ain ies in he modeling p e en us om
making p ecise p edic ions. While he inc easing numbe o g a i a ional-wa e
de ec ions can help cons ain he physical p ocesses o he p ogeni o s (Mandel
& Fa me ,2022), e o s o imp o e he modeling i sel , and unde s anding i s
associa ed physics, a e he main way o p og ess in his g owing ield.
1.5 This hesis
The main goal o his hesis is o ad ance on he heo e ical modeling o massi e
con ac bina y s a s. Gi en he equency o con ac phases, hei pi o al ole in
s ella me ge s and associa ed exo ic phenomena, and occu ence in he chemically
homogeneous e olu ion channel o g a i a ional-wa e p ogeni o s, i is imely o
concen a e e o s on he accu a e modeling o con ac bina ies.
Howe e , we will i s make a b ie bu impo an excu sion in o he obse a ional
cha ac e iza ion o bina y s a s. Mos model-independen masses o ( e y) massi e
s a s come om sho -pe iod eclipsing bina ies. Howe e , such measu emen s
migh be a ec ed by addi ional physical e ec s, such as idal de o ma ion and
uncons ained in e nal mixing. In his sense, young, long-pe iod bina ies, which
a e una ec ed by close bina y in e ac ion, o e an excellen al e na i e o calib a e
massi e-s a models. In Chap. 2, we will analyze he long-pe iod, massi e bina y
s a 9 Sagi a ii (9 Sg ). The cen al ques ion we wan o sol e is whe he he
spec oscopic analysis o Rauw e al. (2012) su e ed om biases using line-p o ile
i ing o he highly blended lines o 9 Sg . We will use ela i e as ome y and
spec al disen angling o de e mine all he o bi al pa ame e s and in so doing
calcula e he dynamical masses o bo h s a s. A e wa ds, we will analyze he
a mosphe es o he s a s, de i ing hei empe a u e and abundances o elemen s
like ca bon, ni ogen and oxygen, and compa e he s ella pa ame e s wi h
e olu iona y models.
58 INTRODUCTION
Figu e 1.25: Pa hway o a bina y black hole me ge h ough he common-en elope
and s able mass- ans e channels. Two phases o in e ac ion a e equi ed o
p oduce me ging black holes in his scena io. Fi s a case A o B mass- ans e
phase occu s, la e ollowed by ei he a common-en elope phase o a s able mass-
ans e phase. Rep esen a i e pe iods a e indica ed a a ious phases in he
e olu ion. Ac onyms a e explained in he p e ace.Adap ed om Ma chan e al.
(2016).
THIS THESIS 59
In Chap. 3, we de elop he me hodology o accu a ely ep esen con ac bina ies in
a one-dimensional s ella -e olu ion code. We will use he me hod o Kippenhahn
& Thomas (1970) and Endal & So ia (1976) o ep esen idally dis o ed s a s
unde he shellula app oxima ion, and adap he equa ions o s ella s uc u e
acco dingly. We will hen quan i y how he su ace p ope ies o a s ella model
change when going om a sphe ically symme ic geome y o a o a ing o idally
de o med geome y.
Chap e 4deals wi h he p ocess o ene gy ans e in con ac bina ies. We will
gi e a b ie o e iew o he heo y o ene gy ans e , a e which we implemen
an ene gy- ans e scheme in o ou s ella -e olu ion models. The e ec s on he
sha ed laye s o he s a s is ca e ully s udied, and di e ences in he e olu iona y
pa hways o a bina y model wi h and wi hou ene gy ans e a e iden i ied.
In Chap. 5, we will compu e a la ge g id o bina y-e olu ion models, whe e, in
one g id, we include he p ocess o ene gy ans e , while in ano he , we exclude
i . In his way, we will ge a measu e o he e ec s o ene gy ans e ac oss he
en i e popula ion o massi e bina y s a s. In pa icula , we aim o answe whe he
including ene gy ans e in heo e ical modeling can esol e he disc epancy on
he mass- a io dis ibu ion in oduced in Sec . 1.3.2.
Finally, Chap. 6concludes his hesis wi h a summa y and ou look.
Chap e 2
The dynamical mass o 9
Sagi a ii
This chap e is mainly based on:
Resol ing he dynamical mass ension o he massi e bina y 9 Sg
M. Fab y, C. Hawc o , A. J. F os , L. Mahy, P. Ma chan , J-B. Le Bouquin, H. Sana
ASTRONOMY & ASTROPHYSICS, 651, A119 (2021)
Au ho s con ibu ions: M. Fab y did he majo i y o he wo k appea ing in his
chap e . M. Fab y, L. Mahy and H. Sana join ly de eloped he me hodology o
he spec al-disen angling app oach. J-B. Le Bouquin and A. J. F os educed
he as ome ic da a, while C. Hawc o pe o med he a mosphe e analyses.
P. Ma chan p o ided c i ical insigh s on he e olu iona y models. The ex was
w i en by M. Fab y, and subsequen ly imp o ed on by all co-au ho s. Wi h espec
o he o iginal publica ion, some o he ex was al e ed o i he chap e o ma o
his hesis.
O iginal abs ac :
Con ex . Di ec dynamical mass measu emen s o s a s wi h masses abo e 30
𝑀⊙
a e a e. This is he esul o he low yield o he uppe ini ial-mass unc ion and he
limi ed numbe o such sys ems in eclipsing bina ies. Long-pe iod double-lined
spec oscopic bina ies ha a e also esol ed as ome ically o e an al e na i e
o eclipsing bina ies o ob aining absolu e masses o s ella objec s. 9 Sg (HD
164794) is such long-pe iod high-mass bina y. Un o una ely, he e exis s la ge
61
62 THE DYNAMICAL MASS OF 9 SAGITTARII
ension be ween i s o al dynamical mass in e ed spec oscopically om adial-
eloci y measu emen s and ha om as ome ic da a.
Aims. Ou goal is o esol e he mass ension o 9 Sg ha exis s in li e a u e, o
cha ac e ize he undamen al pa ame e s and su ace abundances o bo h s a s
as well as o de e mine he e olu iona y s a us o he bina y sys em, hence o h
p o iding a e e ence calib a ion poin o con on e olu iona y models a high
masses.
Me hods. We ob ain he as ome ic o bi om exis ing and new mul i-epoch
VLTI/PIONIER and VLTI/GRAVITY in e e ome ic measu emen s. Using
a chi al and new spec oscopy, we pe o m a g id-based spec al disen angling
sea ch o cons ain he semi-ampli udes o he adial- eloci y cu es. We
compu e a mosphe ic pa ame e s and su ace abundances by adjus ing FASTWIND
a mosphe e models and we compa e ou esul s wi h e olu iona y acks
compu ed wi h he Bonn E olu iona y Code (BEC).
Resul s. G id spec al disen angling o 9 Sg suppo s he p esence o a 53
𝑀⊙
p ima y and a 39
𝑀⊙
seconda y, in excellen ag eemen wi h hei obse ed
spec al ypes. In combina ion wi h he size o he appa en o bi , his pu s 9 Sg a
a dis ance o 1
.
31
±
0
.
06
kpc
. Ou bes - i models e eal a la ge mass disc epancy
be ween he dynamical and spec oscopic masses, which we a ibu e o a i ac s
om epea ed spec al no maliza ion be o e and a e he disen angling p ocess.
Compa ison wi h BEC e olu iona y acks shows he componen s o 9 Sg a e
mos likely coe al wi h an age o oughly 1My .
Conclusions. Ou analysis clea s up he con adic ion be ween mass and o bi al
inclina ion es ima es epo ed in p e ious s udies. We de ec he p esence o
signi ican CNO-p ocessed ma e ial a he su ace o he p ima y, sugges ing
enhanced in e nal mixing compa ed o cu en ly implemen ed le els in he BEC
models. The p esen measu emen s p o ide a high-quali y high-mass ancho
o alida e s ella -e olu ion models, and o es he e iciency o in e nal mixing
p ocesses.
2.1 In oduc ion
Massi e s a s d i e he chemical en ichmen o hea y elemen s and injec la ge
amoun s o kine ic ene gy in o hei neighbo hoods h ough hei s ong, line-
d i en winds and inal explosion as supe no ae and gamma- ay bu s s. Ob aining
accu a e mass measu emen s o s a s in he uppe pa o he HRD has been a
challenge, howe e , as e olu iona y models o high-mass s a s a e iddled wi h
physical unce ain ies. Fu he mo e, spec oscopic masses, ob ained h ough
INTRODUCTION 63
a mosphe ic model i ing, a e in insically inaccu a e. The e o e, di ec mass
measu emen s ha a e independen o a mosphe e o e olu iona y models o e
aluable cons ain s o gauge he quali y o he models.
In Sec . 1.2.1, we in oduced how model-independen masses can be ound h ough
Keple ’s laws, which p o ide so-called dynamical masses. Un o una ely, no single
obse a ional echnique can ully cha ac e ize he o bi and dynamical masses o
he wo componen s. Ei he a double-lined spec oscopic bina y (SB2) has o be
eclipsing, o , as ome ic da a, ei he absolu e o ela i e, mus be a ailable. In
bo h cases, mul i-epoch obse a ions a e equi ed.
T adi ionally, eclipsing SB2s a e conside ed o p o ide he bes cons ain s on
he o bi al pa ame e s. Ye , hey a e a e and unce ain ies abou he e ec s o
idal de o ma ion, mu ual illumina ion and/o bina y in e ac ion may pollu e
he ob ained esul s, making i challenging o con on hese objec s o single-s a
models. This is pa icula ly he case in he ealm o massi e s a s. In his con ex ,
as ome ic bina ies a e a aluable al e na i e o eclipsing SB2s. Recen ad ances
in op ical long-baseline in e e ome y ha e iden i ied a numbe o such sys ems
(e.g.,Sana e al.,2013b,2014;Maye e al.,2014;Maíz Apellániz e al.,2017;Mahy
e al.,2018;Lan he mann e al.,2023), which o e new oppo uni ies o ob ain
dynamical mass cons ain s o s a s in he high-mass egime.
9 Sg (HD164794) is such a long-pe iod, as ome ic SB2 sys em in he Lagoon
Nebula. I was i s s udied by Abbo e al. (1984) in he con ex o i s a iable
synch o on emission, which is in e p e ed o be a esul o wind-wind collisions o
bina ies (Pi a d & Doughe y,2006), hin ing ha he hen p esumed single s a 9
Sg was in ac a bina y. Subsequen s udies by Rauw e al. (2002a,b) and Nazé e al.
(2008) con i med he p esence o ele a ed X- ay emission, a ypical indica ion o
colliding winds in O + O bina ies (Rauw e al.,2002c;Sana e al.,2004,2006;Rauw
& Nazé,2016). Rauw e al. (2012) i s con i med he long-pe iod bina y na u e o
9 Sg h ough adial eloci y (RV) measu emen s and classi ied i s componen s as
O3.5V(( +)) and O5.5V(( )). Rauw e al. (2016) s udied he pe ias on passage o 9
Sg and epo ed a maximum in he X- ay emission coming om shocked gas in
he in e ac ion zone o he s ella winds, as expec ed om a wind-wind collision
in a wide bina y whe e he shocked ma e ial cools adiaba ically (S e ens e al.,
1992).
The sys em was esol ed o he i s ime in 2009 using he As onomical Mul i-
BEam combineR (AMBER) and in 2013 wi h he P ecision In eg a ed-Op ics Nea -
in a ed Imaging ExpeRimen (PIONIER) by Sana e al. (2014). Le Bouquin e al.
(2017) cons ained he as ome ic o bi using mul i-epoch AMBER and PIONIER
in e e ome ic measu emen s, unco e ing a disc epancy wi h he spec oscopic
analyses o Rauw e al. (2012,2016). While he in e e ome ic measu emen s
o Le Bouquin e al. (2017) i mly excluded inclina ions below 80 deg ees, he
64 THE DYNAMICAL MASS OF 9 SAGITTARII
RV cu e semi-ampli udes o Rauw e al. (2012,2016) esul ed in an es ima ed
inclina ion o abou 50 deg ees i he s a s we e o ha e masses ep esen a i e o
hei spec al ypes.
Ano he long s anding issue is he dis ance o 9 Sg , and whe he o no i is
a membe o he young open clus e NGC 6530. P isinzano e al. (2005) and
Kha chenko e al. (2005) measu ed dis ances o he clus e o a ound 1
.
25
kpc
,
while ea lie measu emen s indica ed a dis ance close o 1
.
8
kpc
( an den Ancke
e al.,1997;Sung e al.,2000) as a esul o di e ences in he adop ed eddening
laws. The combina ion o he as ome ic and spec oscopic measu emen s p o ide
a di ec cons ain on he dis ance, allowing us o con i m he cu en Gaia
eDR3 measu emen o 1
.
21
kpc
. Gaia does no su e om sys ema ics owing
o an assumed eddening law, and hus p o ides a powe ul cons ain , bu i s
measu emen can s ill be impac ed by he mul iplici y o he sys em.
In his wo k, we aim o esol e he exis ing disc epan esul s ha cas doub on
he cu en mass es ima es, e olu iona y s a us, and clus e membe ship o 9 Sg .
To do so, we le e age he accu acy o ela i e as ome y and we pe o m a g id
spec al disen angling analysis on spec oscopic da a o ully cons ain he o bi
o 9 Sg . We u he use he a mosphe ic p ope ies o he s a s in he sys em o
de i e i s e olu iona y s a us and es s ella -e olu ion models a high masses.
This chap e is o ganized as ollows. Sec ion 2.2 co e s he obse a ional da a ha
we e used. We p esen and discuss he o bi al analysis and spec al disen angling
in Sec . 2.3, he a mosphe e modeling o bo h componen s in Sec . 2.4, and he
e olu iona y s a us o 9 Sg is discussed in Sec . 2.5. Sec ion 2.6 p esen s ou
conclusions and inal ema ks.
2.2 Obse a ions
We combine a chi al and new op ical spec oscopy wi h nea -in a ed (NIR)
in e e ome y o 9 Sg . Mos o hese measu emen s we e pa o long- e m
moni o ing p og ams.
2.2.1 Op ical spec a
A chi al da a consis o 57 spec a om he High E iciency and Resolu ion
Me ca o Echelle Spec og aph (HERMES, Raskin e al. 2011, used in Rauw
e al. 2016), 20 Fibe - ed Ex ended Range Op ical Spec og aph (FEROS) spec a
(Kau e e al.,1999) and 49 Ul a iole and Visual Echelle Spec og aph (UVES)
spec a (Dekke e al.,2000) in he blue and ed a ms (used in Rauw e al.,2012).
Obse a ions ha a e no p e iously analyzed a e lis ed in Table A.1 and consis
OBSERVATIONS 65
o i e addi ional HERMES spec a and one spec um om he High Accu acy
Radial eloci y Plane Sea che (HARPS) spec og aph (Mayo e al.,2003). The
FEROS spec a co e he spec al domain be ween 3700 and 9000
Å
and ha e a
esol ing powe o
𝑅≈
48000. The UVES spec a co e he wa eleng h ange
𝜆𝜆 =
3500
–
5000
Å
wi h i s blue a m and
𝜆𝜆 =
5000
–
7000
Å
wi h i s ed a m, and
each ha e a esol ing powe o
𝑅≈
40000. The HERMES and HARPS spec a
bo h ha e
𝑅≈
85000 wi h a co e age o
𝜆𝜆 =
3800
–
9000
Å
and
𝜆𝜆 =
3750
–
6900
Å
,
espec i ely. The FEROS, UVES, and HARPS spec a we e ob ained h ough he
ESO a chi e science po al and we e p e- educed wi h hei espec i e pipelines.
The HERMES spec a a e educed using he HERMES Da a Reduc ion So wa e
(DRS) pipeline. Finally, all spec a we e no malized o e hei whole spec al
domain by i ing a cubic spline unc ion h ough selec ed con inuum egions.
2.2.2 Nea -in a ed In e e ome y
We used he p e iously published AMBER and PIONIER da ase ob ained om
Jun 2009 o Aug 2016 (Le Bouquin e al.,2017), along wi h wo new PIONIER
obse a ions ob ained in May and Augus 2017. These new da a show o he
i s ime he sys em u ning back on he appa en ellipse and p o ide an almos
comple e co e age o he nine-yea o bi . They we e ob ained wi h PIONIER
(Le Bouquin e al.,2011) a he Ve y La ge Telescope In e e ome e (VLTI) using
he ou 1.8 me e Auxilia y Telescopes (ATs) in con igu a ions B2-K0-D0-I3
and A0-G1-J2-J3, o e ing a maximal p ojec ed baseline o 120 and 130 me e
espec i ely. The PIONIER da a we e educed and analyzed, as desc ibed in
Le Bouquin e al. (2017), using he
pnd s
package.
1
Each obse a ion p oduces
six isibili ies and ou closu e phases, deli e ing ela i e as ome y wi h sub-
millia csecond p ecision and an H-band lux a io o
𝑓H=
0
.
62
±
0
.
02. The
ull jou nal o in e e ome ic obse a ions is gi en in Table A.2 along wi h he
measu ed as ome ic p ope ies o he sys em.
Addi ionally, h ee VLTI/GRAVITY (GRAVITY Collabo a ion e al.,2017) mea-
su emen s we e aken, o which wo we e ob ained in June 2016 and he o he
in Sep embe 2016. These obse a ions we e pa o he science e i ica ion (SV)
p og am and used he ATs in con igu a ions A0-G1-D0-C1 and A0-G1-J2-K0. As
GRAVITY is a spec o-in e e ome e , i p o ides six isibili ies and ou closu e
phases o each wa eleng h bin in he 2
.
0
–
2
.
4
µm
NIR K band wi h a spec al
esol ing powe o
𝑅≈
4000. These SV da a we e educed wi h he s anda d
GRAVITY pipeline (Lapey e e e al. 2014, e sion 1.0.11) and i ed pa ame ically
o a bina y model wi h PMOIRED.
2
The unce ain ies on he ela i e as ome y
we e es ima ed by adding a boo s apped e o and a sys ema ic e o o 0
.
1
mas
in
quad a u e. In he boo s apping p ocedu e, da a we e d awn andomly o c ea e
1h p://www.jmmc. /pnd s
2h ps://gi hub.com/ame and/PMOIRED
72 THE DYNAMICAL MASS OF 9 SAGITTARII
Figu e 2.3: Two-dimensional his og am o he esul s o he MC sampling in
(𝐾1, 𝐾2)
space. The colo indica es he numbe o samples
𝑁
(ou o a o al o 1500)
a he co esponding
(𝐾1, 𝐾2)
pai s. The esul om he o iginal da a is enci cled
in ed.
2.3.3 Dis ance
Because he semi-ampli udes
𝐾1
and
𝐾2
se he absolu e scale o he o bi , we can
in e he dis ance o 9 Sg by measu ing agains he appa en o bi ound h ough
he in e e ome y (Fig. 2.1), ia he o al mass o he sys em and Keple ’s hi d
law, as ollows:
𝑑=1
𝑎app
3
√︄𝐺(𝑀1+𝑀2)𝑃2
4𝜋2,(2.6)
whe e
𝑎app
is he semimajo axis o he appa en o bi in angula uni s. The alues
(𝐾1, 𝐾2)=(
36
,
49
)km
s
−1
co espond o a o al dynamical mass o 92
𝑀⊙
, which
esul s in a dis ance o
𝑑=1310 ±60pc,(2.7)
whe e we p opaga ed he MC e o s.
This alue lies wi hin he Gaia eDR3 geome ic dis ance epo ed by Baile -Jones
e al. (2021) o 1218
+100
−108 pc
. Wi h hese esul s, i seems likely ha 9 Sg is a
membe o NGC 6530, con i ming he measu emen s o P isinzano e al. (2005)
and especially Kha chenko e al. (2005), who quo ed a dis ance o
𝑑=
1322
pc
.
ORBITAL ANALYSIS 73
Figu e 2.4: Disen angled, eno malized spec a o bo h componen s using
𝐾1=
36
km
s
−1
and
𝐾2=
49
km
s
−1
. The spec um o he p ima y is shi ed up by 0.4
uni s wi h espec o he seconda y o cla i y. In H
𝛼
, he co e is con amina ed by
nebula emission p esen in he FEROS and HERMES spec a. The model spec a
a e hose ob ained wi h he bes - i a mosphe ic pa ame e s om he FASTWIND
analysis (Sec . 2.4).
Con e sely, we can i mly exclude dis ances o 9 Sg o o e 1500
pc
because
ha would equi e a o al mass o o e 130
𝑀⊙
, which ou g id-disen angling
esul s do no suppo . Simila ly, he dis ance o 1780
±
80
pc
o Sung e al. (2000),
adop ed by Rauw e al. (2012), is incompa ible wi h he in e e ome ic and Gaia
measu emen s o he appa en size o he o bi and he de i ed componen masses.
74 THE DYNAMICAL MASS OF 9 SAGITTARII
2.4 A mosphe e modeling
2.4.1 Se up
Using he disen angled spec a, we adjus heo e ical line p o iles compu ed
wi h he FASTWIND non-local he mal equilib ium a mosphe e code sui able
o he expanding a mosphe e o O- ype s a s (San olaya-Rey e al.,1997;Puls
e al.,2005;Ca nei o e al.,2016;Puls,2017;Sundq is & Puls,2018). To educe
he dimensionali y o he pa ame e space, he o a ional and mac o u bulen
eloci ies a e es ima ed using he
iacob-b oad
ool (Simón-Díaz & He e o,
2014). Analyzing (wi h he goodness-o - i me hod) he
OIII 𝜆
5592 and
NV
𝜆
4603 lines o he p ima y and
OIII 𝜆
5592,
He I𝜆
4713 and
He I𝜆
5876 o he
seconda y, we ind a p ojec ed o a ional eloci y
𝑣sin𝑖=
102
+8
−12 km
s
−1
and a
mac o u bulen eloci y
𝑣mac =
77
+23
−20 km
s
−1
o he p ima y s a , while we ake
𝑣sin𝑖=
67
+6
−13 km
s
−1
and
𝑣mac =
48
+21
−14 km
s
−1
o he seconda y. Simila alues a e
ob ained when using he Fou ie ans o m me hod.
The s ella a mosphe e models a e hen i e a ed using a gene ic algo i hm
(Cha bonneau,1995;Mokiem e al.,2005) wi hin a p ede ined pa ame e space o
op imize a
𝜒2
i ness me ic un il a con e gence o he bes i wi h he spec um is
eached. The e sion o he gene ic algo i hm used is de ailed in Abdul-Masih e al.
(2019). We se he
𝛽
exponen o he wind-accele a ion law o 0.85, as app op ia e
o main-sequence s a s (Muij es e al.,2012). The mic o u bulence eloci y is ixed
o
𝑣mic =
10
km
s
−1
in he compu a ion by FASTWIND; in he o mal in eg al i is
selec ed on he c i e ia
𝑣mic =max(
10
km
s
−1,
0
.
1
𝑣wind)
. We include op ically hin
wind clumping, wi h a nea -cons an clumping ac o
𝑓cl
h oughou he wind.
Las ly, we op ed o clip he co e o he H
𝛼
line o a oid i ing he nebula emission
emnan ( isible in he bo om- igh mos panel o Fig. 2.4). The ull lis o i ed
spec al lines is shown in Table A.3.
To p o ide an absolu e magni ude ancho poin o he a mosphe ic model, om
which he componen luminosi ies will be compu ed, we adop he pho ome ic
da a om he Two Mic on All-Sky Su ey (2MASS, Sk u skie e al. 2006), gi ing an
appa en magni ude in he NIR K
S
band o
𝑚KS=
5
.
731
±
0
.
024 o he o al sys em.
Using he in e s ella abso p ion coe icien
𝐴𝑉=
1
.
338
±
0
.
021 om Maíz Apellániz
& Ba bá (2018), he colo co ec ion
𝐴KS/𝐴𝑉=
0
.
116 om Fi zpa ick (1999) and
a dis ance o 1
.
31
±
0
.
06
kpc
calcula ed in Sec . 2.3.3, he absolu e magni ude in
he K
S
band comes o
𝑀KS=−
5
.
01
±
0
.
10. We co ec o he measu ed PIONIER
lux a io
𝑓=
0
.
62 be ween he componen s, whe e we assume ha i emains
unchanged be ween he H and K
S
band (as expec ed o such ho objec s). This
yields absolu e componen magni udes o
𝑀KS=−
4
.
49
±
0
.
10 and
−
3
.
96
±
0
.
10 o
he p ima y and seconda y, espec i ely. We no e ha hese alues co espond o
ain e s a s han hei de i ed spec al ypes sugges . Syn he ic pho ome y o
ATMOSPHERE MODELING 75
Ma ins & Plez (2006) gi e
𝑀K=−
4
.
98 and
𝑀K=−
4
.
39 o he O3V p ima y and
O5V seconda y, espec i ely, which sugges s he s a s a e sligh ly mo e compac .
Finally, using he bolome ic co ec ion in he K-band om Ma ins & Plez (2006),
𝐵𝐶K=28.80 −7.24log(𝑇e ), and he magni ude-luminosi y ela ion
log(𝐿/𝐿⊙)=−0.4(𝑀K+𝐵𝐶K−4.75),(2.8)
we compu e he componen luminosi ies om he K-band absolu e magni udes
and he modeled e ec i e empe a u es.
2.4.2 Resul s and discussion
We lis he spec oscopic pa ame e s o he esul ing bes - i a mosphe ic models
and he esul ing in e ed pa ame e s in he le mos column o Table 2.4. The
co esponding heo e ical spec a a e plo ed in Fig. 2.4 wi h dashed lines. We no e
he ob ious nebula con amina ion o H
𝛼
, as well as he gene al end ha he
disen angled spec a a e sligh ly shallowe han he model spec a in he deep and
b oad lines (like H
𝛿
, H
𝛾
and
He II 𝜆
4686), sugges ing issues in he no maliza ion
o hese b oad lines.
The bes - i pa ame e s depend on which line ea u es we e conside ed in he i
and wi h wha weigh s. Fo example, gi ing mo e weigh o
He I
lines would
esul in lowe in e ed e ec i e empe a u es and ice e sa. Co espondingly,
he in e ed su ace g a i ies would be lowe o lowe
𝑇e
and ice e sa.
The e o e, conse a i e e o s o 1
kK
and 0
.
2
dex
on
𝑇e
and
log𝑔
, espec i ely,
a e adop ed. Fu he mo e, he de e mina ion o he quali y o he CNO abundance
measu emen s is challenging. The ca bon abundance o he seconda y, o
example, is i ed o
𝜀C=log(𝑁C/𝑁H) +
12
=
9
.
12; his is an unusually high
measu emen ha has ( o ou knowledge) ne e be o e been obse ed. We no e
ha his measu emen is d i en by he
CIII 𝜆
5696 line, which is in emission. A
𝑇e =
42
kK
,FASTWIND can only econcile his line in emission by boos ing he
ca bon abundance. Keeping he issues p esen ed by his line in mind, howe e (see
Ma ins & Hillie ,2012), we adop he minimum o a o mal 0
.
1
dex
e o and he
s a is ical e o o he g id o models. Since his o mal e o is somewha a bi a y,
e en hese unce ain ies should be in e p e ed wi h g ea ca e. The e o e we can
only a gue o quali a i e en ichmen o ni ogen in he p ima y and en ichmen
o ca bon in he seconda y. Fo added jus i ica ion, we show in Figs. A.5 and A.6
he compa ison o he model spec a, hei e o anges along wi h spec a using
he B o e al. (2011) CNO baseline abundances in a ious diagnos ic lines o he
CNO elemen s.
The bes - i
log𝑔
alues hen p o ide he spec oscopic masses o he s a s,
which a e ound o be 32
±
16
𝑀⊙
and 19
±
10
𝑀⊙
o he p ima y and
seconda y, espec i ely. These masses a e signi ican ly lowe han hei dynamical
76 THE DYNAMICAL MASS OF 9 SAGITTARII
coun e pa s, albei wi h la ge e o ba s, and a e no ep esen a i e o dwa s a s
o ha luminosi y. The main eason o his disc epancy is he low in e ed
log𝑔
,
which should be aised by abou 0
.
25
dex
o bo h s a s, ha is, sligh ly beyond
he adop ed unce ain y, o ma ch he dynamical masses. The mass disc epancy
p oblem (He e o e al.,1992) is s ill an open issue in massi e-s a spec oscopy,
and while mo e ecen s udies (e.g.,Mahy e al.,2020) show ha o s a s abo e
∼
35
𝑀⊙
, he disc epancy la gely disappea s, in his analysis, i is s ill p esen . The
epea ed no maliza ion o he spec a be o e and a e disen angling could be
he oo cause o his ac , as he econs uc ion plo s in Fig. A.3 and A.4 and he
compa ison o he model spec a (Fig. 2.4) hin owa ds.
2.5 E olu iona y modeling
We compa e ou p e ious esul s wi h he Milky Way e olu iona y acks o B o
e al. (2011), using he Bayesian sea ch ool BONNSAI (Schneide e al.,2014).
5
The
BONNSAI ool allows us o sea ch he o a ing single-s a e olu ion acks o B o
e al. (2011) o he highes -likelihood s ella model ha co esponds o measu ed
quan i ies. We inpu he obse ed
log 𝐿, 𝑇e ,𝑌He
, and
𝑣sin𝑖
om he le column
o Table 2.4 and eques he highes likelihood models o bo h s a s in he g id.
To a oid biasing he Bayesian sea ch, we e ain om using he
log𝑔
due o he
unce ain ies posed by he mass disc epancy. In a i s sea ch, we do no inpu
he CNO abundances ob ained om FASTWIND. The pa ame e s om he highes
likelihood models eplica ed om ou spec oscopic and pho ome ic obse ables
o his sea ch a e gi en in he middle column o Table 2.4.
The compa ison wi h e olu iona y acks poin owa d ela i ely compac
and coe al s a s wi h an age o abou 1
My
. We ind ha he e olu iona y
masses a e wi hin e o o he dynamical masses, which p o ides a u he
indica ion ha he spec oscopic mass likely su e s om sys ema ic e o s.
Addi ionally, he e olu iona y models a o lowe CNO su ace abundances han
a e spec oscopically in e ed, especially ni ogen in he p ima y and ca bon in he
seconda y. The a he modes o a ional eloci ies and young ages do no allow
o o a ional mixing o modi y he su ace composi ion; he CNO abundances
e u ned by BONNSAI co espond o he baseline alue o he B o e al. (2011)
models wi h e y small unce ain ies.
F om he a mosphe e models in Table 2.4, i is clea ha he CNO composi ion
be ween he p ima y and he seconda y is di e en . This is no e lec ed in he
e olu iona y acks because bo h models p e e he baseline alues o he B o
e al. (2011) acks, namely
𝜀C=
8
.
13
, 𝜀N=
7
.
64 and
𝜀O=
8
.
55 (middle column
o Table 2.4). Fu he mo e, i we assume he abundances o he seconda y a e
5The BONNSAI web-se ice is a ailable a www.as o.uni-bonn.de/s a s/bonnsai.
EVOLUTIONARY MODELING 77
Table 2.4: Pa ame e s o he bes - i , gene ically e ol ed FASTWIND a mosphe ic model
(desc ibed in Sec . 2.4), along wi h eplica ed obse ables om he B o e al. (2011) models
using BONNSAI (Sec . 2.5). The e o s co espond o he 1
𝜎
con idence le el. Emp y en ies a e
inde e minable o ha pa ame e .
FASTWIND BONNSAI
P ima y Seconda y P ima y Seconda y P ima y, Scaled CNO
𝑇e /kK 46.0±1.0 42.0±1.0 45.9+0.6
−0.941.9±0.9 46.0+0.6
−1.0
log(𝑔/[cgs]) 3.87 ±0.20 3.87 ±0.20 4.11 ±0.05 4.12+0.06
−0.07 4.10+0.02
−0.06
log ¤
𝑀
𝑀⊙y −1−6.6±0.2−6.6±0.2 ... ... ...
𝑓cl 29 ±5 22 ±3 ... ... ...
𝑣sin𝑖/kms−1a 102+8
−12 67+6
−13 ... ... ...
𝑣 o /kms−1... ... 110+59
−26 70+8
−15 330+26
−30
𝑌He 0.25 ±0.04 0.24 ±0.03 0.26b0.26b0.28+0.08
−0.02
𝜀C8.17+0.60
−0.55 9.12 ±0.10*8.14+0.01
−0.03 8.13b7.12+0.55
−0.05
𝜀N8.45+0.10
−0.29 7.42 ±0.10 7.63+0.09
−0.01 7.64b8.72+0.10
−0.27
𝜀O8.63+0.10
−0.70 8.64+0.10
−0.13 8.55+0.01
−0.02 8.55b8.55+0.01
−0.61
log(𝐿/𝐿⊙)5.68 ±0.08 5.35 ±0.08 5.64+0.07
−0.06 5.33+0.08
−0.06 5.67+0.06
−0.07
𝑅/𝑅⊙10.8±1.0 8.9±1.2 10.45+0.88
−0.59 8.73+0.75
−0.67 10.73+0.79
−0.61
𝑀spec/𝑀⊙32.1±16.0 18.9±10.1 ... ... ...
𝑀e ol/𝑀⊙... ... 53.4+3.2
−3.337.0+2.0
−2.353.8±4.7
Age (My )... ... 0.52+0.32
−0.33 1.00+0.48
−0.58 1.00+0.80
−0.41
a De e mined using IACOB-BROAD, no FASTWIND.
b Ve y small e o , see Sec . 2.5.
*
Highly unce ain measu emen , his o mal e o is likely no ep esen a i e, see Sec . 2.4.2.
baseline o 9 Sg , his sou ce has a di e en CNO baseline han he B o e al.
(2011) acks. I is ha d o jus i y obse a ionally ha he obse ed abundances o
he seconda y a e baseline o 9 Sg o he clus e NGC 6530. Bu we expec he
leas massi e s a wi h he lowes o a ional eloci y o be leas con amina ed by
su ace en ichmen om a heo e ical s andpoin . We hus es i BONNSAI inds
di e en models o he p ima y i we scale down he obse ed CNO abundances
o he B o e al. (2011) baseline. Fo he abundances o he p ima y o main ain
he same ac ional di e ence e sus he seconda y, his amoun s o calcula ing
𝜀′
X=𝜀X,base,B o −𝜀X,base,obs +𝜀X,p im,obs
. Keeping he doub ul C abundance
measu emen o he seconda y in mind (Sec . 2.4.2), we e ain om scaling C and
compu e
𝜀′
N=
8
.
67
+0.14
−0.31
and
𝜀′
O=
8
.
54
+0.14
−0.71
. Using hen again
log 𝐿, 𝑇e ,𝑌He
, and
𝑣sin𝑖 om he FASTWIND models, along wi h hese new N and O abundances as
inpu o BONNSAI, we ob ain o he highes likelihood e olu iona y pa ame e s;
hese a e lis ed in he igh mos column o Table 2.4. These esul s poin o a
di e en scena io. He e he o a ional eloci y is signi ican ly highe , allowing
signi ican o a ional mixing o occu . The p ima y age has inc eased o ma ch
ha o he seconda y as well, while he e olu iona y mass, log𝑔and 𝑇e a e only
sligh ly changed when compa ing o he non-scaled esul s (middle column o
78 THE DYNAMICAL MASS OF 9 SAGITTARII
Table 2.4). The majo implica ion o his is ha ei he he o a ional axis and he
no mal o he o bi al plane a e hea ily inclined, up o an es ima ed 68 deg ees o
explain he obse ed p ojec ed o a ional eloci y, o he e ec o o a ional mixing
is unde es ima ed in he s ella -e olu ion models. We canno exclude ei he ha
a mixing mechanism weakly dependen on o a ion is ope a ing on s a s in his
mass egime. Dis inguishing be ween hese scena ios howe e equi es g ea e
con idence in he quali y o he CNO abundance measu emen s.
We summa ize ou esul s o Sec s. 2.4 and 2.5 in an HRD and a Kiel diag am ha
is o e plo ed by se e al o he e olu iona y acks and isoch ones o B o e al.
(2011). We no e ha while he loca ion o he models on he HR diag am ma ches
well, he e is a misma ch o he spec oscopic mass in e ed om he FASTWIND
models and he e olu iona y masses. In he Kiel diag am, he e is a poo e ma ch
as expec ed om he mass disc epancy discussed in Sec . 2.4.2.
2.6 Conclusions
In his chap e , we ha e ob ained disen angled spec a o 9 Sg using a combina ion
o high angula esolu ion as ome y and spec al g id disen angling wi h he
d3
code. The as ome ic measu emen s solidi y he long pe iod o 8.9 y and ha e a
nea edge-on inclina ion o 86.5 deg ees. Ou esul s con i m he p esence o an
O3V+O5V massi e bina y, which has in e ed dynamical masses o abou 53 and
39
𝑀⊙
, making 9 Sg one o he mos massi e galac ic O+O bina ies e e esol ed.
Fu he mo e, o ou knowledge, his is only he second ins ance o a dynamical
mass es ima e o a galac ic O3V s a ( he o he om Mahy e al. 2018).
By e-de i ing he semi-ampli udes o he RV cu es h ough a spec al
disen angling analysis, we clea up he con adic o y esul s be ween he
p e ious RV measu emen s o Rauw e al. (2012) and he high inclina ion o
he in e e ome ic o bi by Le Bouquin e al. (2017). Fu he mo e, he esul s show
9 Sg is a membe o he young open clus e NGC 6530. The combined dynamical,
a mosphe ic, and e olu iona y modeling shows 9 Sg con ains massi e s a s o
oughly 53
𝑀⊙
and 37
𝑀⊙
o he p ima y and seconda y, espec i ely. 9 Sg is a
unique sys em in he a op le co ne o he HRD, and he e o e p o ides an
equally unique oppo uni y o use i s s ella and sys emic pa ame e s o compa e
wi h massi e s a e olu iona y models as well as bina y o ma ion scena ios.
CONCLUSIONS 79
Figu e 2.5: Top: HRD showing he loca ion o he bes - i FASTWIND models (FW)
and highes -likelihood e olu iona y models om B o e al. (2011) (BONNSAI).
O e plo ed a e e olu iona y acks o di e en masses and ini ial o a ion
eloci ies (g ey lines), along wi h isoch ones o he 100
km
s
−1
ini ial o a ion
eloci y models (blue lines). Bo om: Kiel diag am wi h an equi alen legend as
he HRD abo e. Using he scaled CNO abundances does no app eciably mo e
he model o he p ima y in ei he diag am.
Chap e 3
Modeling idal dis o ion in 1D
s ella s uc u e
“A beginning is he ime o aking
he mos delica e ca e ha he
balances a e co ec .”
— P incess I ulan
This chap e is mainly based on:
Modeling o e con ac bina ies, I. The e ec o idal dis o ion
M. Fab y, P. Ma chan and H. Sana
ASTRONOMY & ASTROPHYSICS, 661, A123 (2022)
Au ho s con ibu ions: M. Fab y did he majo i y o he wo k appea ing in
his chap e . P. Ma chan assis ed in de eloping he me hodology, while H. Sana
p o ided c i ical eedback h oughou he p ojec . The ex was w i en by M. Fab y
and imp o ed on by all co-au ho s. Wi h espec o he o iginal publica ion, some
o he ex was al e ed o i he chap e o ma o his hesis.
O iginal Abs ac :
Con ex . In he ealm o massi e s a s, s ong bina y in e ac ions a e commonplace.
One ex eme case is ha o con ac sys ems, which a e expec ed o be pa o he
e olu ion o all s a s e ol ing owa ds a me ge and hypo hesized as playing a
81
88 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
Figu e 3.1: Illus a ion o he Roche geome y in he equa o ial plane. I shows he
L
1,
L
2
, and L
3
equipo en ials (g ay), as well as he spli ing su aces (blue dashed).
The spli ing su aces ha e hei no mals always pe pendicula o he po en ial
g adien .
3.2.4 Con ac shells
In he con ex o 1D e olu ion models, we ha e o dis inguish he wo s ella
componen s wi hin he geome y o a Roche bina y. Fo equipo en ial shells lying
wi hin hei espec i e RLs, his is i ial, as such laye s a e physically sepa a ed,
bu shells in con ac a e sha ed be ween he wo s a s. We he e o e cons uc h ee
“spli ing su aces,” one h ough each o he colinea Lag angian poin s L
1,
L
2
, and
L
3
, sepa a ing he Roche geome y in h ee main pa s. This will ensu e dis inc ion
be ween he wo componen s, as well as ei he one om egions beyond he ou e
Lag angian poin s L
2
and L
3
, om which ou lows a e expec ed i any componen
o e lows hem. We cons uc hem in such a way ha he condi ion:
∇𝜳·d𝑺=0 (3.18)
holds on all poin s o hese spli ing su aces (see Fig. 3.1, in blue dashed lines).
In o he wo ds, he su ace no mal is always pe pendicula o he g adien o he
po en ial o Eq.
(3.17)
. We no e ha by his cons uc ion o he spli ing su aces,
he scala p oduc d
𝒏·
d
𝑺
is ze o. This ensu es ha Eq.
(3.4)
s ill holds, and i
he con ac laye s a e shellula , he e is no ne lux o adia i e ene gy be ween
he componen s ollowing Eq.
(3.11)
. Howe e , since he condi ion o shellula i y
canno be held exac ly, he e will exis a low
𝐿𝛹 , ans =∫𝑑𝛹 F·
d
𝑺spli
ac oss
equipo en ials, ep esen ing ene gy ans e om one componen o he o he . In
his wo k, we assume his con ibu ion is ze o and de e he modeling o ene gy
METHODS 89
ans e o la e wo k. We no e also ha in he case
𝑞≠
1, he L
1
spli ing su ace
is no a e ical plane h ough L
1
, which di e en ia es ou esul s o equipo en ial
olumes and su ace a eas om hose o Mochnacki (1984) and Ma chan e al.
(2021).
3.2.5 Nume ical calcula ions
In o de o use he modi ied s ella -s uc u e equa ions in a s ella e olu ion
code, we equi e o each shell in he geome y he quan i ies
𝑉𝛹
,
𝑆𝛹
,
⟨𝑔⟩
and
𝑔−1
which speci y
𝑟𝛹, 𝑓𝑃
and
𝑓𝑇
o Eqs.
(3.1)
,
(3.9)
, and
(3.14)
. Addi ionally, he
speci ic momen o ine ia,
𝑖 o
, is needed o calcula e he o a ional eloci y o
he equipo en ial shell om he angula momen um:
𝛺𝛹=𝑗 o /𝑖 o
. We app oach
his p oblem using nume ical me hods. Fi s o all we no e ha he p oblem is
symme ic unde
𝑦→ −𝑦
and
𝑧→ −𝑧
, so we need only explici ly do calcula ions
in he quad an o posi i e 𝑦and 𝑧.
Gi en a se o mass a ios,
𝑞
, and a ge equipo en ial alues,
𝛹0
, we calcula e, in a
sphe ical coo dina e sys em on ays o cons an
(𝜃, 𝜙)
, he poin s
𝑟=𝑟(𝜃, 𝜙)
ha
lie on he co esponding equipo en ial su ace, such ha :
𝛹0−𝛹(𝑟, 𝜃, 𝜙)=0.(3.19)
When sol ing Eq.
(3.19)
, ca e has o be aken in o de o a oid oo s beyond
he spli ing su aces (Sec . 3.2.4), as hese belong o a pa he geome y no
associa ed o he conside ed componen s a , and would ins ead belong o he o he
componen o o he egion beyond he ou e Lag angian poin s. We he e o e
es ic he sol e o alues
𝑟
smalle han he alue
𝑟spli (𝜃, 𝜙)
on he spli ing
su ace ha he ay
(𝜃, 𝜙)
c osses. I i inds no oo s in he in e al
(
0
, 𝑟spli )
,
𝑟spli
is eco ded ins ead as he spli ing su aces ma k he maximal ex en o he
componen s.
90 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
Nex , o hese poin s 𝑟(𝜃, 𝜙), we nume ically in eg a e he ollowing quan i ies:
𝑉𝛹=∫𝛹
𝑟3
3sin𝜃d𝜃d𝜙, (3.20)
𝑆𝛹=∫𝛹
1
ˆ𝒏·ˆ𝒓𝑟2sin𝜃d𝜃d𝜙, (3.21)
⟨𝑔⟩=1
𝑆𝛹∫𝛹
𝑔
ˆ𝒏·ˆ𝒓𝑟2sin𝜃d𝜃d𝜙, (3.22)
𝑔−1=1
𝑆𝛹∫𝛹
1
𝑔ˆ𝒏·ˆ𝒓𝑟2sin𝜃d𝜃d𝜙, (3.23)
𝑖 o =∫𝛹(𝑥2+𝑦2)d𝑚𝛹
∫𝛹d𝑚𝛹
=∫𝛹𝑟2sin2𝜃𝑔−1d𝛹d𝑆
𝑆𝛹𝑔−1d𝛹,
=1
𝑆𝛹𝑔−1∫𝛹
𝑟2sin2𝜃
𝑔ˆ𝒏·ˆ𝒓𝑟2sin𝜃d𝜃d𝜙, (3.24)
whe e
ˆ𝒏·ˆ𝒓=d𝛹
d𝑟.|∇𝜳|
and we used he no a ion d
𝑚𝛹=𝜌
d
𝑛
d
𝑆
o signi y a small
mass elemen on an equipo en ial shell, such ha d
𝑀𝛹=∫𝛹
d
𝑚𝛹
. F om hese,
𝑓𝑃
and 𝑓𝑇can be compu ed o each equipo en ial shell wi h Eqs. (3.9) and (3.14).
3.2.6 In eg a ion esul s
We conside he componen o a Roche bina y a he o igin wi h mass
𝑀1
and
a companion wi h mass
𝑀2
, and we compu ed all he necessa y in eg als o
Eqs.
(3.20)
-
(3.24)
in he wo-dimensional pa ame e space o mass a io
𝑞≡𝑀2/𝑀1
and Roche equipo en ials,
𝛹0
. We emphasize he e ha we calcula e solely
he p ope ies o he componen wi h mass
𝑀1
and ha he p ope ies o he
companion
𝑀2
can be ob ained by in e ing he mass a io. We sampled 280
equally spaced mass a ios in loga i hmic space in he in e al
log𝑞∈ [−
7
,
7
]
, as
well as he equal mass a io
log𝑞=
0 case while in po en ial space, we sampled
158 alues in se e al loga i hmic anges om
𝛹0=
50
𝛹L1
o
𝛹0=1
2(𝛹Lou +𝛹𝐿4)
o di e en densi ies. Wi h L
ou
we deno e he ou e Lag angian poin o he
conside ed componen o he bina y (always ha ing
𝑥Lou <
0, which is L
3
i
𝑞≤
1
and L
2
i
𝑞≥
1). As quan i ies exhibi impo an a ia ions a ound he o e low
po en ials
𝛹L1
and
𝛹Lou
, hose will be he egions we sample mo e densely. We
no e ha 𝛹L1< 𝛹L2≤𝛹L3< 𝛹L4=𝛹L5, as in ou con en ion𝛹is s ic ly nega i e.
METHODS 91
Figu e 3.2: S uc u e co ec ion ac o
𝑓𝑃
as a unc ion o ac ional RL adius
𝑟𝛹/𝑅RL
o a ious mass a ios. The discon inui ies in all g aphs co espond o
he c ossing o he Lag angian poin s L
1
(a
𝑟𝛹/𝑅RL =
1, by cons uc ion) and
L
ou
a la ge adii. Fo e e ence, he equi alen single o a ing s a in e ed
𝑓𝑃
is
o e plo ed in dashed lines.
Figu e 3.2 shows he in eg a ion esul s o
𝑓𝑃
o a ious mass a ios as a unc ion
o hei adius ela i e o he RL adius
𝑟𝛹/𝑅RL
, whe e we de ined
𝑟𝛹L1≡𝑅RL
. We
obse e discon inui ies a he Lag angian o e low poin s, which is a esul o
he spli ing su aces limi ing he conside ed olume o one componen (Fig. 3.1).
Fo a compa ison wi h he single o a ing s a de o ma ion model, we o e plo
he equi alen
𝑓𝑃
alues as i he bina y s a was in e p e ed as a single o a ing
s a wi h he same olume and o a ional eloci y (see Appendix C.2 o u he
de ails).
3.2.7 Compa ison o li e a u e
As e i ica ion o ou in eg a ions, we compa e ou esul s o calcula ions
pe o med by Mochnacki (1984). As his s udy conside ed e ical spli ing
su aces h ough L
1
a all mass a io’s, o con ac laye s we a e limi ed o
compa ing he mass a io uni y case. Figu e 3.3 shows excellen ag eemen
be ween ou
𝑓𝑃
alues and hose compu ed ia Eq.
(3.9)
using he esul s om
92 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
Figu e 3.3: Compa ison o he calcula ions o he s uc u e co ec ion ac o
𝑓𝑃
om
Mochnacki (1984) and hose in his wo k, as a unc ion o he ac ional RL adius
𝑟𝛹/𝑅RL
. We pe o med a esolu ion con e gence es wi h double, quad uple and
oc uple he de aul spa ial esolu ion, and conclude ou in eg a ions ha e well
con e ged a an eigh - old esolu ion inc ease.
Mochnacki (1984), wi h he ela i e di e ence being less han abou 1 pa in 10
3
.
A esolu ion con e gence es o ou in eg a ions show ou calcula ions con e ge
o he 10
−5
le el a eigh imes ou o iginal esolu ion. The inal compu a ions o
all in eg als o use in s ella e olu ion ins umen s a e done wi h his eigh - old
esolu ion inc ease, and co esponds o di iding he in e al
𝜙=[
0
, 𝜋]
in
𝑛𝜙=
5280
pa s and cos𝜃=[0,1]in 𝑛𝜃=𝑛𝜙/2 pa s so ha 𝛥𝜙 =𝜋/𝑛𝜙and 𝛥cos 𝜃=2/𝑛𝜙.
3.3 The 𝛺𝛤 limi
Thanks o he adia ion p essu e o he pho on lux gene a ed in he co e, a s a
gains adia i e suppo agains i s sel -g a i y. In he si ua ion whe e hese e ec s
a e canceled ou exac ly, he s a has eached he Edding on limi , which is gi en
as ( o a single, non- o a ing and hus sphe ical s a ):
𝐿=𝐿Edd ≡4𝜋𝑐𝐺𝑀
𝜅.(3.25)
THE 𝛺𝛤 LIMIT 93
De ining hen he Edding on ac o 𝛤as:
𝛤≡𝐿 ad
𝐿Edd ,(3.26)
wi h
𝐿 ad
he adia i e luminosi y, equal o
𝐿
in a adia i e en elope,
𝛤=
1 is hen
an equi alen s a emen o he Edding on limi . I
𝛤 >
1, he sum o adia i e and
g a i a ional accele a ion is di ec ed ou wa d. In he in e io o he s a his can in
p inciple be compensa ed by an in e sion o he gas p essu e (Joss e al.,1973), bu
i a s a app oaches
𝛤=
1 a he su ace, s ong ou lows a e expec ed o de elop
(G ä ene e al.,2011).
A e adding a cen i ugal con ibu ion o he o ce balance, we ha e:
𝒈 o =𝒈g a +𝒈 o +𝒈 ad =𝒈+𝒈 ad =0,(3.27)
whe e we w i e
𝒈=∇𝜳
as he e ec i e g a i y. Lange (1997) hen conside ed
he Edding on limi on he equa o o a o a ing (bu non-de o med) s a , and,
assuming ha he adia ion ield is iso opic, he adia i e lux is
F=𝐿/
4
𝜋𝑅2
, so
ha he o ce balance educes o
1−𝜔2−𝛤=0,(3.28)
whe e we ha e de ined
𝜔=𝛺/𝛺c i ,class
as he ac ional classical c i ical o a ion
a e
𝛺c i ,class =√︃𝐺𝑀
𝑅3
. I is impo an o make he dis inc ion he e as he o mal
c i ical eloci y is lowe ed o:
𝑣c i =√︂𝐺𝑀
𝑅(1−𝛤)=√︁𝑣c i ,class(1−𝛤).(3.29)
The e o e, he close a s a is o he Edding on limi , he lowe he c i ical eloci y
will be.
Maede & Meyne (2000) la e no ed ha his ea men does no ake on Zeipel’s
heo em in o accoun , namely ha he adia i e lux is dependen on local e ec i e
g a i y, Eq.
(3.11)
. Exp essing his o a o a ing s a o lowes o de in o a ional
eloci y 𝛺gi es
𝒈 o =𝒈(1−𝛤𝛺(𝜗)),(3.30)
whe e
𝜗
signi ies a pola dependence and
𝛤𝛺≡𝛤×𝑓(𝛺, 𝜗)
is now a o a ion-
dependen Edding on ac o . Since he s eng h o he adia i e lux is a unc ion
o he e ec i e g a i y h ough he on Zeipel e ec , Maede & Meyne (2000)
showed ha , neglec ing he
𝜗
dependence, Eq.
(3.30)
bi u ca es when
𝛤=
0
.
639.
Below his,
𝒈=
0 is he only solu ion, so ha , independen o luminosi y, he
b eak up eloci y is he classical eloci y
𝑣c i ,class =√︁𝐺𝑀/𝑟e
, wi h
𝑟e
he equa o ial
adius o he o a ing s a . Abo e he bi u ca ion poin , bo h
𝒈=
0 and, impo an ly,
1−𝛤𝛺=0,(3.31)
94 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
de e mine he c i ical eloci ies, he second o which is lowe han he classical
c i ical eloci y. The lowes o hese wo eloci ies hen de e mines he physical
b eak-up eloci y o a o a ing s a .
We u he expand on he no ion o b eak-up limi s by conside ing he case o
de o med s a s due o a conse a i e po en ial. As a s a ing poin , we conside
he same c i e ion o ze o ne accele a ion,
𝒈 o =
0
,
as he s abili y bounda y.
The con ibu ions o his accele a ion a e he e ec i e g a i y,
𝒈
, as a esul o
he po en ial, and he adia i e accele a ion. Using on Zeipel’s exp ession o
adia i e lux (Eq. 3.11), combined wi h he equa ion o hyd os a ic equilib ium
esul s in:
𝒈 ad =−4𝑎𝑇3
3
d𝑇
d𝑃𝒈,(3.32)
which, upon using Eq.
(3.13)
as he adia i e g adien in he non-sphe ical
geome y, gi es he s abili y c i e ion:
𝒈1−𝛤𝑓𝑇
𝑓𝑃=0.(3.33)
We ecognize howe e ha on Zeipel’s law o adia ion can b eak down close o
c i ical eloci y due o signi ican ba oclinici y, ha is, depa u e om shellula i y
(see, e.g.,Espinosa La a & Rieu o d,2011). Con inuing on ou assump ion o
shellula i y, Eq.
(3.33)
is o a e y simila o m as ha o Maede & Meyne
(2000), namely,
𝒈(
1
−𝛤
𝛹)=
0, whe e
𝛤
𝛹=𝛤×𝑓
is again a p oduc o he
classical Edding on ac o wi h a unc ion dependen on he geome y. Fo he
single o a ing s a de o ma ion,
𝑓=𝑓(𝜔)
, dependen on he ac ional c i ical
o a ion a e, while in he idal de o ma ion case he unc ion
𝑓
is o he o m
𝑓=𝑓(𝑟𝛹/𝑅RL, 𝑞)
, dependen on he mass a io and deg ee o Roche illing. In
Eq.
(3.33)
, he classical solu ion
𝒈=
0 is p esen o cou se, whe e he e ec i e
g a i y anishes as a esul o o a ional (and idal) suppo only. As abo e, in his
case he e is no co esponding Edding on luminosi y as i is independen o any
adia i e accele a ion due o he on Zeipel e ec . The o he solu ion is gi en by:
1−𝛤𝑓𝑇
𝑓𝑃
=0,(3.34)
which ansla es o a modi ied exp ession o he Edding on luminosi y:
𝐿Edd,𝛹 =𝐿Edd 𝑓𝑃
𝑓𝑇
,(3.35)
o he Edding on ac o (see also Sanyal e al.,2015):
𝛤
𝛹=𝐿𝛹
𝐿Edd,𝛹
=𝜅𝐿𝛹
4𝜋𝑐𝐺𝑀𝛹
𝑓𝑇
𝑓𝑃
.(3.36)
THE 𝛺𝛤 LIMIT 95
Figu e 3.4: Maximal Edding on ac o
𝛤max
ha a o a ing s a can ha e as a
unc ion o ac ional classical c i ical o a ion a e o
𝜔=𝛺/𝛺c i ,class
. Ou model
o shellula s a s has a maximal educ ion o 𝛤max =0.639.
These exp essions, as in he case o he equa ions de i ed in Sec . 3.2.1, a e
applicable in any conse a i e po en ial𝛹.
As he c i e ia o ins abili y, Eqs.
(3.28)
,
(3.31)
, and
(3.34)
all exp ess a depa u e
o he Edding on limi o a s a om he classical Edding on limi o
𝛤=
1. In he
model de eloped he e, i amoun s o a educ ion o he maximal Edding on ac o
by he a io
𝑓𝑃/𝑓𝑇
, which is smalle han one in bo h he single o a ing s a o
synch onized bina y case. In Fig. 3.4, we show he compa ison o his educ ion
om he esul s o Lange (1997) o his wo k in he case o a single o a ing s a .
Mo eo e , consis en o Maede & Meyne (2000), we ind a bi u ca ion in he
c i ical o a ion eloci y a a
𝛤=
0
.
639. Below his numbe , only he classical
c i ical eloci y,
𝜔=
1, will make he s a uns able, while abo e i , his eloci y
is educed. The case o a synch onized Roche bina y is shown in Fig. 3.5. Fo
example, a bina y wi h a mass a io o uni y, o e lowing o i s ou e Lag angian
poin , has i s Edding on limi educed o abou 84% i s classical alue.
96 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
Figu e 3.5: Reduc ion,
𝑓𝑃/𝑓𝑇
, o he maximal Edding on ac o as a unc ion o
mass a io,
𝑞
, and ac ional o e low adius,
𝑟𝛹/𝑅RL
. The equi alen adius o a
s a illing up o i s ou e Lag angian poin is ma ked wi h he g een line.
3.4 A mosphe ic bounda y condi ions
The equa ions o s ella s uc u e mus be supplied wi h app op ia e bounda y
condi ions (BCs) a he cen e and a mosphe e o he s a . In he cen e , he
condi ions a e such ha
𝑟=𝑀=𝐿=
0, which ans o m o
𝑟𝛹=𝑀𝛹=𝐿𝛹=
0 in
ou equipo en ial shell model. A he su ace, an a mosphe e model is in eg a ed o
supply he p essu e
𝑃su
and empe a u e
𝑇su
. A common choice is he so-called
g ay, Edding on a mosphe e, which is a plane-pa allel model in eg a ed om
𝜏=0 o some p ede e mined a mosphe ic op ical dep h 𝜏su , and is p esen ed in,
o example, Cox & Giuli (1968, hence o h CG).
In sphe ical s a s, he su ace op ical dep h
𝜏su
is eached a he same physical
pe pendicula dep h,
𝑧
, a all poin s in he s a . Fo de o med shellula s a s,
howe e , he op ical dep h
𝜏
ac oss an equipo en ial depends on local e ec i e
g a i y, as ollows:
d𝜏=𝜅𝜌 d𝑧=−𝜅𝜌 d𝑛
d𝛹d𝛹=−𝜅𝜌𝑔−1d𝛹, (3.37)
wi h
𝜅
and
𝜌
as he opaci y and densi y o he equipo en ial, espec i ely.
Consequen ly, equipo en ial su aces do no coincide wi h su aces o cons an
op ical dep h. Fo example, in single o a ing s a s, a ixed
𝜏su
is eached a
a deepe equipo en ial on he poles compa ed o on he equa o , esul ing in a
ATMOSPHERIC BOUNDARY CONDITIONS 97
di e en obse ed e ec i e empe a u e be ween hose egions (which can u he
be modula ed by a g a i y da kening law). The same is ue o highly de o med
bina y s a s like con ac sys ems. The a mosphe e p ope ies ou pu by s ella
e olu ion codes should he e o e also ake in o accoun his e ec o posi ion
dependen a mosphe ic dep h.
Wi hin he ea men o de o med s a s in a conse a i e po en ial, we p oceeded
as ollows. We s a by de ining he global e ec i e empe a u e
𝑇e ,𝛹
o an
equipo en ial by he law o S e an-Bol zmann:
𝐿𝛹=∫𝛹
F·d𝑺≡𝑆𝛹𝜎𝑇4
e ,𝛹 ,(3.38)
ha is, he global e ec i e empe a u e o an equipo en ial shell is ha o a black
body adia ing a luminosi y
𝐿𝛹
o e an a ea
𝑆𝛹
, which need no be sphe ical.
Nex , using he equa ion o adia i e lux (Eq. 3.11), we can de ine a local e ec i e
empe a u e 𝑇e ,ℓ ha a ies ac oss he equipo en ial:
F=𝑔
⟨𝑔⟩𝜎𝑇4
e ,𝛹 =𝜎𝑇4
e ,ℓ .(3.39)
We now conside a g ay, plane-pa allel, Edding on-app oxima ed a mosphe e
o cons uc app op ia e BCs. A each poin on he su ace equipo en ial, he
empe a u e p o ile can be w i en as (CG):
𝑇4(𝜏)=1
2𝑇4
e ,ℓ 1+3
2𝜏.(3.40)
This exp ession is de i ed using wo s anda d assump ions ha can be simila ly
applied o he case o de o med s a s. Fi s , i is assumed ha he adia ion
p essu e a all op ical dep hs can be w i en as
𝑃𝑟(𝜏)=1
3𝑎𝑇4(𝜏)
. Second, he
in ensi y o adia ion on op o he a mosphe e is assumed o be iso opic in he
ou wa d di ec ions, meaning ha we igno e limb da kening, so ha he p essu e
a he 𝜏=0 su ace is compu ed o be (CG):
𝑃𝑟(0)=2F
3𝑐.(3.41)
The i s BC we conside cons ains he empe a u e
𝑇su
o he ou e mos
bounda y by equi ing ha
𝑇su =𝑇e ,𝛹
. To ind he second BC on he su ace
p essu e 𝑃su , we conside he equa ion o hyd os a ic equilib ium:
𝑑𝑃
𝑑𝜏 =𝑔
𝜅.(3.42)
Taking a poin on he su ace equipo en ial whe e
𝑇e ,ℓ =𝑇e ,𝛹
, hen by Eq.
(3.39)
,
we ha e ha
𝑔=⟨𝑔⟩
. As we also equi e ha
𝑇su =𝑇e ,𝛹
, Eq.
(3.40)
implies ha
he su ace op ical dep h is
𝜏su =
2
/
3. The su ace p essu e can hen be in eg a ed
104 MODELING TIDAL DISTORTION IN 1D STELLAR STRUCTURE
Figu e 3.9: E ec i e empe a u e e olu ion o he 32
𝑀⊙
con ac model wi h he
di e en assump ions on BC and o a ion model. We no ice ha he models
including o a ion a e up o 5% coole du ing con ac . Be ween he di e en
o a ion models and BCs, we ind only mino di e ences. The op panel shows
he RL e olu ion, which explains he di e ence in age a onse o o e low o he
Lag angian poin s h ough spin-o bi coupling.
o he s a s, his p ocess causes he so-called Da win ins abili y (Da win,1879;
Hu ,1980), whe e he s a s ca as ophically spi al in on each o he on he o bi al
imescale.
3.6 Discussion and conclusions
We de eloped he me hodology needed o accoun o idal de o ma ion in bina y
s a s in 1D s ella e olu ion codes. They a e ep esen ed in he s uc u e-co ec ion
ac o s o
𝑓𝑃
and
𝑓𝑇
ha mul iply he equa ions o hyd os a ic equilib ium and
adia i e ene gy anspo , espec i ely. Addi ionally, modi ied exp essions o
he Edding on limi and a mosphe ic bounda y condi ions as a esul o he
de o ma ion ha e been calcula ed.
We show ha he adii p edic ed by s ella -e olu ion models can shi by
∼
5% i
cen i ugal de o ma ion is included, while also including idal de o ma ion leads
o a smalle
∼
1% e ec . This means compa ing obse ed s ella pa ame e s o
non- o a ing models could esul in a misma ch and his could ha e implica ions
DISCUSSION AND CONCLUSIONS 105
o high-p ecision as ophysics. In a s udy whe e obse ed su ace p ope ies
wi h unce ain ies on he o de o 5% a e used, cen i ugal de o ma ion has o
be included in he s ella models. In bina ies, i he p ecision o measu emen is
be e han 1%, hen he idal de o ma ion has o be aken in o accoun . We no e
ha he e ec obse ed he e is only due o geome ical conside a ions, ha is, he
equa o bulges ou due o apid o a ion o he p esence o a companion. Mixing
as a esul o apid o a ion was no included and will u he a ec he adius
e olu ion (and age). Gene ally, he e ec o o a ional mixing is o keep he s a s
compac as esh uel is in oduced in o he co e, so his may somewha cancel he
geome ical e ec . Howe e , he exac in e play and ou come o hese compe ing
e ec s is a om es ablished.
The me hods de eloped he e ha e a wide ange o applicabili y as hey can be used
o any (semi-)de ached bina y o a bi a y mass a io. This can be applied, o
ins ance, in s udying he e ec o idal o ces in mass ans e s abili y, which can
ha e a po en ial impac on he o ma ion o g a i a ional-wa e sou ces h ough
s able mass ans e ( an den Heu el e al.,2017;Ma chan e al.,2021). The e
a e, howe e , limi a ions o ou model, mos no ably ha he s uc u e-co ec ion
ac o s compu ed om he bina y Roche po en ial only apply o ully idally locked
sys ems. The gene al case, whe e he o a ion o he componen s is ee, canno be
desc ibed wi h a conse a i e po en ial, and is hus no sui able o he ea men o
Kippenhahn & Thomas (1970) and Endal & So ia (1976) used he e. De eloping a
global model o ee o a ion including ides is a complex p oblem, bu a po en ial
wo ka ound would be o conside ee o a ion o s a s well wi hin hei RLs and
use he cen i ugal de o ma ion co ec ions o single s a s, and hen implemen ing
a swi ch o he synch onized bina y co ec ions once a s a ills an app eciable
amoun o i s RL.
Fo con ac con igu a ions, cu en ly only he mass a io o uni y can be conside ed
consis en ly, as ene gy ans e in he con ac laye s, is ze o. When we include he
p ocesses o mass and ene gy ans e in con ac laye s, which is he subjec o he
nex chap e , he me hods de eloped he e enable he consis en modeling o all
con ac bina y sys ems.
Chap e 4
Modeling ene gy ans e in
con ac bina ies
“E.T. phone home...”
— E.T., he Ex a-Te es ial
This chap e is mainly based on:
Modeling con ac bina ies, II. The e ec o ene gy ans e
M. Fab y, P. Ma chan , N. Lange and H. Sana
ASTRONOMY & ASTROPHYSICS, 672, A175 (2023)
Au ho s con ibu ions: M. Fab y did he majo i y o he wo k appea ing in his
chap e . M. Fab y, P. Ma chan and N. Lange join ly de eloped he me hodology,
while H. Sana p o ided c i ical eedback h oughou he p ojec . The ex was
w i en by M. Fab y, wi h addi ions om N. Lange , and subsequen ly imp o ed
on by all co-au ho s. Wi h espec o he o iginal publica ion, some o he ex was
al e ed o i he chap e o ma o his hesis.
O iginal Abs ac :
Con ex . I is common o massi e s a s o engage in bina y in e ac ions. In
close bina ies, he componen s can en e a con ac phase, when bo h s a s
simul aneously o e low hei espec i e Roche lobes. While obse a ional
cons ain s on he s ella p ope ies o such sys ems exis , he mos de ailed s ella
107
108 MODELING ENERGY TRANSFER IN CONTACT BINARIES
e olu ion models ha ea u e a con ac phase a e no ully econcilable wi h hose
measu emen s.
Aims. We aim o consis en ly model he con ac phases o bina y s a s in a 1D
s ella e olu ion code. To his end, we ha e de eloped a me hodology o accoun
o ene gy ans e in he common con ac laye s.
Me hods. We implemen ed an app oxima i e model o ene gy ans e be ween
he componen s o a con ac bina y based on he on Zeipel heo em in he s ella
e olu ion code
MESA
. We compa ed s uc u e and e olu ion models bo h wi h
and wi hou his ans e . We hen analyzed he implica ions o he obse able
p ope ies o he con ac phase.
Resul s. Implemen ing ene gy ans e helps in elimina ing ba oclinici y in he
common en elope be ween he componen s o a con ac bina y, which (i p esen )
would d i e s ong he mal lows. We ind ha accoun ing o ene gy ans e
in massi e con ac bina ies signi ican ly al e s he mass- a io e olu ion and can
ex end he li e ime o an unequal mass a io con ac sys em.
4.1 In oduc ion
The modeling o ene gy ans e (ET) goes back o he ’60s in an e o o
esol e Kuipe ’s pa adox. Lucy (1968) i s conside ed ET in common con ec i e
en elopes, and compu ed he i s app oxima ed s uc u e models o W UMa s a s.
La e , Lucy (1976); Flanne y (1976); Hazlehu s (1985) and Kähle (1989) calcula ed
models ha a e ou o he mal equilib ium and show cyclic beha io . Shu e al.
(1976,1979) and Lubow & Shu (1977,1979) (collec i ely SLA) cons uc ed models
o con ac bina ies whe e hey d opped he equi emen o a con inuous s uc u e
a he laye coinciding wi h he equipo en ial su ace o he i s Lag angian poin
(
𝐿1
). These models esol e Kuipe ’s pa adox ega dless o he he mal s uc u e
o he en elopes, namely, whe he hey a e adia i e o con ec i e. Despi e much
c i icism (see e.g., Hazlehu s ,1993;Kähle ,1989), his is he simples model o ET
in adia i e en elopes a ailable in he li e a u e.
Accu a e modeling o massi e, long-li ed con ac sys ems has been a emp ed
in he pas . Ma chan e al. (2016) compu ed de ailed e olu ion model g ids o
massi e bina ies wi h ini ial pe iods down o 0
.
5d, which includes he egime
o con ac bina ies a he ze o-age main sequence. Those models howe e did
no include ET be ween con ac componen s, as he models conce ned bina ies o
mass a io close o uni y,
𝑀2/𝑀1=
0
.
8
–
1, and he e ec was hough o be minimal.
Sen e al. (2022) used he models o Ma chan (2018) o s udy he semi-de ached
Algol sys ems, al hough hese models also included con ac phases. Menon e al.
(2021) compu ed models wi h ini ial mass a ios down o
𝑀2/𝑀1=
0
.
6 wi h he
THEORY OF ENERGY TRANSFER 109
s udy o massi e, long-li ed con ac bina ies in mind. They ound, also wi hou
including ET in con ac phases, a s ong co ela ion be ween obse ed mass a io
and pe iod in con ac sys ems ha is b oadly in ag eemen wi h obse a ions.
Howe e , he mass- a io dis ibu ion hey de i e is hea ily skewed owa d alues
close o uni y, which is no suppo ed obse a ionally. These au ho s sugges ed
ha including ET in con ac phases o unequal mass componen s could alle ia e
his disc epancy.
In his chap e , we apply he ET model o SLA in common s ella laye s o mode n
s ella s uc u e models o u he ad ance ou e olu iona y modeling o massi e
con ac bina ies. In Sec . 4.2, we desc ibe and discuss he heo y o ET. Sec ion
4.3 desc ibes he physical se up o he s ella e olu ion code, along wi h ou ET
implemen a ion. In Sec . 4.4, we compa e models compu ed wi h and wi hou
ET in con ac laye s and discuss he di e ence in he obse able p ope ies o he
models. Las ly, in Sec . 4.5, we p o ide ou concluding ema ks.
4.2 Theo y o ene gy ans e
4.2.1 Simple conside a ions
Theo e ical modeling o con ac s a s s a ed wi h Kuipe (1941), who s a ed ha
s able con ac sys ems o uni o m composi ion wi h unequal masses canno exis
as a esul o di e ing mass- adius ela ionships. Fo he galac ic ZAMS models
o B o e al. (2011), we de i e he mass- adius ela ion o single s a s o be
app oxima ely:
𝑅2
𝑅1ZAMS
=𝑀2
𝑀10.57
=𝑞0.57.(4.1)
Howe e , he condi ion o con ac in a bina y ollowing he Roche geome y
cons ains he su ace o he s a s o he same equipo en ial, which leads o:
𝑅2
𝑅1Roche ≈𝑀2
𝑀10.46
=𝑞0.46,(4.2)
o s a s no o e lowing hei RL oo much. Clea ly, Eqs.
(4.1)
and
(4.2)
canno be
sa is ied simul aneously unless
𝑀1=𝑀2
. Howe e , while he con ac condi ion
needs o be sa is ied om dynamical a gumen s, he s ella s uc u e does no ,
a p io i, need o ollow a single-s a model. E en hough he dense s ella co e
will be la gely unpe u bed due o he companion, he ou e laye s o con ac
componen s a e highly dis o ed, causing a ia ions in he o al adii wi h espec
o sphe ical models, as seen explici ly in Chap. 3. This esul had no ye aken
he possibili y o ET in o accoun and we expec u he changes unde he
110 MODELING ENERGY TRANSFER IN CONTACT BINARIES
conside a ion ha a ho e gas unde simila p essu e akes up mo e olume.
The e o e, he ZAMS mass- adius ela ion in Eq.
(4.1)
is no expec ed o be sa is ied
unde gene al con ac condi ions, and so Kuipe ’s pa adox can be esol ed by
p o iding al e na i e s ella models ha ha e a mass- adius ela ion close o he
con ac condi ion in Eq. (4.2).
Following he Roche-lobe geome y, Lucy (1968) inds he a io o he su ace
a eas
𝑆2/𝑆1
o wo s a s in con ac o be p opo ional o
(𝑀2/𝑀1)𝛽
, wi h
𝛽=
0
.
96.
Using he app oxima ion
𝛽≃
1, combined wi h on Zeipel’s heo em o g a i y
da kening,
𝑇4
e ∝𝑔
( on Zeipel,1924), and he S e an-Bol zmann law,
𝐿∝𝑆𝑇4
e
,
esul s in he simple expec a ion ha in con ac bina ies, he luminosi y a io
ollows he mass a io (Lucy,1968;Tassoul,2000):
𝐿2
𝐿1≃𝑀2
𝑀1
=𝑞. (4.3)
Single main-sequence s a s, on he o he hand, ollow he well-known mass-
luminosi y ela ion:
𝐿s,2
𝐿s,1≃𝑀2
𝑀1𝛼
=𝑞𝛼,(4.4)
wi h
𝛼≃
2
–
3 o he uppe main sequence (G ä ene e al.,2011;Köhle e al.,2015).
We use he symbol “
≃
” o deno e hese ela ions a e app oxima ions o simple
powe laws. We see ha his leads o a di e ence be ween he luminosi y o a
single s a
𝐿s,1
and he luminosi y
𝐿1
o a s a o he same mass in a con ac bina y
o
𝛥𝐿1=𝐿1−𝐿s,1=−𝑓 𝐿1,(4.5)
and o he companion
𝛥𝐿2=𝐿2−𝐿s,2=𝑓 𝐿1≃𝑓
𝑞𝐿2,(4.6)
since
𝐿2≃𝑞𝐿1
and we equi e
𝛥𝐿1+𝛥𝐿2=
0 o conse e ene gy. This de ines
𝑓
as
𝑓≃𝑞−𝑞𝛼
1+𝑞𝛼.(4.7)
The e o e, he wo s a s in a con ac bina y can ul ill he single-s a mass-
luminosi y ela ion in Eq.
(4.4)
in hei co es and he con ac bina y mass-
luminosi y condi ion in Eq.
(4.3)
a hei su aces i he amoun o ene gy pe
ime, gi en by Eqs.
(4.5)
and
(4.6)
, is ans e ed om he mo e massi e o he less
massi e s a in hei common en elope.
THEORY OF ENERGY TRANSFER 111
4.2.2 Models o ene gy ans e
The gene al solu ion o Kuipe ’s pa adox is o conside de ailed s ella models
wi h he inclusion o ET be ween he bina y componen s. Se e al models o ET
a e gi en in he li e a u e.
Lucy (1968) and Bie mann & Thomas (1972) p o ided a i s solu ion by adjus ing
he adiaba ic cons an s o con ec i e en elopes in con ac componen s. Howe e ,
his is an unsa is ac o y solu ion, since his se up equi es he s a s o be bu ning
hyd ogen h ough di e en nuclea chains o cycles in he case o Lucy (1968)
o ha he models exhibi inaccu a e ligh cu es, as in Bie mann & Thomas
(1972). O he models, such as hose o Lucy (1976) o Flanne y (1976), elaxed he
equi emen o he mal equilib ium and cons uc ed models o W UMa s a s ha
exhibi ed he mal cycles. Kähle (1989) p esen ed a de ailed model ha equi ed
u bulen mo ions in he common en elope o explain ea ly- ype ( adia i e) W
UMa bina ies.
Meanwhile, SLA p esen ed he con ac discon inui y model o con ac bina ies, by
elaxing he equi emen o con inuous s uc u al quan i ies ac oss he RL. This is
he only model ha ea s he common en elope as a single olume o he bina y
s uc u e, a he p ice o hiding a hea engine in a e y hin egion a ound he
RL. One peculia ea u e is ha his model necessi a ed a empe a u e in e sion
a he RL laye in one o he componen s as o he wise hey would no be able o
cons uc he mally s able con ac models o uni o m composi ion (in o de o
model bina ies a ze o age). This ea u e has ecei ed c i icism in ha he p oposed
hea engine iola es he second law o he modynamics and canno be s able o e
he mal imescales (see Hazlehu s ,1993;Kähle ,1989, and e e ences he ein).
E en ually, Kähle (2004) d ew he ollowing conclusion based on all collec ed
heo e ical a gumen s: in e nal ci cula ion cu en s mus exis in he less luminous
componen o educe he adia i e empe a u e g adien , since he luminosi y
ca ied by adia ion is educed by he ci cula ion luminosi y.
Gi en he complexi y o he heo e ical p oblem o he s uc u e o con ac bina ies,
especially wi h adia i e en elopes, i is beyond he scope o his wo k o u he
de elop he analy ic heo y. Ins ead, we apply an ET model in mode n s ella -
s uc u e calcula ions. Using shellula i y as ou base assump ion o he s ella
s uc u e (see Sec . 4.3.2 o he p ecise de ini ion), he model o SLA is a na u al
choice, al hough we ecognize ha his comes wi h he appa en he modynamical
p oblems s a ed abo e. Howe e , we belie e we a oid he mos undamen al
one, as we do no explici ly equi e a empe a u e in e sion in ou models. We
only used he model o SLA o compu e he amoun o ene gy ans e ed (see
Sec . 4.2.3).
112 MODELING ENERGY TRANSFER IN CONTACT BINARIES
4.2.3 Ene gy ans e in he Roche geome y
The wo k o SLA p o ides a gene al model o ET by in oducing he no ion o an
ene gy low a he base o he common en elope. I he con ac laye s a e shellula ,
and hey sa is y on Zeipel’s g a i y da kening, conse a ion o ene gy a he RL
implies:
𝐿′
1=(𝐿1+𝐿2)𝑆1⟨𝑔⟩1
𝑆1⟨𝑔⟩1+𝑆2⟨𝑔⟩2
,(4.8a)
𝐿′
2=(𝐿1+𝐿2)𝑆2⟨𝑔⟩2
𝑆1⟨𝑔⟩1+𝑆2⟨𝑔⟩2
.(4.8b)
He e,
𝑆
is he su ace a ea o he RL,
⟨𝑔⟩
is he su ace a e aged e ec i e g a i y a
he RL, and he p imed quan i ies speci y he s a e jus abo e he ET laye , while
unp imed hose jus below. This equa ion speci ies ha he ac ion o he o al
luminosi y ha each componen adia es is p opo ional o
𝑆⟨𝑔⟩
. The ans e ed
luminosi y hen equals
𝐿 ans =𝐿1−𝐿′
1=𝐿1𝑆2⟨𝑔⟩2−𝐿2𝑆1⟨𝑔⟩1
𝑆1⟨𝑔⟩1+𝑆2⟨𝑔⟩2
.(4.9)
Compa ing Eqs. (4.8) agains Eqs. (4.5)-(4.6), we ind o he ac ion 𝑓:
𝑓=𝐿1𝑆2⟨𝑔⟩2−𝐿2𝑆1⟨𝑔⟩1
(𝐿1+𝐿2)𝑆1⟨𝑔⟩1
,(4.10)
which is consis en wi h Eq. (4.7) as 𝑆2⟨𝑔⟩2
𝑆1⟨𝑔⟩1≃𝑞and 𝐿2
𝐿1≃𝑞𝛼.
4.3 Me hods
To in es iga e he e ec o ET on he e olu ion o con ac bina ies, we compu ed
bina y e olu ion models using he s ella e olu ion code
MESA
(Pax on e al.,
2011,2013,2015,2018,2019;Je myn e al.,2023), e sion 22.11.1. These models
a e he i s massi e bina y e olu ion models ha include ET in con ac laye s.
We ollowed he e olu ion o bina ies om he ZAMS un il he leas massi e
componen o e lows he second Lag angian poin .
4.3.1 Physical assump ions in MESA
The mic ophysical se up o ou s ella e olu ion models emains mos ly he same
as in Chap. 3, wi h some addi ions om he newe
MESA
e sion. We use he same
nuclea mic ophysics as in Chap. 3. The EOS has an addi ional able a ailable om
METHODS 113
he Skye p ojec (Je myn e al.,2021), i s blend speci ied in Je myn e al. (2023).
Radia i e opaci ies a e also blended om CO-enhanced ables o OPAL opaci ies
(Iglesias & Roge s,1993,1996) and Fe guson e al. (2005) o lowe empe a u es.
A high empe a u es, Comp on sca e ing opaci y is om Pou anen (2017), and
elec on conduc ion opaci ies a e aken om Cassisi e al. (2007) and Blouin e al.
(2020). Like Chap. 3, we se he me allici y o s a s o he sola alue,
𝑍⊙
, whe e
𝑍⊙=0.0142, wi h me al ac ions as de e mined om Asplund e al. (2009).
Mass loss h ough winds is accoun ed o by ollowing he p esc ip ion o B o
e al. (2011). I he su ace hyd ogen ac ion is
𝑋 >
0
.
7, he mass loss a e is
aken ei he om Vink e al. (2001), o empe a u es abo e he i on bi-s abili y
jump (calib a ed also by Vink e al.,2001), o he maximum o he a es om Vink
e al. (2001) and Nieuwenhuijzen & de Jage (1995), below his empe a u e. Fo
𝑋 <
0
.
4, he wind p esc ip ion o Hamann e al. (1995) is used, albei dec eased
by a ac o o en. When 0
.
4
< 𝑋 <
0
.
7, he wind is linea ly in e pola ed be ween
he abo e esul s. We allow pa o he wind launched by a s a o be acc e ed
by i s companion using he Bondi-Hoyle mechanism (Bondi & Hoyle,1944) as
implemen ed by Hu ley e al. (2002).
Fo in e nal mixing and o a ion, we use he same se up as in Sec . 3.2.
The ea men o mass ans e (MT) is explained in Sec . 4.3.3, while he
implemen a ion o ene gy ans e (ET) is shown in Sec . 4.3.4.
4.3.2 Shellula i y and Roche lobe geome y
Since we deal wi h highly idally de o med s a s, we use he modi ica ions o he
s ella s uc u e equa ions om Sec . 3.2.1 o inco po a e he RL geome y in o
a one-dimensional (1D) s ella e olu ion code. The s a s a e he e o e modeled
as hyd os a ic s uc u es li ing in he Roche po en ial
𝛹
o a ully synch onized
bina y. Fu he mo e, we assume s ella laye s o be shellula . Shellula i y is
eached when all in ensi e quan i ies (in pa icula he empe a u e, p essu e, and
mass densi y) a e cons an along a s ella laye and when such a laye coincides
wi h a unique equipo en ial su ace.
I should be emphasized ha in he con ex o 1D models, he ull shellula i y o
laye s o a (single) s a is an assump ion – no a sel -consis en ly modeled ea u e
as, o cou se, he e is no 3D s uc u e o explici ly es he shellula i y o a s ella
laye . Howe e , o e lowing laye s o componen s in a con ac bina y can be es ed
whe he hey a e shellula wi h espec o each o he i he empe a u e, densi y, and
so on a he 1D cells a e equal o equal alues o Roche po en ial. Gi en he
me hod we used o spli he common en elope o he con ac bina y (Sec . 3.2.4),
he compu a ion o he idal de o ma ion co ec ions based on equipo en ial
su aces, and he usage o he co esponding ou e bounda y condi ion (Sec . 3.4),
120 MODELING ENERGY TRANSFER IN CONTACT BINARIES
Figu e 4.3: Con ac du a ion dis ibu ions as unc ion o obse ed mass a io
¯𝑞=min 𝑀1
𝑀2,𝑀2
𝑀1
.Top: Di e en ial du a ion (in bins o
𝛥¯𝑞=
0
.
005). Bo om:
Cumula i e du a ion in con igu a ions mo e ex eme han ¯𝑞.
seen in he le column o Fig. 4.4 by he joining o he empe a u e and densi y
p o iles o he p ima y and seconda y. In pa icula , a he su ace, he densi y o
he ET componen s ag ee o wi hin 0
.
2%, while he empe a u e o wi hin 0
.
5%,
whe eas he su ace p ope ies o he no-ET componen s a y mo e han 10% (as
expec ed o models o di e ing mass). While he o me di e ences a e ha d o
cons ain om measu emen , cu en da a-analysis echniques a e able o measu e
su ace empe a u es wi h accu acies be e han 10%, so ha he no-ET model can
be uled ou obse a ionally.
A signi ican sho coming o ou model is he assumed hickness o he ET laye .
Shu e al. (1979) a gued ha he hickness 𝑑o he ET laye is on he o de o :
𝑑
𝑎∼𝛿0.4, 𝛿 =𝐿 ans/𝑎2
𝜌ℎ𝑐𝑠
,(4.17)
wi h
𝑎
as he bina y sepa a ion, and
𝜌
,
ℎ,
and
𝑐𝑠
as he densi y, speci ic en halpy
and local sound speed e alua ed a he RL, espec i ely. Lubow & Shu (1979)
compu ed his numbe o be
𝑑/𝑎∼
10
−2
o s a s o masses a ound 4
–
8
𝑀⊙
which
jus i ies SLA in modeling he laye as a discon inui y in he s ella p o ile, loca ed
a he RL adius.
STELLAR MODELS 121
Figu e 4.4: P o iles o ou e laye s o he bina y componen s a he onse o in e se
MT. I shows ( op o bo om) he empe a u e, densi y, and luminosi y p o iles o
bo h componen s in he ET ( ed, le column) and no ET (blue, igh column) cases,
all as a unc ion o he scaled adius coo dina e ˜𝑟=𝑟−𝑅RL
𝑅−𝑅RL . The g ay e ical lines
show he loca ion o he RL.
Howe e , di ec compu a ion o Eq.
(4.17)
o ou 25
𝑀⊙+
20
𝑀⊙
model show ha
his es ima ion b eaks down o highe masses, see he ed line in Fig. 4.5. This
sugges s ha he ene gy edis ibu ion low, modeled as a discon inui y by SLA,
is no su icien in hese highe mass s a s. Excep when he bina y has eached
conside able o e low, whe e
𝑑/𝑎≲
10
−2
, we ind ha he hickness needed can
be a signi ican ac ion o he bina y sepa a ion, e en su passing i in he ea ly
s ages o he con ac phase.
Ano he way o compu e he hickness o he ET laye is o use Be nouilli’s
equa ion. I we conside luid mo ion om a away om L
1
on he p ima y
s a (loca ion 𝑖) owa d L1(loca ion 𝑓), we ha e
1
2𝑣2
𝑓+∫𝑓
𝑖
𝑑𝑃
𝜌=0,(4.18)
122 MODELING ENERGY TRANSFER IN CONTACT BINARIES
whe e we ha e al eady canceled he po en ial e ms
𝛹𝑖,𝛹𝑓
since we mo e along
an equipo en ial su ace, and he ini ial eloci y,
𝑣𝑖
, is assumed o be negligible.
Making he es ima ion:
∫𝑓
𝑖
𝑑𝑃
𝜌≈1
𝜌𝑓+1
𝜌𝑖𝑃𝑓−𝑃𝑖,(4.19)
and assuming ha he ans e ed ene gy h ough he bina y neck o wid h,
𝑏
, by
a mass low o ¤
𝑀=𝜌𝑖𝑣𝑓𝑏2is:
𝐿 ans =𝜌𝑖𝑣𝑓𝑏2(𝑐𝑝,1+𝑐𝑝,2)(𝑇𝑓−𝑇𝑖),(4.20)
we compu e o he minimal hickness o he ET laye :
𝑏≈
u
u
u
𝐿 ans
𝜌𝑖(𝑐𝑝,1+𝑐𝑝,2)(𝑇𝑓−𝑇𝑖)√︂1
𝜌𝑓+1
𝜌𝑖𝑃𝑓−𝑃𝑖.(4.21)
Finally he hickness o he laye a he neck is ela ed o he hickness a away
om L1 ia:
𝑑
𝑎≈𝑏
𝑎2
,(4.22)
since he Roche po en ial a ies quad a ically nea L
1
and linea ly elsewhe e.
Equa ion
(4.21)
gi es a lowe limi on he wid h o ET laye so ha a balanced
mass low,
¤
𝑀
, in he con ac bina y can ca y a o be ans e ed luminosi y,
𝐿 ans
.
Con e sely, i can be in e p e ed as
𝐿 ans
being he maximal luminosi y he mass
low can ca y in a laye o ixed wid h, 𝑏.
We plo he Be nouilli-compu ed hickness o Eq.
(4.22)
in Fig. 4.5. Simila ly, we
see ha he equi ed hickness,
𝑑/𝑎
, is much la ge han wha he adius o he
p ima y s a allows oom o . I is only a la e imes, when he mass a io has
equilib a ed and deepe con ac is engaged, ha he es ima ed wid h becomes
smalle han he o e low a e, 𝑅−𝑅RL, o he p ima y.
4.4.2 Mass e sus luminosi y a ios
Du ing he nuclea - imescale, in e ed MT phase, con ac is engaged so ha ou ET
scheme ac s o mo e luminosi y om one componen o he o he . As men ioned
in Sec . 4.2, we expec he luminosi y a io o con ac bina ies o ollow he mass
a io,
𝐿∝𝑀
, as opposed o de ached s a s ollowing a single-s a mass-luminosi y
ela ion,
𝐿∝𝑀𝛼
, wi h
𝛼≃
2
–
3. Figu e 4.6 shows he e olu ion o he luminosi y
a io as unc ion o he mass a io du ing he slow MT phase o he 25
𝑀⊙+
20
𝑀⊙
explo ed in Sec . 4.4.1. In his g aph, he models e ol e om he op igh a
STELLAR MODELS 123
Figu e 4.5: Thickness o he ET laye ,
𝑑
, wi h espec o he bina y sepa a ion,
𝑎
, as
a unc ion o he adius o he p ima y s a du ing he long li ed con ac phase.
The g ay line gi es he physical size o he o e lowing laye s, and co esponds o
he maximal wid h he ET laye can assume.
𝑞≈
1
.
4 o nea -equal mass a io on he le . We see ha he model no including
ET ollows closely a
𝑞2.2
ela ion, app op ia e o single s a s in he mass ange o
10
–
30
𝑀⊙
. As he models including ET engage in o deep con ac howe e , hei
mass-luminosi y a io changes d as ically om he
𝑞2.2
line in nea -con ac o he
𝑞1 ela ion in ull con ac .
We gi e (o e plo ed on Fig. 4.6) he measu emen s o se e al obse ed massi e
con ac bina ies o Abdul-Masih e al. (2021); Yang e al. (2019) and Lo enzo e al.
(2014) (see also Fig. 2 o Lange ,2022). Cu iously, he luminosi y a ios om
Abdul-Masih e al. (2021), while hey do ollow an
𝐿∝𝑀
end, a e o se o lowe
𝐿2/𝐿1
han p edic ed. Ei he he luminosi y o he p ima y is o e es ima ed o he
unce ain ies a e unde es ima ed. The sys ems included om Mahy e al. (2020)
we e ca ego ized as ‘unce ain con igu a ions’ since he measu emen o he adius
was consis en wi h being abo e as well as below he RL. In he con ex o he
mass-luminosi y ela ion howe e , we expec ha he sys em om Mahy e al.
(2020) a
𝑞≈
1
.
3 (VFTS 563) is a ue con ac sys em, while he one a
𝑞≈
1
.
2 (VFTS
217) is no (al hough wi hin 1𝜎i could be ei he ).
124 MODELING ENERGY TRANSFER IN CONTACT BINARIES
Figu e 4.6: Luminosi y a io e sus mass a io du ing he nuclea - imescale MT o
he 25
𝑀⊙+
20
𝑀⊙
sys ems o Sec . 4.4.1. Measu emen s om obse ed massi e
(nea -)con ac sys ems om Abdul-Masih e al. (2021), Yang e al. (2019), and
Lo enzo e al. (2014) a e o e plo ed. The sys ems om Mahy e al. (2020) we e
classi ied as ‘unce ain con igu a ions’.
4.5 Conclusions
In his chap e , we ha e aken a s ep o wa d in he de ailed modeling o con ac
bina ies, by implemen ing a model o ET in de ailed s ella -s uc u e and e olu ion
models. F om Fig. 4.4, we see ha his ela i ely simple model (i.e., he inclusion o
a hea sou ce o sink in he s ella model as p oxy o he ET) is capable o making
he common laye s o s ella componen s in a con ac con igu a ion shellula .
Models wi hou such ET do no exhibi hei common laye s o be shellula , which,
om heo e ical a gumen s, would d i e s ong ho izon al lows equilib a ing all
g adien s, in pa icula p essu e.
F om Figs. 4.2 and 4.3, we see ha he ime spen in deep con ac a mass a ios
be ween
¯𝑞=
0
.
7
–
0
.
8 is ex ended when ET was included, e sus when i was
igno ed. This is a p omising esul , in ha i his end pe sis s ac oss he pa ame e
space o he o al mass, ini ial mass a io, and ini ial pe iod, ET could p o ide
an answe o he disc epancy be ween he obse ed mass- a io dis ibu ion o
massi e con ac sys ems and i s p edic ed dis ibu ion. The compu a ion and
analysis o a ull g id o models o a popula ion syn hesis s udy is he opic o he
nex chap e .
Chap e 5
Ene gy ans e in a popula ion
o massi e con ac bina ies
“I know my app ehensions migh
ne e be allayed, and so I close,
ealizing ha pe haps he ending
has no ye been w i en.”
— A us
This chap e is mainly based on:
Modeling con ac bina ies, III. P ope ies o a popula ion o close, massi e
bina ies.
M. Fab y, P. Ma chan , N. Lange and H. Sana
in p epa a ion o submission o ASTRONOMY & ASTROPHYSICS
5.1 In oduc ion
In Chaps. 3and 4, we ha e de eloped he me hodology o consis en ly ake in o
accoun idal de o ma ion o close bina y componen s and ene gy ans e in he
common laye s o con ac bina ies. In he in oduc ion, we ha e also es ablished
ha he s a e-o - he-a models ha e ouble ep oducing he obse ed mass- a io
dis ibu ion o massi e con ac sys ems. While he models indica e ha , once
nuclea - imescale con ac is engaged, he masses should equalize apidly (Menon
125
126 ENERGY TRANSFER IN A POPULATION OF MASSIVE CONTACT BINARIES
e al.,2021), long- e m moni o ing o he pe iod de i a i e shows obse ed sys ems
a e s able on he main-sequence li e ime (Abdul-Masih e al.,2022). S ill, in Chap. 4,
we showed ha o pa icula sys ems, ene gy ans e in adia i e en elopes can
al e he mass- a io e olu ion o massi e con ac sys ems, and has he possibili y
o elie e he disc epancy be ween models and obse a ions.
In his chap e , we compu e a g id o sho -pe iod, massi e bina y-e olu ion
models and in es iga e he e ec o ene gy ans e on he mass- a io dis ibu ion
o con ac sys ems. In Sec . 5.2, we desc ibe he modi ica ions o he s ella
e olu ion code wi h espec o he p e ious chap e s, he ini ializa ion o he
bina y models, e mina ion condi ions, and we de ail he calcula ions used o
making popula ion p edic ions. Nex , Sec . 5.3 p esen s and discusses he esul s
ob ained om he popula ion syn hesis, while Sec . 5.4 gi es concluding ema ks.
5.2 Me hodology
We use he de ailed bina y e olu ion code
MESA
, e sion 22.11.1 (Pax on e al.,
2011,2013,2015,2018,2019;Je myn e al.,2023), o compu e g ids o bina y
models ha unde go case A mass ans e , om which we assemble a syn he ic
popula ion. The pa ame e s o be a ied in each g id a e he ini ial p ima y mass,
𝑀1,ini
, ini ial mass a io,
𝑞ini =𝑀2,ini /𝑀1,ini
, and he ini ial pe iod,
𝑝ini
. Fo
ini ial p ima y mass, we sample
𝑀1,ini ={
8
,
9
,
10
, ...,
19
}∪{
20
,
22
,
24
, ...,
48
} ∪
{
50
,
52
.
5
,
55
, ...,
70
}𝑀⊙
. The ini ial mass a io is
𝑞ini =
0
.
6
–
0
.
975, spaced uni o mly
wi h
𝛥𝑞ini =
0
.
025, while he ini ial pe iod is
𝑝ini =
0
.
5
–
8d, spaced loga i hmically
wi h
𝛥log(𝑝/
d
)=
0
.
04. We compu e wo g ids wi h hese pa ame e a ia ions:
one ha includes he e ec o ene gy ans e (ET), while he o he igno es i . This
amoun s o a o al o 35712 de ailed bina y-e olu ion models. Th oughou he
discussion, we will e e o he models ha include ET as he “ET models,” while
hose ha do no he “no-ET models.”
5.2.1 Physical assump ions in MESA
We use a simila se up o he
MESA
code as in Chap. 4,i.e., we use he same
mic ophysics, con ec ion and o e shoo pa ame e s (wi h wo excep ions, see
below), wind p esc ip ion, de o ma ion geome y, and mass- and ene gy- ans e
(MT, ET) calcula ions. In con as o Chaps. 3and 4we make he ollowing
modi ica ions o he modeling se up:
•
We do no assume igid-body o a ion h oughou he e olu ion, so
ha di e en ial o a ion is allowed. Howe e , we do include as idal
synch oniza ion by uni o mly modi ying he angula momen um o he s a
METHODOLOGY 127
on he imescale o he o bi al pe iod, 𝑝,
𝛥𝑗 ides =1−exp −𝛥𝑡
𝑝(𝜔o b𝑖 o −𝑗).(5.1)
He e
𝛥𝑡
is he imes ep o he e olu ion,
𝑗
and
𝑖 o
a e he cu en angula
momen um and he momen o ine ia o he s ella laye , espec i ely, and
𝜔o b =2𝜋
𝑝
is he o bi al angula eloci y. We model spin-o bi coupling o
conse e o al angula momen um, so ha he equi ed angula momen um
o synch onize bo h s a s in his ashion is hen sub ac ed om he o bi al
angula momen um. Gi en ha
𝛥𝑡
is much longe han he o bi al pe iod in
e olu iona y calcula ions (yea s o housands o yea s e sus days), Eq.
(5.1)
speci ies e icien idal synch oniza ion. We hus expec li le depa u e om
solid-body o a ion in ou e y close bina y simula ions, since p ocesses ha
cause depa u es om i ope a e on longe imescales han he o bi al one.
•
We also allow o o a ional mixing in ou models. We include Edding on-
Swee ci cula ion, he Gold eich-Schube -F icke ins abili y and bo h he
dynamical- and secula -shea ins abili y, all ollowing he implemen a ion o
Hege e al. (2000).
•
We educe he pa ame e s o Mixing Leng h Theo y o
𝛼MLT =
1
.
5, and he
semicon ec i e e iciency o 𝛼sc =1, o be in line wi h Menon e al. (2021).
5.2.2 Bina y ini ializa ion
Once ini ial masses and a pe iod a e selec ed, he bina y model is ini ialized as
ollows. Non- o a ing, single-s a models o mass
𝑀1,ini
and
𝑀2,ini =𝑞𝑀1,ini
a e in e pola ed om p e-compu ed ze o-age main sequence (ZAMS) models o
sola me allici y (
𝑍⊙=
0
.
0142, wi h me al ac ions ollowing Asplund e al. 2009)
and a helium con en o
𝑌=
0
.
2703 and loaded in o a bina y model. We se he
eccen ici y o ze o as we expec ci cula o bi s o close bina ies (Zahn,1975). The
single s a models a e hen spun up o he app op ia e Keple ian angula eloci y
𝜔o b
. Since he models a e b ough ou o he mal equilib ium due o he spin up,
wha ollows is a he mal elaxa ion pe iod on o he ue ZAMS o he models,
du ing which MT, ET o any o he mass loss is u ned o . We also keep he pe iod
cons an o he ini ial alue and en o ce igid body o a ion a he o bi al angula
eloci y. We de ine he ZAMS as he poin whe e he luminosi y o bo h s a s is
wi hin 1% o hei nuclea luminosi y. When his condi ion is eached, we make a
decision depending on he a e o o e low o he s a s.
Fi s , i bo h s a s a e wi hin hei RL, he simula ion p oceeds as no mal, wi hou
u he in e en ion. Second, i a leas one s a o e lows i s RL, his means
ha MT mus ha e s a ed in he p e-main sequence (PMS) phase. To simula e
128 ENERGY TRANSFER IN A POPULATION OF MASSIVE CONTACT BINARIES
his, we a i icially boos he o bi al pe iod o pu he componen s in a de ached
con igu a ion, and hen, on a he mal imescale o he mos massi e s a (
𝜏=
0
.
75
𝐺𝑀2
1/𝑅1𝐿1
), d ain he added angula momen um om he sys em o again
each he ini ial pe iod. Du ing his p ocess, he componen s each he RL, and MT
and ET a e allowed. Finally, i one o he s a s o e lows he second Lag angian
poin (L2OF) a he ZAMS, we e mina e he simula ion immedia ely as we assume
a me ge would ha e happened on he PMS.
5.2.3 Ou comes and e mina ion
The ou come o ou simula ions b oadly alls in h ee ca ego ies.
•
The bina y eaches L
2
in a con ac con igu a ion. In his si ua ion, we assume
mass loss will s a om L
2
, ca ying away angula momen um, which we
expec esul s in a me ge on a dynamical imescale. We s op he simula ion
a his poin and classi y hese sys ems as “me ge s.” A a ian o his
ou come is when he MT calcula ion de e mines ha
¤
𝑀
is a e y high alue.
We choose o ake a limi ing alue o
¤
𝑀=
10
−1𝑀⊙y −1
ac oss he whole
g id. MT a es highe han his alue a e a leas wo o de s o magni ude
abo e he he mal imescale MT a es o massi e main-sequence s a s (excep
pe haps o he highes masses in ou g id, whe e
¤
𝑀 he mal ≈
5
×
10
−3𝑀⊙y −1
),
and we expec MT o e ol e owa d dynamical imescale MT.
•
One o he s a s (mos likely he ini ially mo e massi e one) eaches co e
hyd ogen exhaus ion, wi hou igge ing a me ge . A his poin , he s a
will s a unde going apid e olu ion, and we s op he simula ion calling
hese “(main-sequence) su i o s.”
•
The simula ion can encoun e nume ical di icul ies, by ailing o each
con e gen models du ing he e olu ion. In his case, we hal he simula ion
and signal his sys em encoun e ed an “e o .”
5.2.4 Popula ion syn hesis compu a ions
When doing popula ion syn hesis calcula ions, each model in ou g id needs
o be assigned a ela i e weigh o accu a ely p edic he occu ence a e o a
ce ain sys em in a popula ion. Because he p obabili y,
P
, o inding sys ems in
a ce ain con igu a ion
𝜗0
is p opo ional o he ime he sys em spends in ha
con igu a ion, we ha e
P(𝜗0) ∝ ∫𝛿(𝜗−𝜗0)WNdN,(5.2)
METHODOLOGY 129
whe e we sum he ime each o he models,
N
, spends a he con igu a ion
𝜗0
, using
he Di ac del a unc ion,
𝛿
. The in eg and is hen mul iplied wi h i s s a is ical
weigh ,
WN
, coming om he likelihood o his model being c ea ed om he s a
and bina y o ma ion p ocesses. Since we a y he ini ial p ima y mass, ini ial
mass a io and ini ial pe iod, and ake in o accoun he s a - o ma ion a e (SFR)
his o y, he o al ( ela i e) weigh is going o be,
WNdN=W𝑀1,ini d𝑀1,ini W𝑞ini d𝑞ini W𝑝ini d𝑝ini WSFR d𝑡. (5.3)
The indi idual weigh s a e aken om models o obse ed young popula ions.
We ake
W𝑀1,ini d𝑀1,ini =𝑀−2.35
1,ini d𝑀1,ini ,(5.4a)
W𝑞ini d𝑞ini =𝑞0
ini d𝑞ini =d𝑞ini ,(5.4b)
W𝑝ini d𝑝ini =(log 𝑝ini )0dlog 𝑝ini ,(5.4c)
WSFR d𝑡=d𝑡, (5.4d)
coming om he Salpe e ini ial-mass unc ion (Salpe e ,1955;K oupa,2001;
Bas ian e al.,2010) and he modeled mass- a io dis ibu ions ound om a sample
o O- ype s a s om Sana e al. (2012), which a e shown o be simila o B-
ype s a s (Villaseño e al.,2021;Banya d e al.,2022). No sample o obse ed
bina ies co e s pe iods sho e han 1d, so we adop Öpik’s law (Öpik,1924),
W𝑝ini =log 𝑝ini
, which is close o he indings o Almeida e al. (2017). Unless
he index o he pe iod-dis ibu ion powe -law is posi i e and high, which is no
suppo ed by any obse a ional sample o young, massi e s a s, sho -pe iod
bina ies a e a o ed o e longe ones. We use no win bina y excess in ou se up,
since high-mass s a s ha e small win ac ions (Sana e al.,2012;Moe & Di S e ano,
2017). The age p io indica es we assume a cons an s a - o ma ion his o y.
Gi en we ha e a g id o a disc e e numbe o models, each g id poin ep esen s a
ini e a ea o he bi h dis ibu ions in Eqs. (5.4). In pa icula , we compu e:
∫uppe
lowe 𝑀−2.35
1,ini d𝑀1,ini ∝𝑀−1.35
1,ini ,uppe −𝑀−1.35
1,ini ,lowe ,(5.5)
o he con ibu ion o he model o ini ial p ima y mass
𝑀1,ini
, and he uppe
and lowe bounds a e aken in he middle be ween ou g id poin s. Fo poin s
on he edge o he g id, we linea ly ex apola e he spacing ou wa d and hen
use Eq.
(5.5)
as usual. In his way, he model wi h ini ial p ima y mass 8
𝑀⊙
has
𝑀uppe =
8
.
5
𝑀⊙
and
𝑀lowe =
7
.
5
𝑀⊙
, while he model o highes ini ial mass has
𝑀uppe =
71
.
125
𝑀⊙
and
𝑀lowe =
68
.
875
𝑀⊙
. A simila compu a ion o he bins o
𝑞ini and 𝑝ini is pe o med.
