Cygnus X-1 contains a 21-solar mass black hole -- implications for massive star winds
James C.A. Miller-Jones, Arash Bahramian, Jerome A. Orosz, Ilya Mandel, Lijun Gou, Thomas J. Maccarone, Coenraad J. Neijssel, Xueshan Zhao, Janusz Zió?kowski, Mark J. Reid, Phil Uttley, Xueying Zheng, Do-Young Byun, Richard Dodson, Victoria Grinberg, Taehyun Jung, Jeong-Sook Kim, Benito Marcote, Sera Markoff, María J. Rioja, Anthony P. Rushton, David M. Russell, Gregory R. Sivakoff, Alexandra J. Tetarenko, Valeriu Tudose, Joern Wilms
CCygnus X-1 contains a 21-solar mass black hole –implications for massive star winds
James C.A. Miller-Jones, ∗ Arash Bahramian, Jerome A. Orosz, Ilya Mandel, , , Lijun Gou, , Thomas J. Maccarone, Coenraad J. Neijssel, , , Xueshan Zhao, , Janusz Zi ´ołkowski, Mark J. Reid, Phil Uttley, Xueying Zheng, , Do-Young Byun, , , Richard Dodson, Victoria Grinberg, Taehyun Jung, , Jeong-Sook Kim, Benito Marcote, Sera Markoff, , Mar´ıa J. Rioja, , , Anthony P. Rushton, , David M. Russell, Gregory R. Sivakoff, Alexandra J. Tetarenko, Valeriu Tudose, Joern Wilms
1: International Centre for Radio Astronomy Research – Curtin University, Perth, Western Aus-tralia 6845, Australia2: Astronomy Department, San Diego State University, San Diego, CA 92182-1221, USA3: School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia4: OzGrav: The Australian Research Council Centre of Excellence for Gravitational Wave Dis-covery, Australia5: School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B152TT, United Kingdom6: Key Laboratory for Computational Astrophysics, National Astronomical Observatories, Chi-nese Academy of Sciences, Beijing 100012, China7: University of Chinese Academy of Sciences, Beijing 100012, China8: Department of Physics & Astronomy, Texas Tech University, Lubbock, TX 79409-1051,USA9: N. Copernicus Astronomical Center, PL-00-716 Warsaw, Poland1 a r X i v : . [ a s t r o - ph . H E ] F e b
0: Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA11: Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1098 XH Amsterdam,The Netherlands12: Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea13: University of Science & Technology, Daejeon 34113, Republic of Korea14: International Centre for Radio Astronomy Research – The University of Western Australia,Crawley, Western Australia 6009, Australia15: Institut f ¨ur Astronomie und Astrophysik, Universit¨at T ¨ubingen, 72076 T ¨ubingen, Germany16: Joint Institute for Very Long Baseline Interferometry European Research InfrastructureConsortium, 7991 PD Dwingeloo, The Netherlands17: Gravitation & Astroparticle Physics Physics Amsterdam Institute, University of Amster-dam, NL-1098 XH Amsterdam, The Netherlands18: Commonwealth Scientific and Industrial Research Organisation Astronomy and Space Sci-ence, Perth, Western Australia 6102, Australia19: Observatorio Astron´omico Nacional, Instituto Geogr´afico Nacional, 28014 Madrid, Spain20: Department of Physics, Astrophysics, University of Oxford, Oxford OX1 3RH, UK21: School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ,UK22: Center for Astro, Particle and Planetary Physics, New York University Abu Dhabi, AbuDhabi, United Arab Emirates23: Department of Physics, Centennial Centre for Interdisciplinary Science, University of Al-berta, Edmonton, AB T6G 2E1, Canada24: East Asian Observatory, Hilo, Hawaii 96720, USA25: Institute for Space Sciences, 077125 Bucharest-Magurele, Romania26: Dr. Karl Remeis-Sternwarte and Erlangen Centre for Astroparticle Physics, Friedrich-2lexander-Universit¨at Erlangen-N¨urnberg, 96049, Bamberg, Germany ∗ To whom correspondence should be addressed; E-mail: [email protected].
The evolution of massive stars is influenced by the mass lost to stellar windsover their lifetimes. These winds limit the masses of the stellar remnants (suchas black holes) that the stars ultimately produce. We use radio astrometry torefine the distance to the black hole X-ray binary Cygnus X-1, which we findto be . +0 . − . kiloparsecs. When combined with previous optical data, thisimplies a black hole mass of . ± . solar masses, higher than previous mea-surements. The formation of such a high-mass black hole in a high-metallicitysystem constrains wind mass loss from massive stars. Gravitational wave detections of black hole merger events have revealed a population ofblack holes with masses ranging from 7 to 50 solar masses ( M (cid:12) ) ( ). Black holes that interactwith a companion star are visible to electromagnetic observations as an X-ray binary. Radialvelocity measurements of these companion stars have shown that black holes in X-ray binariesall have masses below M (cid:12) ( ). The highest measured black hole mass in an X-ray binary is . ± . M (cid:12) for the extragalactic system M33 X-7 ( ).The mass of a black hole is initially set by the properties of its progenitor star, augmented byaccretion or mergers over its lifetime. The relevant properties of the progenitor include its initialmass and abundance of heavy elements (referred to as its metallicity), the mass lost in stellarwinds over its lifetime, and the evolutionary pathway that it followed, which can be stronglyinfluenced by a binary companion. Mass measurements for massive stellar-mass black holesconstrain stellar and binary evolution models ( ) and predictions of the expected black holemerger rates.The X-ray binary Cygnus X-1 (V1357 Cyg; coordinates in Table S1) contains a black hole ina 5.6-day orbit with a more massive supergiant donor star, of spectral type O. Previous estimatesof its component masses were based on a parallax measurement, whereby the apparent annualangular shift in the source position relative to more distant objects was measured using radiovery long baseline interferometry. Via trigonometry, this gave a distance of . +0 . − . kpc ( ).When combined with optical radial velocity measurements of the system, these data yielded ablack hole mass of . ± . M (cid:12) ( ). However, the derived system parameters are then inconsis-tent with the expected mass-luminosity relation for the donor, if it is a hydrogen-burning mainsequence star ( ). The optical parallax measurement of . ± . milliarcseconds (mas) usingthe Gaia space telescope ( ), after correction for the known zero-point offset of ≈ . mas(with estimates ranging from 0.03–0.08 mas; ( )), becomes . ± . mas. This is inconsis-tent with the radio value of . ± . mas ( ), and is unlikely to be due to orbital displacementof the donor star because the Gaia value is the average over 119 orbital periods.Between May 29 and June 3, 2016, we performed six observations (one per day) of Cygnus X-1 with the Very Long Baseline Array (VLBA) at 8.4 GHz, sampling one full orbital period. To3educe systematic uncertainties in our position measurements, we phase referenced the data toa nearby calibrator source . ◦ from Cygnus X-1 ( ). These data resolve the orbital motionof the black hole as projected onto the downstream surface from which the jet emission can es-cape, known as the photosphere. We find the orbit is clockwise on the plane of the sky (Fig. 1),in agreement with previous observations ( ).Combining the orbital phase coverage of our VLBA data with the archival observations ( ),we simultaneously fitted the entire data set (covering a 7.4-year baseline) with a full astrometricsolution incorporating linear motion across the sky (proper motion), parallax and orbital motion.A fit to both right ascension and declination co-ordinates showed an orbital phase dependence inthe direction of the residuals. We attribute this to the effect of free-free absorption in the stellarwind, which is known to modulate the radio emission of Cygnus X-1 on the orbital period ( ).Electrons in the ionised wind of the O star can absorb radio photons in the presence of atomicnuclei, preventing the radio emission from the inner parts of the jet from reaching us. This free-free absorption is reduced as the stellar wind density decreases on moving away from the O star,allowing radiation to escape from further downstream. As the black hole (from which the jet islaunched) moves around its orbit, the varying path length through the stellar wind imprints anorbital periodicity on the apparent radio position along the jet axis. When the black hole is onthe far side of the donor star, the path length and hence absorption are maximised, pushing theradio photosphere downstream along the jet axis ( ).To negate the impact of the stellar wind absorption, we therefore repeated our astrometricmodel fitting in one dimension only, perpendicular to the known jet axis. This removed theorbital phase dependence of the fit residuals perpendicular to the jet axis (Fig. 2). Our measuredsemi-major axis of the black hole orbit is ± microarcseconds ( µ as), and our revisedparallax measurement is . ± . mas, consistent with the optical value from Gaia aftercorrection for the zeropoint. After converting our measured parallax to a distance using anexponentially-decreasing space density prior ( ), we find a distance of . +0 . − . kiloparsecs(kpc).This revised distance affects the system parameters derived from the optical modelling ( ).We reanalysed the optical light curve ( ) and radial velocity curves ( ), adopting our reviseddistance and additional constraints on the effective temperature, surface gravity and heliumabundance of the donor star ( ). We find substantially higher masses for both the black hole( . ± . M (cid:12) ; Fig. S8) and the donor star ( . +7 . − . M (cid:12) ). The black hole orbital semi-majoraxis derived from this reanalysis of the optical data is . ± . astronomical units (au)(Table 1), which equates to ± µ as at our best-fitting distance of . +0 . − . kpc. This isconsistent with the value derived directly from our VLBA astrometry (Fig. 2, Table S3).The higher donor mass and greater luminosity inferred from the larger distance (Table 1)bring the system into closer agreement with the mass-luminosity relationship for main-sequencehydrogen-burning stars of solar composition (
7, 10 ). However, the measured surface composi-tion shows that the helium-to-hydrogen ratio is enhanced by a factor of 2.6 relative to the solarcomposition ( ). This would imply a slightly different mass-luminosity relationship if the sur-face abundance is indicative of the overall composition of the donor star, but remains broadly4onsistent with our revised values (Fig. 3).The higher mass and distance could also affect the black hole spin determined from spec-tral fitting of the X-ray continuum ( ). We therefore reanalysed archival X-ray data using acontinuum-fitting method, assuming the black hole spin axis is aligned with the orbital plane.We find the black hole dimensionless spin parameter a ∗ > . , close to the maximum pos-sible value of 1, although this could be affected by systematic uncertainties ( ). Even if thetrue value is less extreme, it would still be very high, consistent with previous results derivedfrom both the continuum and iron line fitting methods (
16, 17 ). While the spin derived from ouranalysis would be reduced if the black hole spin axis were not aligned with the orbital plane,even a ◦ misalignment would still require a high spin, a ∗ = 0 . .As an X-ray binary system with a high-mass donor star, the black hole in Cygnus X-1 cannothave been spun up by accretion from its companion at the maximum theoretical rate (known asthe Eddington limit; ∼ × − M (cid:12) yr − for Cygnus X-1), as that cannot occur for longerthan the lifetime of the donor ( ∼ Myr for our inferred mass ( )). The accretion time maybe close to the age of the jet-inflated nebula surrounding the source (a few tens of kyr; ( )).The current spin must therefore reflect the angular momentum of the core of the progenitor star.An evolutionary pathway that could explain this is main sequence mass transfer from the blackhole progenitor to the secondary star (Case A mass transfer; ( )) with the core of the progenitortidally locked and hence rapidly rotating, which can produce high black hole spins ( ). Thisevolutionary pathway for Cygnus X-1 would imply a spin axis of the black hole progenitor thatis aligned with the orbital angular momentum. Given the very low velocity kick of ± km s − imparted to the system on black hole formation (as determined by VLBA astrometry; ( )),the black hole spin should still be aligned with the orbital inclination, as we assumed above.This scenario is also consistent with the low orbital eccentricity ( ), and the lack of strongquasi-periodic variability seen in the X-ray power density spectra of the source ( ).This evolutionary pathway is associated with enhanced nitrogen abundances, as observed inthe donor spectrum ( ). The transfer of enriched material from the black hole progenitor couldalso explain the high helium abundance in the spectrum of the donor star ( ). In the absenceof convection, which is expected to be limited to a very thin surface layer in the envelope ofa M (cid:12) star of solar (or possibly super-solar; ( )) metallicity, some fraction of this materialwould be retained on the surface for up to ∼ yr (see Supplementary Text). As a surfacecontaminant, this would not be reflective of the overall composition of the donor. Our revisedvalues for the mass and luminosity of the donor star are consistent with theoretical expectations(Fig. 3, ( )).The increase in the inferred black hole mass makes Cygnus X-1 more massive than previ-ously observed black holes in X-ray binaries ( ), surpassing M33 X-7. M33 X-7 has a substan-tially sub-solar metallicity ( ∼ . times the solar metallicity, Z (cid:12) ; ( )). However, the metallicityof the mass donor in Cygnus X-1 is much higher. It has been estimated to be approximatelytwice solar ( ), although the complexity of the system makes precise measurements challeng-ing, and the true value may be closer to solar ( ). The existence of a M (cid:12) black hole atsolar (or super-solar) metallicity implies that mass loss rate prescriptions ( ) over-estimate the5ass loss during the luminous blue variable or Wolf-Rayet stages of stellar evolution (
25, 26 ).Assuming solar metallicity for the system, we find that either the mass loss rates in Wolf-Rayetwinds from naked helium stars are reduced by a factor of three compared to current models ( ),or those in luminous blue variable winds are reduced by at least a third, or both (see Supple-mentary Text).Reduced mass loss can lead to higher progenitor masses at the time of the supernova. It mayalso allow massive stars at moderate metallicities to retain hydrogen as they evolve (thoughprobably not in the evolutionary history of Cygnus X-1, where the black hole’s progenitorlikely had most of its hydrogen stripped off by the companion while the former was still on themain sequence; see Supplementary Text). This would change the observational signatures ofsupernovae, for example producing hydrogen-rich (pulsational) pair-instability supernovae ( ).Enrichment of the interstellar medium by stellar winds could be reduced, and, depending onwhich phases of the stellar evolution are most impacted by reduced winds, the contribution ofmassive stars to the re-ionization of the Universe may be affected (see Supplementary Text).The black hole mass distribution inferred from gravitational wave events favours largermasses than predicted by stellar and binary evolution models. Variations in metallicity-specificstar formation history that favour greater star formation at lower metallicity have been proposedas an explanation (
28, 29 ). Reduced stellar winds would increase the mass of black holes thatcould be produced at all metallicities, such that massive gravitational wave sources could format intermediate (not just very low) metallicities. This would imply that the progenitors of somegravitational wave events could have formed at correspondingly lower redshift, with a shorterdelay time between binary formation and merger.The high spin of Cygnus X-1 (in common with most high-mass black hole X-ray binaries,which appear to be predominantly rapidly spinning; ( )) implies that it followed a differentevolutionary pathway to the majority of black holes detected in gravitational wave events, whichhave spins that are either low or misaligned ( ). Given the current orbital separation, we donot expect Cygnus X-1 to undergo a binary black hole merger in a timescale comparable to theage of the Universe. 6 eferences
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Acknowledgments
Acknowledgments:
We acknowledge Guy Pooley’s contribution to the radio observing cam-paign. The Very Long Baseline Array is a facility of the National Science Foundation operatedunder cooperative agreement by Associated Universities, Inc. This work made use of the Swin-burne University of Technology software correlator, developed as part of the Australian MajorNational Research Facilities Programme and operated under licence. This work has made useof data from the European Space Agency (ESA) mission
Gaia ( ), processed by the Gaia
Data Processing and Analysis Consortium (DPAC, ). Funding for theDPAC has been provided by national institutions, in particular the institutions participating inthe
Gaia
Multilateral Agreement.
Funding:
JCAM-J and IM are recipients of Australian Re-search Council Future Fellowships (FT140101082 and FT190100574, respectively) funded bythe Australian government. LJG acknowledges the support from the National Program on KeyResearch and Development Project through grant No. 2016YFA0400804, and from the NationalNSFC with grant No. U1838114, and by the Strategic Priority Research Program of the ChineseAcademy of Sciences through grant No. XDB23040100. VG is supported through the Mar-garete von Wrangell fellowship by the ESF and the Ministry of Science, Research and the ArtsBaden-W¨urttemberg. BM acknowledges support from the Spanish Ministerio de Econom´ıa yCompetitividad (MINECO) under grant AYA2016-76012-C3-1-P and from the Spanish Minis-terio de Ciencia e Innovaci´on under grants PID2019-105510GB-C31 and CEX2019-000918-Mof ICCUB (Unidad de Excelencia “Mar´ıa de Maeztu” 2020–2023). SM was supported by theNetherlands Organization for Scientific Research (NWO) VICI grant (no. 639.043.513). GRSacknowledges support from an NSERC Discovery Grant (RGPIN-2016-06569). VT is sup-ported by programme Laplas VI of the Romanian National Authority for Scientific Research.JW acknowledges funding from the Bundesministerium f¨ur Wirtschaft und Technologie underDeutsches Zentrum f¨ur Luft- und Raumfahrt grant 50 OR 1606. JZ acknowledges the supportfrom the Polish National Science Centre grant 2015/18/A/ST9/00746.
Author contributions:
JCAM-J analyzed the VLBA data, and led the manuscript preparation. AB conducted the as-trometric fitting. JAO conducted the optical light curve and radial velocity curve fitting. IMand CJN performed the stellar wind modelling and led the discussion of the system’s evolu-tion, with input from TJM. LG, XZha and XZhe performed the analysis of the black hole spin.JZ calculated the mass-luminosity tracks. MJRe provided the archival VLBA data and guid-ance on precision astrometry. PU, VG, JCAM-J and JW co-ordinated the VLBA observing12ampaign, with theoretical input from SM. JCAM-J, PU, TJM, VT, APR, JW and DMR wrotethe observing proposal, and D-YB, RD, TJ, J-SK, BM, MJRi, GRS, and AJT contributed tothe design and setup of the observations. All authors provided input and comments on themanuscript.
Competing interests:
The authors declare no conflicts of interest.
Data andmaterials availability:
The raw VLBA data are available from the NRAO archive ( https://archive.nrao.edu/archive/advquery.jsp/ ), under project codes BR141 andBM429. Our measured positions are listed in Table S1. The COMPAS population synthesiscode we used is available at http://github.com/TeamCOMPAS/COMPAS . The softwarefor performing the astrometric and optical model fitting, the spin fitting, and for calculating themass-luminosity relationships, together with our COMPAS input and output files, is available inour code repository at https://github.com/bersavosh/CygX-1_JMJ2020 , whichis archived at https://zenodo.org/record/3961240 ( ).13 .20.10.00.10.2 RA offset (mas) D e c o ff s e t ( m a s ) A B C DE F GHIJ K J e t a x i s A BH line-of-sight distance behind centre of mass (au)
Brightness (mJy beam )
15 10 5 0 -5 -10151050-5-10
RA Offset (mas) D e c O ff s e t ( m a s ) B Figure 1 : Cygnus X-1 and its best fitting model orbit on the plane of the sky. (A) astrometricmeasurements from the VLBA (red points) and the archival data from ( ) (blue points). Errorbars show the 68% confidence level. The letter labels reflect the chronological ordering of theobservations, as detailed in Table S1. Dashed lines link the measured positions to the locationon the fitted orbit, shown as the colored ellipse. Color bar indicates the location of the blackhole along the line of sight, relative to the centre-of-mass of the system (shown as the blackstar), with positive values being behind the centre of mass. Arrows indicate the direction oforbital motion and the jet axis. (B) stacked radio image of the jet in color, with white contoursevery ± ( √ times the rms noise level of 23 microJanskys ( µ Jy) per beam. Red ellipse showsthe synthesised beam. Although the measured positions scatter along the jet axis, the motionperpendicular to the jet axis is reproduced by the astrometric model (see Fig. 2). Coordinatesare given in right ascension (RA) and declination (Dec), J2000 equinox.14 igure 2 : Orbital displacements relative to the best-fitting one-dimensional astrometricmodel.
Parallax and proper motion signatures have been subtracted. (A) The measured dis-placements perpendicular to the jet axis. Red points are our VLBA data, and blue points arethe archival observations ( ), with error bars showing the 68% confidence level. Labels reflectthe chronological ordering of the observations, as listed in Table S1. Black and magenta linesshow 500 random draws from the posterior probability distribution of the orbital parameters, forradio and optical models, respectively (with the posterior medians indicated by thicker lines),which are consistent within the uncertainties. The data were only fitted perpendicular to thejet axis. (B) The measured displacements parallel to the jet axis show that the measured corepositions are primarily downstream of the model predictions when the black hole is close tosuperior conjunction (behind the donor star; phases close to 0.0), and upstream when the blackhole is close to inferior conjunction (phases close to 0.5), as expected for wind absorption.15 Stellar mass ( M fl ) S t e ll a r l u m i no s i t y [ l og ( L / L fl ) ] X=0.70, Z=0.02X=0.52, Z=0.02X=0.47, Z=0.03X=0.52, Z=0.03X=0.57, Z=0.03
Figure 3 : Predicted mass-luminosity relations for high-mass main sequence stars.
Massesare given in solar masses, and luminosities relative to the solar luminosity L (cid:12) . The black solidline shows the predicted relation for a standard composition (hydrogen mass fraction X = 0 . ;mass fraction of heavy elements Z = 0 . ( )). The grey dot-dashed line is for an enhancedhelium abundance, X = 0 . , Z = 0 . (as inferred for the surface abundance of the donorstar; ( )). The red solid line shows the effect of an increased metallicity, with Z = 0 . , X = 0 . . The magenta dotted and dashed lines show the effect of the uncertainty on thehelium abundance ( ). The mass and luminosity determined from previous observations ( )are shown as the cyan triangle. The values from this work are shown as the blue circle, whichlies closer to the theoretical relations, irrespective of composition or metallicity. Error barsshow 68% confidence levels. 16arameter Median Mode Lower bound Upper bound i (deg) 27.51 27.33 26.94 28.28 e ω (deg) 306.6 306.3 300.3 313.1 M ( M (cid:12) ) 40.6 39.8 33.5 48.3 f T eff (K) 31,138 31,158 30,398 31,840 K (km s − ) 75.21 75.18 74.80 75.63 φ Ω rot M BH ( M (cid:12) ) 21.2 21.4 18.9 23.4 R ( R (cid:12) ) 22.3 22.2 20.6 24.1 log L/L (cid:12) log( g / cm s − ) a (au) 0.244 0.243 0.231 0.256 a (au) 0.0838 0.0840 0.0816 0.0856 a BH (au) 0.160 0.159 0.147 0.173 Table 1 : Fitted and derived physical parameters for Cygnus X-1.
The inclination i , eccen-tricity e , and argument of periastron ω of the orbit; the mass M , Roche-lobe filling factor f ,effective temperature T eff , and semi-amplitude of the radial velocity curve K for the O-star;a phase shift parameter φ to account for ephemeris errors; and the ratio Ω rot of the rotationalfrequency of the O-star to the orbital frequency (the nine parameters above the horizontal line)were directly fitted in the model. The black hole mass M BH ; the radius R in solar radii R (cid:12) ,luminosity L in solar luminosities L (cid:12) , and surface gravity g of the O-star; and the semi-majoraxes of the full orbit a , stellar orbit a , and black hole orbit a BH (the seven parameters below thehorizontal line) were derived from the fitted parameters, using the known orbital period ( ).Lower and upper bounds encompass the 68% confidence interval.17 upplementary Materials This PDF file includes
Materials and MethodsSupplementary TextFigures S1 to S11Tables S1 to S3References (33-118) Materials and Methods
Our radio observations were made using the VLBA, as part of the Cygnus X-1 Hard-state Ob-servations of a Complete Binary Orbit in X-rays (CHOCBOX) project; a multi-wavelengthobserving campaign to monitor Cygnus X-1 around an entire binary orbit.
The closest two known compact extragalactic radio calibrator sources ( ) to Cygnus X-1are NVSS J195330+353759 (hereafter J1953+3537) and NVSS J195740+333827 (hereafterJ1957+3338), which lie 1.08 ◦ and 1.57 ◦ away, respectively (see Fig. S1). We chose J1957+3338as our primary phase reference calibrator. Despite being further from Cygnus X-1, it is less re-solved at our observing frequency of 8.4 GHz, allowing better calibration of the longer baselinesto stations at Mauna Kea and St Croix, and hence better astrometric precision.Because systematic errors in astrometry scale with the distance between the target and thephase reference source ( ), we conducted a calibrator search to find a closer phase referencesource and thereby reduce the systematic uncertainties. We observed with the VLBA on 2016April 9, targeting 44 sources from the National Radio Astronomy Observatory (NRAO) VeryLarge Array (VLA) Sky Survey (NVSS) catalog ( ) within (cid:48) of Cygnus X-1. Observing at1.64 GHz, the primary beam was large enough to cover the 44 targets in 4 pointings. We usedthe multi-phase centre capability of the DiFX software correlator ( ) to provide individualdata sets for each target. Our observing bandwidth was 64 MHz, and we achieved roughly8 min of observations on each target source. Due to the scattering along the line of sight to theCygnus region we did not use the antennas at Mauna Kea or St Croix. The brightest sourcedetected (NVSS J195754+353513, hereafter J1957+3535), at . σ , was located at co-ordinates(J2000) 19 h m s + d (cid:48) (cid:48)(cid:48) . ◦ from Cygnus X-1. It appeared unresolved to ourobservations, with a peak flux density of 2.8 mJy beam − . We therefore selected this source asa secondary phase reference source for our observing campaign. We observed Cygnus X-1 daily from 2016 May 29 through June 3, using the VLBA underproject code BM429. Our observations covered an entire 5.6-day binary orbit, allowing usto track the orbital motion on the plane of the sky. Each observation was taken at a centralfrequency of 8.416 GHz, with a bandwidth of 256 MHz. We cycled between Cygnus X-1,our primary phase reference calibrator J1957+3338 and our secondary phase reference sourceJ1957+3535, spending 100, 60 and 60 s per cycle, respectively, on each source.Our observing runs were 12 hr in duration, with half-hour geodetic blocks at the beginning,middle and end of each observation. In each of these blocks, we observed a range of brightcalibrator sources spread across the sky to solve for unmodeled tropospheric delay and clockerrors, to improve our astrometric precision. 19ur data were correlated using the DiFX software correlator ( ), and reduced accordingto standard procedures within the 31DEC17 version of the Astronomical Image ProcessingSystem ( AIPS ; ( )). Following initial fringe fitting on J1957+3338, we stacked the data fromall six epochs to provide a global model that was used for the final fringe fitting. The finalphase, delay and rate solutions were interpolated to both the target source and the secondaryphase reference calibrator, J1957+3535. To improve our amplitude calibration, we also self-calibrated the data on J1957+3338 from each epoch in amplitude and phase, on a 10-minutetimescale, using our global model. We interpolated the resulting amplitude self-calibrationsolutions only (having zeroed the phases to prevent them from overriding the shorter-timescalephase solutions derived from the fringe fitting) to both the target source and J1957+3535. Wethen imaged both Cygnus X-1 and J1957+3535 at each epoch, and measured their positions byfitting the core regions with elliptical Gaussians in the image plane. Stacked images of the twosources made with all six epochs of data are shown in Figs. 1 and S2, respectively. Because our own data were taken over a period of just six days, they could not constrain theparallax and proper motion of Cygnus X-1. However, they fully sample the 5.6-day orbit,so when combined with archival data from 2009–2010 ( ) (VLBA program code BR141), canmeasure the parallax and refine the proper motion via the extended time baseline. These archivaldata also provide an additional five samples of the binary orbit. With a signal-to-noise > , our astrometric measurements were limited by systematic ratherthan statistical uncertainties. These include contributions from the ionosphere ( ) and the tro-posphere ( ), both of which scale with angular separation between target and phase calibratorsource.Ionospheric wedges can produce parallax gradients across the sky, leading to systematicshifts of order 50 µ as ( ν/ . GHz) − per degree of offset between target and phase referencesource, where ν is the observing frequency ( ). While this effect can in principle be minimizedby fitting an ionospheric wedge using multiple phase reference sources surrounding the target,none of our Cygnus X-1 data sets contained sufficient well-situated calibrators to enable suchan approach, so we adopted the above estimate of the likely systematic uncertainty.For our VLBA data, we used the measured offsets of the nearby secondary phase referencecalibrator, J1957+3535, from its mean position to correct the astrometry, combining the statisti-cal uncertainties from each source in quadrature. This effectively phase referenced Cygnus X-1to J1957+3535, with the 0.4 ◦ calibrator offset implying expected systematic uncertainties fromthe troposphere of 23 and 29 µ as in RA and Dec., respectively ( ). We flagged local sunriseat each station to prevent ionospheric wedges from affecting the astrometric measurements,which, together with the long (12-hour) tracks, reduces the effects of any ionospheric wedge.20ith a calibrator offset of 0.4 ◦ , the estimated ionospheric systematic uncertainty was 13 µ as,negligible compared to the tropospheric and combined statistical uncertainties.The archival observations ( ) used two different phase calibrator sources, J1953+3537 andJ1957+3338, located . ◦ and . ◦ from Cygnus X-1, respectively (see Fig. S1). We usedtheir position measurements ( ), taking a mean of the two measurements (i.e. the positionsmeasured relative to the two different phase reference sources) at each epoch to minimise theeffect of any ionospheric wedge, and adding the statistical uncertainties from the two differentmeasurements in quadrature. To this we added in quadrature both the estimated troposphericsystematics (38 and 47 µ as in RA and Dec., respectively; ( )), and the estimated ionosphericsystematic uncertainties of 34 µ as in both dimensions. While the dependence of ionosphericsystematic uncertainties on observing duration and time of year is not yet understood, we regardthese estimates as conservative. The final positions of Cygnus X-1 and their uncertainties aregiven in Table S1.The above discussion outlines the systematic uncertainties affecting the relative astrometrymeasurements on which our results are based. To enable comparisons with other work, we alsoestimate the uncertainties affecting our absolute astrometric positions. J1957+3338 is one of thesources making up the third realization of the International Celestial Reference Frame (ICRF3).However, all our positional measurements were derived assuming the position of J1957+3338to be 19 h m s ◦ (cid:48) (cid:48)(cid:48) µ as in RA and 323 µ as in Decfrom the ICRF3 positions ( ). The uncertainties on the ICRF3 positions are 151 µ as in RAand 181 µ as in Dec. The uncertainties on the absolute positions of Cygnus X-1 (Tables S1 andS3) and J1957+3535 ( ) will be further augmented by the systematic uncertainties due to theoffset between phase reference calibrator and target positions, as discussed above. Adding thesecontributions in quadrature, our absolute astrometric positions are uncertain by 181 µ as in RAand 375 µ as in Dec. The orbital motion of the black hole can be measured in astrometric VLBI observations ( ).Those observations showed that the sense of the black hole orbit was clockwise on the sky, butonly measured the size of the orbit to σ , as . ± . au. Our VLBA data, corrected for thenominal parallax and proper motion over the six days of the campaign, showed a clockwise orbitthat was resolved on the plane of the sky (Fig. 1). Our measured source positions are those ofthe radio core of the jet, which is presumably launched from close to the black hole. However,the radio photosphere is located ≈
40, 41 ), sowe see the projection of the true black hole orbit on the radio photosphere, potentially modifiedby orbitally- or wind-induced bending of the jet axis (
40, 42 ).To sample both the orbit and proper motion of the system, we combined our VLBA data withthe archival observations ( ), and fitted the combined data set with a model incorporating propermotion, parallax and orbital motion. A full astrometric solution includes the reference positionin both co-ordinates ( α and δ ), the proper motion in both co-ordinates ( µ α cos δ and µ δ ), the21arallax ( π ), plus the seven orbital parameters (orbital period P , epoch of periastron passage T , eccentricity e , inclination i , semimajor axis a BH , argument of periastron ω , and longitude ofthe ascending node, Ω ). We adopted the orbital period P from the photometric ephemeris ( ).Constraints on the inclination of the binary orbit, the eccentricity, the argument of periastron,and the masses of the two components were taken from previous work ( ), which, with the or-bital period, can be converted to a constraint on the semimajor axis via Kepler’s Third Law. Forthe known inclination angle and argument of periastron ( ), periastron passage (which definesorbital phase zero in their formalism) occurs at phase 0.109 in an alternative ephemeris ( ),which assumed a circular orbit and defined orbital phase zero to occur at superior conjunctionof the black hole ( ). We therefore defined our reference epoch T to be the reference date forsuperior conjunction determined by ( ), augmented by this phase shift.To fit the motion of the system on the sky, we used a Markov chain Monte Carlo (MCMC)approach, using the PYMC ). We adopted the Hamiltonian Monte Carlo formalism(HMC; (
45, 46 )) with a No-U-Turn Sampler (NUTS; ( )). We used Gaussian priors on i and ω based on the optical modelling work of ( ). Because optical modelling cannot determine thesense of the orbit as seen on the plane of the sky, optically-determined inclinations of i and (180 ◦ − i ) are degenerate, with values of ◦ < i < ◦ corresponding to the clockwise orbitseen in our VLBA data. Furthermore, the two components of the binary each have their ownargument of periastron, ω ∗ and ω BH , which are separated by ω ∗ = ω BH + 180 ◦ . Using theconstraints from ( ), we therefore adopted a prior of . ± . ◦ for the orbital inclination, . ± . ◦ for the argument of periastron of the black hole ω BH , and uniform priors on sixof the other seven parameters ( α , δ , µ α cos δ , µ δ , π , and a BH ). For the final parameter, Ω , weassumed that the jet was aligned with the orbital angular momentum vector. The jet positionangle is known on the plane of the sky, so with this constraint and the inclination and argumentof periastron from the optical modelling work, we infer the longitude of the ascending node tobe ◦ . We therefore used a Gaussian prior on Ω with this mean, and a standard deviation of ◦ .A summary of our adopted priors is given in Table S2.This astrometric model fitting gave a parallax of π = 535 ± µ as and an orbital semi-majoraxis for the black hole of a BH = 89 ± µ as (see Table S3 for full results). The fit residualsare shown in Fig. S3, and are substantially larger than we expected from our (conservative)estimates of the uncertainties on the data. To achieve a reduced χ value of 1 for their astrometric model fitting, ( ) found that they had toadopt error floors of 80 and 160 microarcseconds ( µ as) in RA and Dec., respectively, which isagain substantially larger than expected from the calibrator throw. This suggests an extra sourceof systematic uncertainty in addition to those modelled by simulations ( ).Cygnus X-1 has a strong stellar wind, which attenuates the radio emission at frequenciesabove 2 GHz ( ). The strong free-free absorption by the wind is enhanced at superior conjunc-tion of the black hole, which is then seen through the maximum path length through the wind.22t inferior conjunction, the path length is minimized and the radio emission is least absorbed.Because we are observing at a frequency of 8.4 GHz, where the orbital modulation of the ra-dio emission is . ± . % ( ), our data are affected by wind absorption. Close to superiorconjunction of the black hole, this would have the effect of pushing the optical depth τ = 1 surface out to larger distances downstream. Changes in the location of the τ = 1 surface dueto intrinsic variations in the electron number density or magnetic field strength in the jet wouldlead to additional, non-orbital shifts in the measured position of the radio core.The jet axis in Cygnus X-1 is well-known, oriented at − ◦ east of north (Fig. 1), and stableover time (
5, 48 ). We can therefore determine the positions measured parallel and perpendic-ular to the jet axis, for each epoch. Because the wind absorption and intrinsic jet variabilityshould only affect the parallel co-ordinate, we can conduct astrometric fits only on the per-pendicular component to provide an estimate of the true astrometric and orbital parameters ofthe system ( ). However, when fitting in only one dimension, the reference position ( α , δ )and proper motion ( µ α cos δ , µ δ ) become degenerate in the two sky co-ordinates. To overcomethis problem, we used Gaussian priors for both α and µ α cos δ , with the means and standarddeviations taken from the posterior probability distributions of the original model fitting.Using this model gives a substantially smaller parallax of π = 458 ± µ as (Fig. S4), and asemi-major axis for the black hole orbit of a BH = 58 ± µ as. Our final best-fitting parametersare given in Table S3. Fig. S5 shows the one-dimensional histograms of the posterior probabilitydistribution of each of the nine fitted parameters, together with two-dimensional scatter plotsshowing the covariances of the parameters. The correlations between parameters arise due tothe use of the one-dimensional fit. There is a degeneracy in position along the jet axis, asreflected in the mean value of the residuals parallel to the jet axis, which we found to be µ asupstream (i.e. towards the south south-east on the sky). This is well within the peak of thetwo-dimensional posterior probability distributions of ( α , δ ) shown in Fig. S5. This shift hasbeen corrected in Fig. 1, and also in Fig. 2, which shows the best-fitting displacements bothperpendicular and parallel to the jet axis.The reduction in the amount of information available (fitting in one co-ordinate rather thantwo) increases the uncertainties in the fitted parameters. However, this model is based on well-characterised properties of the system, and reduces the systematic uncertainty caused by theradio photosphere moving up and down the jet axis as a function of orbital phase. Figs. 1 and2 show that, as we expected, the measured position is scattered further downstream along thejet axis at superior conjunction of the black hole, and back towards the black hole at inferiorconjunction.Our measured parallax of ± µ as is consistent with the zero-point corrected opticalvalue of ∼ ± µ as from the second data release (DR2) from the Gaia space telescope(
8, 9 ), which was recently refined to ± µ as in the early version of the third data release(eDR3; (
49, 50 )). Unless the uncertainty is very small, simple inversion of a measured parallaxto determine a source distance can introduce a non-negligible bias in the distance estimation.With a 7.7% parallax uncertainty, we therefore adopted the standard Bayesian formalism ( )to convert our measured parallax to a probability density function for the source distance, using23n exponentially-decreasing space density prior ( ). We find a median distance of 2.22 kpc,with a σ range of 2.05–2.40 kpc, and a 90% Bayesian credible interval of 1.96–2.54 kpc. The proper motion of Cygnus X-1 relative to its likely parent stellar association Cygnus OB3 is . ± . km s − ( ), indicating that it formed with little to no natal kick, and possibly withouta supernova explosion ( ). Our revised distance does not change this conclusion, because itstill falls within the distance range of . ± . kpc determined for Cyg OB3 ( ).In the absence of an asymmetric supernova kick, we would expect the angular momentumvector of a rapidly-spinning black hole like Cygnus X-1 (
16, 53 ), and hence the jet axis, to bealigned with the orbital angular momentum vector, which motivates our choice of a highly-constrained Gaussian prior for the longitude of the ascending node, Ω . From the observed driftvelocity of Cygnus X-1 relative to Cygnus OB3, we estimate the maximum asymmetric natalkick that the system could have received as 10–20 km s − . The maximum likely misalignmentangle should scale as the ratio of kick velocity to the pre-supernova orbital velocity. For theobserved mass ratio between black hole and donor star, and assuming a pre-supernova orbitalvelocity of ∼ km s − , then unless the asymmetric kick was directly opposed to the Blaauwkick from mass ejection ( ), we estimate a maximum misalignment angle of ∼ ◦ . Whileinner disk misalignments of this size have been inferred from some X-ray spectral fitting results( ), other work suggests that this conclusion is sensitive to the assumed electron density inthe accretion disk, with no misalignment being required in the case of the higher densities thatwould be appropriate for X-ray binary disks ( ).To quantify the effect of any misalignment, we explored the effect of increasing the standarddeviation on the Gaussian prior on Ω to ◦ . While the the posterior distribution for Ω increasedto ± ◦ , all other changes were well within the uncertainties. The parallax increased byless than one percent, to ± µ as. The semi-major axis of the black hole orbit increased to ± µ as, the argument of periastron increased by a degree to ± ◦ , and the inclinationdid not change. Even for the most extreme case of a uniform prior on Ω (0–360 ◦ ), when theposterior distribution on Ω increased to ± ◦ , there was minimal further change in theparallax, with a median and 68% confidence interval of ± µ as. Even then, the case ofalignment still falls within the 90% Bayesian confidence interval. We conclude that our distancedetermination is therefore insensitive to the prior on Ω , and that the data are consistent with analigned jet axis. The increase in the distance of Cygnus X-1 compared to the previous results ( ) implies that theabsolute magnitude of the donor star is larger, in turn increasing the inferred stellar radius. Theradius is an input to the dynamical model ( ), which used optical radial velocity and light curves,a measurement of the projected rotational velocity of the O-star and a measurement of the radius24f the O-star, to determine the physical parameters of the system. Given the revised stellar radiusarising from our distance determination, we therefore reanalysed the optical photometric andvelocity curves of ( ) and ( ) to update the estimated system parameters.We retained the previous stellar rotational velocity estimate (taken from ( )). The radiusand luminosity of the O-star can be computed as a function of temperature ( ), as shown inFig. S6, where for simplicity we have adopted the revised distance with a conservative andsymmetric uncertainty, . ± . kpc. The larger distance implies larger radii and higher lu-minosities as compared to the previous results ( ). We also used optical spectra ( ) to improveprevious estimates of the effective temperature, surface gravity and helium abundance of theO-star, which further constrain the stellar radius and luminosity.To model the light curve and radial velocity curve we used the ELC code ( ), choosing theoptimizer code based on the differential evolution (DE-MCMC) algorithm ( ). We adoptedthe “Model D” framework from ( ), which incorporates nonsynchronous rotation of the O-star and an eccentric orbit. We modified Model D ( ) to fit for the Roche lobe filling factorrather than the radius of the O-star. We define this filling factor as the ratio of the distancebetween the centre of the star and the point on its surface closest to the black hole, to thedistance between the centre of the star and the L1 Lagrange point. Our model therefore has 9free parameters ( i, K , M , f , φ, e, ω, Ω rot , T eff ) . We used the same table of model atmospherespecific intensities as ( ), and hence no parameterized limb darkening law is needed.Because the donor star is close to Roche lobe filling ( ), the Roche geometry places strongconstraints on the surface gravity, which is close to log g = 3 . ( ). Stellar surface gravity andtemperature are partly degenerate in optical spectral fitting ( ), so this places constraints on thelikely effective temperature. While there are several estimates of the effective temperature ofthe donor star in the literature (
15, 60, 63, 64 ), we discarded ( ), because it did not include theeffects of line blanketing, which has a substantial impact on the effective temperature. Of theremainder (i.e., (
15, 60, 64 )), all suggested the effective temperature to be in the range 30,000–31,000 K (where we used the green spectrum of ( ), as their blue spectrum gave an inconsistentestimate of the stellar surface gravity). We consider a broad weighted mean of these threeeffective temperature determinations, giving T eff = 30 , ± K.We used uniform priors for the 9 fitting parameters, including a broad prior on T eff of27,500–36,000 K. Our likelihood function is based on the χ statistic and has two parts χ = χ + χ . (S1) χ applies to the U , B , and V light curves and the radial velocity curve, [ ( ), their equation(2)]. Five additional measured properties of Cygnus X-1 are independent of orbital phase,namely the radius of the O-star, R ( T eff ) , which is computed from the parallax for a giventemperature; the rotational velocity of V rot sin i = 96 ± km s − ( ); our weighted mean forthe O-star temperature of , ± K; the O-star surface gravity of log g = 3 . ± . ( );and the lack of an X-ray eclipse. A given vector of the 9 model parameters will produce valuesfor all of these observed parameters, and the second part of the likelihood function is then found25y χ = (cid:18) R − R ( T eff ) σ R ( T eff ) (cid:19) + (cid:18) V rot sin i − (cid:19) + (cid:18) T eff − , (cid:19) + (cid:18) log g − . . (cid:19) + Θ χ , (S2)where Θ χ = 10 if there is an X-ray eclipse, and zero otherwise. The uncertainty of R ( T eff ) depends on the effective temperature, hence we used σ R ( T eff ) in the expression above. Theinclusion of T eff in the expression for χ effectively imposes a Gaussian prior for thatparameter. ELC
Model Fitting
To find the uncertainties on the fitted and derived parameters, we implemented the DE-MCMCalgorithm ( ). The code was run 8 times using 40 chains each. The chains were initializedusing slightly tweaked copies of the nearly optimal model (based on the results of ( )), and eachrun had a different initial seed in the random number generator. The individual runs were doneon computers with different CPU speeds, with between 2,500 and 26,000 generations for eachrun.To determine the burn-in time (how long the chains need to disperse from the initial state),we visually inspected plots of the chains over the first few hundred generations, finding that ittook between about 20 and 30 generations for the chains to initially disperse. Thereafter, thechains reached a steady-state by generation 150 or so. To be conservative, we therefore set theburn-in period to be 200 generations.The posterior samples for the fitted and derived parameters were made by sampling thechains starting at generation 201 in each run, skipping enough generations to allow the chainsto cross most of the parameter space. The eight individual posterior samples for each parameterfor each run were combined.The distributions for the fitted parameters are shown in Fig. S7, and summarized with thederived parameters in Table 1, which gives the posterior sample median, the mode (determinedby using 75 bins), and the σ confidence ranges. For our adopted parameters we quote thesample medians.With the exception of the inclination i and the Roche lobe filling factor f , the posteriorprobability distributions are close to symmetric (Fig. S7). In the case of the filling factor f , thedistribution is peaked towards the maximum value of 1.0. The median value of the distributionis about 0.963 and the value for the best-fitting model is 0.993. The 95% confidence lower limitis 0.917.We find masses of . +7 . − . M (cid:12) and . ± . M (cid:12) (Fig. S8) for the O-star and black hole,respectively, as compared to the values of . ± . M (cid:12) and . ± . M (cid:12) , respectively,from ( ). The best-fitting radius of the O-star is . ± . R (cid:12) , which when combined with its26emperature of T eff = 31 . ± . kK, gives a luminosity of log( L/L (cid:12) ) = 5 . +0 . − . . Figs. S9 andS10 show the light curve and radial velocity fits using these median values. We find rotation ofthe donor star consistent with being synchronous with the orbit, unlike the previous solution ( )which had a donor rotating at 1.4 times the orbital frequency.The strong dependence of the component masses on the distance results from the nearlyRoche-lobe filling donor, which is at a known orbital period with well constrained flux andtemperature. The angular size of the donor is set from the combination of temperature andluminosity. The donor radius is thus proportional to the distance to the system. Then, sincethe donor is constrained to be nearly filling its Roche lobe, and the Roche lobe size at a givenperiod is set by the star’s mean density, the donor mass scales as its radius cubed, and hence thedistance cubed. This leads to the nearly doubling of the donor mass based on the 20% increasein distance.The size of the derived black hole orbit, a BH = 0 . ± . au, corresponds to ± µ as atour best-fitting distance of . +0 . − . kpc. This is consistent with the size of the black hole orbitderived independently from our astrometric fit ( ± µ as), providing confidence in the result.This indicates that the jet cannot undergo a high degree of bending as it propagates outwardfrom the black hole to the radio photosphere, or we would have measured a substantially largerprojected orbit when fitting our VLBA astrometric data. The continuum-fitting method ( ) is one of the two common techniques used to estimate thedimensionless spin parameter of a black hole, a ∗ = cJ/GM , where J is the black holeangular momentum, c is the speed of light, and G is the gravitational constant. This methodrelies on determining the inner radius R in of the thin accretion disk, by fitting the continuumX-ray spectrum to the Novikov-Thorne thin disk model ( ). R in is assumed to be locatedat the innermost stable circular orbit (ISCO) around the black hole, R ISCO . R ISCO is directlyrelated to the dimensionless spin parameter a ∗ , which for a prograde equatorial orbit decreasesfrom 6 GM/c to GM/c as the spin goes from a ∗ = 0 to 1. Fitting the inner radius of theaccretion disk then in principle constrains the black hole spin ( ). However, the spins derivedby the continuum-fitting method are sensitive to the black hole mass, the disk inclination, andthe source distance, and the uncertainties on these parameters tend to dominate the uncertaintieson the derived spins ( ).The continuum fitting method was originally applied only to spectra from the high/soft X-ray spectral state (
68, 69 ), to ensure that the inner radius of the disk was located at the ISCO,and to avoid confusion introduced by the strongly Comptonized component that is present inthe hard and intermediate X-ray spectral states. While Cygnus X-1 does not reach the canonicalhigh/soft state, ( ) have shown that the inner disk radius remains within a few percent of theISCO for X-ray spectra in which < % of the thermal seed photons are Compton upscattered.We applied this criterion to select six archival spectra (
16, 53 ), and reanalysed them to constrainthe spin parameter. We used our estimates for the distance, black hole mass and orbital inclina-27ion of Cygnus X-1. We used Monte Carlo simulations to quantify the uncertainty on the spinintroduced by the combined observational uncertainties on D , M , and i (
32, 53 ).Assuming that the binary orbital angular momentum vector is aligned with the spin axis ofthe black hole (as discussed above, and as expected under the evolutionary scenario discussed inthe Supplementary text), we find that, consistent with the previous spin results ( ), a ∗ > . at the 3 σ confidence level. While the value of a ∗ monotonically decreases with increasinginclination, we find that even given a misalignment angle of ◦ (equivalent to i = . ◦ ), thefitted spin value would still be as high as a ∗ = 0 . .While the best-fitting spin is above the maximum value expected from classical modelling( ), this could be attributed to systematic effects in the model arising from factors such as afinite disk thickness, or the inner radius not being exactly at the ISCO. Regardless, the true spinis likely high, as also found by studies fitting the Fe K α line profile (
17, 55–58 ). A detailedanalysis of our spin-fitting results and a discussion of the inherent systematics is presented in acompanion paper ( ). We computed a mass-luminosity relation using the Warsaw evolutionary code (
32, 64 ). Wemade three changes relative to the model described in ( ). First, we adopted a default effectivetemperature of 30,500 K ( ), which did not lead to any appreciable modification of the mass-luminosity relation.Next, we used a different method for choosing the parameter f sw (defined in ( ) as a simplemultiplicative factor applied to the stellar wind mass loss rate prescriptions of ( ), to accountfor uncertainties in our knowledge of the winds in massive stars; see ( ) for an alternativeapproach). Two example relations are available ( ) for fixed values of f sw equal to 2 and 5(fixed for each mass along the relation). We calibrate the mass-luminosity relation specificallyfor the mass donor in Cygnus X-1. We use the fact that the stellar wind mass outflow rate fromthe star is known: ˙ M w = − (2 . ± . × − M (cid:12) yr − ( ). While this estimate does notinclude systematic uncertainties, considerations of wind clumping for comparable late O-typesupergiants ( ) suggest that these should be below a factor of two. For each stellar mass alongthe mass-luminosity sequence, we adjusted the value of f sw to ensure that when the star evolvesto reach the effective temperature T eff = 30 , K, the calculated mass outflow rate wouldmatch the measured value ( ). The relation was calculated for masses in the range – M (cid:12) .This procedure resulted in values of f sw varying from 4.15 to 0.415 as log( M/M (cid:12) ) varied from1.345 to 1.668.Finally, we modified the assumed chemical composition. For the hydrogen mass fractionand metallicity, we adopted a set of default values ( ) of X = 0 . ± . (corresponding to [He / H] = 0 . ± . ) and Z = 0 . (a super-solar value set by the level of precision available( )). Given the uncertainty on the metallicity, we also computed a relation for Z = 0 . .The assumed donor composition was found to have the largest effect on the computed mass-luminosity relation, as shown in Fig. 3. However, these models do not include the effects of28tellar rotation or tidal distortion on temperature and luminosity.Adopting instead the σ upper bound on the effective temperature found above of T eff =31 , K, we found that the predicted luminosity change for a given mass was < . Toaccount for systematic uncertainties in the assumed wind mass loss rate, we also tested theeffect of changing the wind mass loss rates by 50%, finding the predicted luminosity change fora given mass to be < %. Our calculations are therefore insensitive to uncertainties in the windmass loss rate and effective temperature.As shown in Fig. 3, our revised parameters of the donor star are consistent with all threeof the assumed compositions, within the uncertainties. Should the “classical” composition of X = 0 . , Z = 0 . be correct, then the true hydrogen content would be larger than themeasured surface value of X = 0 . , possibly suggesting enrichment of the donor star duringmass transfer (see Supplementary text). Wind-driven mass loss rates of massive stars are difficult to accurately measure empiricallyand to model theoretically (see (
75, 76 ) for reviews). We use simplified phenomenologicalprescriptions with scaling parameters to encode existing uncertainty, following the formalismof ( ).We focus on two stellar evolution phases that determine the maximum mass of black holes:the luminous blue variable and Wolf-Rayet phases, whose wind mass loss rates are uncertain.Luminous blue variable winds are conjectured to prevent the star from crossing the Humphreys-Davidson limit ( ). In our parametrised model, these are independent of metallicity (exceptindirectly, through conditions determining their onset) and eject mass at a rate ˙ M LBV = f LBV × − M (cid:12) yr − . (S3)Meanwhile all stars that strip themselves of their envelope through winds beyond the mainsequence – or, in our preferred evolutionary scenario, are stripped by the companion while stillon the main sequence – are subject to strong Wolf-Rayet winds with a metallicity-dependentrate of ˙ M WR = f WR × − (cid:18) LL (cid:12) (cid:19) . (cid:18) ZZ (cid:12) (cid:19) . M (cid:12) yr − , (S4)where L is the star’s luminosity ( ).We use single stellar evolution and wind models as implemented in the binary populationsynthesis code COMPAS ( ) to explore the impact of Wolf-Rayet and luminous blue variablewind prescriptions on the final black hole mass ( ). We require that black holes with a massequal to that inferred for Cygnus X-1 can be produced at a metallicity of Z = 0 .
02 = 1 . Z (cid:12) (po-tentially somewhat lower than the metallicity of Cygnus X-1, so our wind reduction is conser-vative ( )). In Fig. S11 we show the possible solutions for single stars in the two-dimensional29arameter space of f LBV and f WR . We find that the mass loss rates of either luminous bluevariable winds, or Wolf-Rayet winds, or both, must be reduced relative to the ( ) defaults (cali-brated to the previous, lower estimates of the Cygnus X-1 black hole mass under the assumptionthat it formed at solar metallicity), which are also used in COMPAS ( ). To form black holeswith a mass consistent with that of the black hole in Cygnus X-1 from single stars at this metal-licity, either the mass loss rates in luminous blue variable winds have to be reduced by a third,or those in Wolf-Rayet winds must be reduced by two thirds. We also investigated the effectof the standard main sequence winds from O-stars, finding that the effect on the final remnantmass of switching them off entirely was marginal (when compared to the effects of varyingeither luminous blue variable or Wolf-Rayet winds), and therefore do not consider O-star windsfurther.Cygnus X-1 is unlikely to contain the highest-mass black hole that can be reached at itsmetallicity, so this provides only a conservative constraint on the maximum wind mass loss rate.This simplified analysis assumed single stars; if the progenitor of the Cygnus X-1 black holewas stripped of its hydrogen envelope during the main sequence as conjectured below, Wolf-Rayet winds will kick in earlier, again making these constraints conservative. Regardless, eventhe moderate wind strength reduction assumed here has consequences for the mass distributionof compact objects (
81, 82 ) and supernova models. For example, ( ) find that varying wind as-sumptions, in addition to affecting the core mass, substantially impact the compactness param-eter of stellar cores, which is connected with the ‘explodability’ in a supernova. Reduced windswould also impact the enrichment of the interstellar medium by massive stars ( ), and theirrole in the re-ionization of the universe; while ionising radiation from massive stars made theuniverse transparent ( ), it strongly depends on stripping by winds or binary interactions ( ). Dynamical black hole mass measurements of the high-mass X-ray binaries IC10 X-1 ( ) and NGC 300 X-1 ( ) favour higher masses than our revised measurement of theCygnus X-1 black hole. However, the source of the emission lines used for those measurementsis unknown. For example (
92, 93 ) point out that in the case of Wolf-Rayet companions, theemission lines likely originate in the wind and thus indicate the wind velocity rather than thebinary orbital velocity, which would invalidate the black hole mass measurements. IC10 X-1and NGC 300 X-1 reside in galaxies with sub-solar metallicities, so the evolution of their pro-genitors is likely to differ from that of the black hole progenitor in Cygnus X-1.LB-1 (LS V +22 25) in the Milky Way has been suggested to contain a M (cid:12) blackhole ( ). However, other work has challenged the interpretation of the broad H α line usedto determine the black hole mass, and suggested that the mass of the unobserved companion(which may not even be a black hole; ( )) is closer to M (cid:12) ( ).30 .3 Formation history We investigate a possible formation channel for Cygnus X-1. The close separation indicatesthat the binary must have interacted, with mass transfer from the progenitor of the black hole,the primary, onto its companion, the secondary. Such mass transfer would typically removethe envelope of the progenitor, which would dominate its moment of inertia and, assumingreasonably efficient angular momentum transport within the star, would contain the bulk of theangular momentum. Therefore, removing this envelope would leave behind a slowly spinningblack hole ( ).However, observations point to a rapidly rotating black hole, aligned with the orbital angularmomentum ( ). The black hole must therefore retain its birth spin, because the mass of a blackhole would have to be doubled in order to appreciably change spin after birth ( ), whichis inconsistent with the short lifetime of the system ( ); even Eddington-limited accretionover the entire 4 Myr lifetime of the donor (both assumptions that are likely to be substantiallyoverestimated) would only spin up the black hole to a maximum of a ∗ = 0 . . The observedseparation is too large for tides to spin up the stellar core of the primary after its envelope wasremoved. Chemically homogeneous evolution ( ) could be responsible for the rapid spinof the black hole; however, given the observed mass ratio, it appears unlikely that the primaryevolved chemically homogeneously but the secondary did not, as evidenced by its factor of ∼ expansion from its zero-age main sequence radius. Spin-up is possible during the supernovaitself through tidal torquing by the companion of ejecta which eventually fall back ( ), butthis generally yields lower spins than complete fallback.A channel for forming a rapidly rotating black hole in a close binary was proposed by ( ):gradual case A mass transfer from the late main-sequence primary allows it to be spun upthrough tides while removing enough hydrogen-rich material to prevent post-main sequenceexpansion and loss of angular momentum through envelope stripping. While the details areuncertain, this channel appears to explain the similar properties of the high-mass X-ray binariesCygnus X-1, LMC X-1, and M33 X-7, including orbital separations of ∼ R (cid:12) , and highspins ( ) (see ( ) for previous evolutionary studies of these systems).Assuming evolution through this channel, the primary would have lost some mass throughwinds during the Wolf-Rayet phase, widening the binary. However, this mass loss cannot havebeen too great, due to the mass of the black-hole remnant, and to avoid excessive spin-down.Eventually, the primary collapsed into an aligned, rapidly spinning black hole, likely throughnearly complete fallback with little to no natal kick ( ), consistent with the few km s − driftvelocity of the binary relative to its Cygnus OB3 birth association ( ). The lack of substantialmass loss and natal kick is supported by the very low binary eccentricity ( ), although the factthat the eccentricity is inconsistent with zero may argue for a minimal mass loss of ∼ M (cid:12) (perhaps through neutrino emission) and imperfect tidal circularisation through tides operatingon the expanding secondary.The combination of case A mass transfer and Wolf-Rayet winds is presumably responsi-ble for the unusual surface abundances of the secondary, including surface helium abundance31nhanced by more than a factor of two ( ). This is different from the model of ( ), whoassumed that the surface abundances are due to partial stripping of the secondary during a pre-vious Roche lobe overflow episode onto the black hole; however, in their model the secondaryis now less massive than the black hole, which is inconsistent with the present measurements.Enrichment during the supernova itself may have occurred in the black hole X-ray binary GROJ1655-40, rather than during previous mass transfer ( ). But as discussed above wefavour nearly complete fallback in Cygnus X-1. Moreover, if the current secondary were, infact, a partially stripped evolved star, we would expect it to have a higher core mass and there-fore a higher luminosity than a normal main sequence star of its current mass, which is notsupported by observations (Fig. 3).Our scenario relies on a small quantity of enriched material being responsible for the ob-served secondary surface abundance. This demands that the material accreted from the primaryis retained in a thin surface layer on the secondary, rather than being mixed throughout the star.Late main-sequence stars with the mass and metallicity of the secondary should only have avery thin convective surface layer, preventing efficient mixing (although the impact of tides androtation could enhance convection). While temperature inversion in the accreted material wouldsuppress the Rayleigh-Taylor instability, the more massive helium-rich material would be sus-ceptible to thermohaline mixing ( ), which would eventually distribute the accreted ma-terial through the envelope, reducing the observed surface abundance. Therefore, the observedenhancement points to fairly recent enrichment and collapse of the black hole, likely (cid:46) years ago. This requirement is also necessary to avoid the enriched material being blown off bystellar winds from the secondary. This timescale is consistent with the few × yr estimate forthe age of the jet ( ), and hence the age of the black hole if the jet switched on promptly afterthe black hole was formed, as expected for wind mass transfer given the proximity of the twocomponents.The formation scenario and its inherent uncertainties are discussed in detail elsewhere ( ).32 h m m h m m ◦ ◦ ◦ ◦ Right Ascension (J2000) D ec li n a t i o n ( J ) Cygnus X-1NVSS J195740+333827NVSS J195330+353759NVSS J195754+353513
Figure S1 : Locations of our observed sources on the plane of the sky.
We show the posi-tions of our calibrator sources, NVSS J195330+353759 (J1953+3537), NVSS J195754+353513(J1957+3535), and NVSS J195740+333827 (J1957+3338), as well as that of Cygnus X-1. Hor-izontal and vertical axes are Right Ascension and Declination, respectively. North is upwards,and east is to the left. The calibrator throw from J1957+3338 is similar to both Cygnus X-1and the check source, J1957+3535, constraining the systematic uncertainties affecting our mea-surements of Cygnus X-1. All coordinates are in the International Celestial Reference System(ICRS). 33 .0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012
Brightness (Jy beam )19 h m s s s s D e c li n a t i o n ( J ) Figure S2 : The image was made from a stack ofall six epochs of data from 2016 May and June. The source is comprised of three compo-nents, all slightly extended relative to the VLBA beam. Contours are at levels of ± n times90 µ Jy beam − , where n = 0 , , , ... The rms noise level in the image is 26 µ Jy beam − . Thefitted locations of the central component have a standard deviation of 14 and 30 microarcseconds( µ as) in R.A. and Dec., respectively, over our six epochs of observation. The red ellipse in thebottom left corner represents the synthesised beam.34 . . . ∆ R A ( m illi a r c s ec ) A - -
01 2010 - -
01 2011 - -
01 2012 - -
01 2013 - -
01 2014 - -
01 2015 - -
01 2016 - - Date54700 55200 55700 56200 56700 57200 57700Modified Julian Date (days) − . . . ∆ D ec ( m illi a r c s ec ) B Figure S3 : Residuals from the two-dimensional astrometric fit.
Our model incorporatedproper motion, parallax, and orbital motion, fitted in both Right Ascension (A) and Declination(B). Residuals exceed 0.1 mas, substantially larger than we expected from our estimated sys-tematic and statistical uncertainties, which are reflected in the sizes of our error bars, shown atthe 68% confidence level. 35 .40.20.00.20.4 P a r a ll a x pe r pend i c u l a r t o j e t a x i s ( m a s ) A - - - - - - - - - - - - - - - - Date
Modified Julian Date (days) R e s i dua l ( m a s ) B Figure S4 : The best-fitting parallax signal from our one-dimensional model fitting.
Wefitted the data perpendicular to the jet axis on the plane of the sky. (A) The parallax signal afterremoving the reference position, proper motion and orbital signatures. (B) The residuals aftersubtracting the best-fitting parallax of π = 458 ± µ as. Error bars are shown at the 68%confidence level. 36 e l a t i v e p r o b a b ili t y . +21 . − . − δ ( µ a s ) . +59 . − . π ( µ a s ) . +34 . − . − − − − µ α c o s δ ( µ a s / y r ) − . +4 . − . − − − − µ δ ( µ a s / y r ) − . +17 . − . ω o r b ( ◦ ) . +5 . − . . . . . . i o r b ( ◦ ) . +0 . − . Ω o r b ( ◦ ) . +0 . − . − −
40 0 40 80 α ( µas ) a o r b ( µ a s ) −
150 0 150 δ ( µas )
320 400 480 560 π ( µas ) − − − − µ α cos δ ( µas/yr ) − − − − µ δ ( µas/yr )
110 120 130 140 150 ω orb ( ◦ ) . . . . . i orb ( ◦ )
60 62 64 66 68 Ω orb ( ◦ )
30 60 90 120 a orb ( µas ) . +19 . − . Figure S5 : Results of our PyMC astrometric model fitting for Cygnus X-1.
We fitted theposition ( α , δ ; relative to the best-fitting position, in µ as), proper motions ( µ α cos δ , µ δ ; in µ as yr − ), parallax ( π , in µ as), inclination angle of the orbit ( i , in degrees), argument of perias-tron ( ω , in degrees), longitude of the ascending node ( Ω , in degrees), and semi-major axis of theblack hole orbit ( a BH , in µ as). Histograms show the one-dimensional posterior probability dis-tributions, and contour plots show the two-dimensional posterior probability distributions fromour nine-dimensional parameter space. This shows both the spread of results and their covari-ances. The dashed vertical lines in the histograms represent the 1 σ credible intervals, and thecontour lines in the contour plots show the 1, 2 and 3 σ regions (from innermost to outermost,respectively). The red dots in the contour plots represent posterior realizations outside the 3 σ contour. 37 B Figure S6 : Stellar parameters of Cygnus X-1 as a function of its effective temperature.
We show both the radius (A) and luminosity (B) of the O-star. The lines joining the points arenot fits, and are shown only to highlight the broad trends in the data. The dependence of thebolometric corrections (see ( )) on temperature leads to the luminosity variation.38 i (deg) N A .
010 0 .
015 0 .
020 0 .
025 0 . e B
290 300 310 320 330 ω (deg) C
20 40 60 M ( M (cid:12) ) N D .
85 0 .
90 0 . f E T eff (K) F
74 75 76 K (km s − ) N G .
000 0 .
002 0 .
004 0 . φ H .
75 1 .
00 1 .
25 1 .
50 1 . Ωrot I Figure S7 : Posterior probability distributions for all nine parameters from our opticalmodel fitting. N is the number of posterior samples in a given bin, out of a total of 24,360. Theonly two non-symmetric distributions are for the orbital inclination angle i (A), and the RocheLobe filling factor for the secondary, f (E). The inclination angle is well-constrained, and thesecondary appears very close to Roche lobe filling. We are only able to place a 95% confidencelower limit on the filling factor, of > . . Blue solid line shows the sample medians, andpink dashed lines show the lower and upper 15.87% points, which between them encompassthe σ parameter ranges. These values are listed in Table 1. Orange dotted lines show the 90%Bayesian credible intervals. 39 M BH (M (cid:12) ) N Figure S8 : Posterior probability distribution of the black hole mass determined from ouroptical light curve and radial velocity modelling. N is the number of posterior samples in agiven bin, out of a total of 24,360. Blue solid line shows the sample median, and pink dashedlines show the lower and upper 15.87% points, which between them encompass the σ rangefor the black hole mass, of 18.9–23.4 M (cid:12) . Orange dotted lines show the 90% Bayesian credibleinterval of 17.4–24.8 M (cid:12) . 40 igure S9 : Light curve results from our
ELC model fitting . (A) The U -band light curveand the best-fitting model. (B) The U -band residuals. (C) The B -band light curve and thebest-fitting model. (D) The B -band residuals. (E) The V -band light curve and the best-fittingmodel. (F) The V -band residuals. Error bars are shown at the 68% confidence level. The dataare duplicated over two orbital cycles for clarity.41 igure S10 : Radial velocity results from our
ELC model fitting. (A) The phase-folded radialvelocity measurements (data points) and the best-fitting model (red line). (B) The radial velocityresiduals (dots). The filled circles are the averaged residuals in 30 bins, with uncertaintiesrepresenting the standard deviation within each bin. For context, the fitted semi-amplitude ofthe radial velocity curve is . ± . km s − . Data are duplicated over two orbital cycles forclarity. 42 igure S11 : The inferred wind strength in Cygnus X-1, relative to standard prescriptions.
We show the maximum remnant mass from the evolution of single stars at Z = 0 . as calcu-lated using COMPAS , as a function of two parameters describing the strength of luminous bluevariable winds, f lbv , and Wolf-Rayet winds, f wr . The solid and dashed white lines indicate themedian values and 68% confidence uncertainty on our Cygnus X-1 mass measurements. Thestar denotes default choices calibrated to the previous Cygnus X-1 mass measurement ( ). Pa-rameter choices all fall below and to the left of it. Triangles show the isolated effects of reducingthe mass loss rates in Wolf-Rayet winds by a factor of three, and reducing those in luminousblue variable winds by a third. 43 able S1 : Measured positions of the radio core of Cygnus X-1.
Our images were fitted withan elliptical Gaussian function in the image plane (appropriately corrected for the measuredoffsets of the secondary phase reference calibrator for epoochs F–K). Tabulated uncertaintiesinclude both statistical and systematic uncertainties, as described in ( ). Epochs A–E weretaken in 2009–2010 under proposal code BR141 ( ), and epochs F–K are our VLBA measure-ments from 2016, taken under proposal code BM429.Epoch Modified Right Ascension Uncertainty Declination UncertaintyJulian Date(days) ( µ s) ( µ as)A 54854.793 19 h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) h m s ◦ (cid:48) (cid:48)(cid:48) Table S2 : MCMC priors adopted for the two-dimensional astrometric model fitting. ( α , δ ) are the reference positions in RA and Dec. on the reference date, MJD 56198.0. U denotes a uniform distribution with the given range, and N denotes a normal (Gaussian) distri-bution with mean µ and standard deviation σ .Parameter Description Prior distribution Units α R.A. reference position U ( − . , . ) ∗ arcseconds δ Dec. reference position U ( − . , . ) ∗ arcseconds µ α cos δ R.A. proper motion U ( − , ) mas yr − µ δ Dec. proper motion U ( − , ) mas yr − π Parallax U ( . , . ) mas i Orbital inclination N ( µ = 152 . , σ = 0 . ) † degrees ω Argument of periastron N ( µ = 127 . , σ = 5 . ) † degrees Ω Longitude of ascending node N ( µ = 64 . , σ = 1 . ) ‡ degrees a BH BH orbital semimajor axis U ( . , . ) mas ∗ Relative to mean of measured positions in R.A. and Dec.. † Taken from optical fitting results of ( ). ‡ Assuming jet axis and orbital angular momentum are aligned.44 able S3 : Astrometric and orbital parameters of Cygnus X-1 from our one-dimensionalMCMC model fitting.
We used the priors described in Table S2. ( α , δ ) are the referencepositions in RA and Dec. on the reference date, MJD 56198.0. The stated value is the median ofthe posterior distribution, with the σ uncertainty (in brackets, applied to the final digits) definedas the spread between the median and percentiles 15.9 and 84.1 of the posterior probabilitydistribution. The final two columns show the 90% Bayesian credible interval (the 5th and 95thpercentiles of the posterior distribution) for the 1-D fit perpendicular to the jet axis.Parameter 2-D fit ∗ † † † α (19 h m ) ‡ s s s s δ (35 ◦ (cid:48) ) ‡ (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) µ α cos δ (mas yr − ) − . − . − . − . µ δ (mas yr − ) − . − . − . − . π (mas) . . .
399 0 . i ( ◦ ) § . . . . ω ( ◦ ) ( ◦ ) . .
0) 64 . .
0) 62 . . a BH ( µ as) ∗ Two-dimensional fit to the measured Right Ascension and Declination values, as de-scribed in ( ). † One-dimensional fit perpendicular to the jet axis, adopting priors for α and µ α cos δ takenfrom the posterior distribution of the standard fit, as described in ( ). ‡ Positions given for the reference date, MJD 56198.0. § Inclinations of 90–180 ◦◦