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Understanding Black Hole Formation in Cygnus X-1 Tsing-Wai 1 Center 1,2 Wong , Francesca 1 Valsecchi , Tassos 2 Fragos , and Vassiliki 1 Kalogera for Interdisciplinary Exploration and Research in Astrophysics (CIERA) & Department of Physics and Astronomy, Northwestern University 2 Harvard-Smithsonian Center for Astrophysics Background and Purpose Cygnus X-1 is a luminous X-ray binary, hosting a 14.81 ± 0.98 M⊙ black hole (BH) and a 19.16 ± 1.90 M⊙ O supergiant star in a Keplerian orbit of 5.6 days (Orosz 2011). This system is a persistent X-ray source that emits in two states: low-hard or high-soft state. Although the supergiant is close to filling its Roche-lobe, the observed X-ray emission comes from the partial accretion of the supergiant’s stellar wind onto the BH. The purpose of this study is to estimate the mass of the BH immediate progenitor and the magnitude of a possible natal kick imparted to the BH. Formation scenario of Cygnus X-1 We adopt the simplest formation scenario that can explain all the observations. • assume both stars, the BH progenitor and the O supergiant were born at the same time • assume there was no mass transfer at all in the history of Cygnus X-1. (This means we can model the O supergiant as a single isolated star.) • the BH immediate progenitor is a He star. • during the core-collapse event, the binary orbit was altered by mass loss from the system and the possible natal kick imparted to the BH. • the BH has accreted negligible amount of mass since its birth. > 120 M⊙ MS 14.81 ± 0.98 M⊙ He Black Hole Core Collapse Cygnus X-1 MBH = 14.81 ± 0.98 M⊙ M2 = 19.16 ± 1.90 M⊙ L2 = (2.33 ± 0.42) ! 105 L⊙ natal kick? MS MS Teff = 31000 ± 1000 K Lx = (1.3 " 2.1) ! 1037 erg/s MS Porb = 5.599829(16) days eorb = 0.018(3) Fig 1. The adopted formation scenario of Cygnus X-1. The notation MS and He stands for main sequence star and helium star, respectively. Methodology Step 1: create stellar models that matched all the observed BH companion’s properties > 120 M⊙ 15.0 ! 20.0 M⊙ • used a modified stellar evolution code, which was originally developed by Paxton (2004) • computed the X-ray luminosity by Bondi-Hoyle accretion model the companion was747:111 a 20 to(12pp), 23 M2012 at 10 zero-age main sequence • found MS (ZAMS) ⊙ star He The Astrophysical Journal, March • estimated the BH was born 4.8 to 7.6 million years ago Step 2: find the systemic peculiar velocity (Vpec_postSN) right after the BH’s formation 14.81 ± 0.98 M⊙ • traced the motion of Cygnus X-1 in a galactic potential backwards in time Cygnus X-1 • derived Vpec_postSN = 22 to 32 km/s Black Wong et al. Hole Core 4.8 ! 7.6 Myr ± 0.98 M⊙ Step 3: constrain the binary properties right before the core-collapseMevent via Monte BH = 14.81 Collapse The Astrophysical Journal, 747:111 (12pp), 2012 March 10 Wong et al. Carlo simulations The Astrophysical Journal, 747:111 (12pp), 2012 March 10 Wong et al. Companion Age vs. ZAMS mass Kinematics of Cygnus X-1 in the past Mc = 19.16 ± 1.90 Vk " 77 km/s The Astrophysical Journal, 747:111 (12pp), 2012 March 10 Wong et al. • important free parameters: BH immediate progenitor mass (MHe), orbital semi-major the Galactic plane and the end points of all trajectories fall within esti + current location mat axis (ApreSN) & eccentricity (epreSN), and natal kick magnitude (Vkick) ×from Cyg OB3the Galactic plane, we consider each end point as ed c 110 pc omp anio MS MSof the BH. The post-SN MS a possible birth site peculiar velocity • properties of the companion star was given by stellar models created in Step 1. n ag e Vpec,postSN of the binary is obtained by subtracting the local Galactic rotational velocity of 20 ! 23 M⊙ from the center-of-mass 19.8 ! 22.6 velocity M⊙ the binary at the birth sites. We find Vpec,postSN ranges from + current location 22 to 32 km s−1 and is time independent. The distribution of potential BH birth sites × Cyg OB3 Vpec,postSN against the time expired since the formation of the BH are displayed in Figure 5. 6. ORBITAL DYNAMICS AT CORE COLLAPSE • conservation of orbital energy and angular momentum gave the binary parameters 19.8 ! 22.6 M⊙ right after the core-collapse event • applied the Vpec_postSN = 22 to 32 km/s constraint derived in Step 2 • evolved the binary The Astrophysical Journal, 747:111 (12pp), 2012 March 10 Binary Orbital Evolution orbit forward in time to the current epoch • if the orbital period and eccentricity matched the measured values, we used that data to construct probability distribution functions (PDFs) M⊙ For each successful sequence, we perform a Monte Carlo simulation which consists of 20 million pre-SN binaries. estimated life time of BH progenitor The properties of the BH progenitor’s companion are taken from the stellar model of that sequence, at the time when the age of the star is equal to tBH . During a supernova (SN) explosion, the mass loss from the system and possibly the kick imparted to the BH change the binary’s orbital parameters. The preand post-SN component masses, orbital semimajor axis, Fig 4. One possible past evolution of Cygnus X-1’s orbit. Right after Upper panels: the gray dots3. illustrate the possible locations of Cygnusbirth X-1 at the birth of timethe of theBH. BH, obtained Figure of tof against companion M2,zams . TheFigure gray-shaded region Fig4.2.Variations Variations the estimate age5.against 2 (circles) Fig Upper panels: potential sites Lowerfrom 3000 integrations of its trajectory in time. The initial of the integrations are generated randomly using the methodology described in Section 5.laws The plus and orbital eccentricity are related by the conservation ofsigns indicate the current the core-collapse event at t = 3.8 Myr, this binary system consists of a sequences, which is conditions indicates themass. range of tBH for the corresponding successfulbackward ZAMS panel: the distribution of systemic peculiar velocities against location of Cygnus X-1, derived from the mean distance of 1.86 kpc. The crosses represent the current location of Cyg OB3 center, with an adopted distance of 2 kpc. the orbital energy and angular the following, we calculated by Equation (12). The difference between t2 andLower tBH panel: givesthetsys . 14.8 M⊙ BH and a 21.7 M⊙ main-sequence companion. distribution ofthe post-SN peculiar velocities Vpec,postSN against time momentum. expired since the BHIn formation. time expired since the BHtheformation. add the subscripts “preSN” and “postSN” to the notation of the orbital elements to the distinguish between their etvalues just prior VHe,preSN is the relative pre-SN orbital velocity of BH formulae of Sepinsky al. (2007). When calculating the pre-SN 5. KINEMATIC HISTORY IN THE GALAXY progenitor, Roche lobe we assume that the pre-SN orbit is pseudoto, and right after, the SN explosion thatradii, formed the BH. Figure 6. OrbitalFigure evolution of a selected winning binary. Right after the formation of the BH (t = 3.8 Myr), this binary consists a a14.8 MM BH and astellar 21.7MM! synchronized. Again, dueconsists to theofdifference in!!calculated 6. Orbital evolution of a selected winning binary. Right after the formation of the BH (t = 3.8 Myr), this binary of 14.8 BH and a 21.7 ! $ ! We start with seven free parameters: the BH immediate 1/2 Here, we assume that Cygnus X-1 formed in the Galactic disk. # " main-sequence star. The top panels show timeshow evolution orbital period eccentricity. The bottom panels show the evolution rates of change ofof the axisAAand and Results and Discussion radii among stellar codes, we consider an uncertainty main-sequence star. The topthe panels the timeofevolution of orbital period The bottom panels show the rates of change thesemimajor semimajor axis 1 and eccentricity. 2 and − accretion = G(M +M . the (17) VHe,preSN (He-rich) progenitor mass (M ), Rpre-SN orbital semimajor He 2 ) wind eccentricityof e due to OB3 tidal effects (solid line), mass and wind accretion onto theonto BH (dashed line), and gravitational radiation (dotted ofHe ±2.5 the companion radius (Valsecchi et al. 2010). The The consideration Cyg being the association of The Astrophysical Journal, 747:111 (12pp), 2012 March 10 eccentricity e due to tidalparent effectswind (solid line),loss wind mass loss and BH (dashed line), and radiation (dottedline). line). " ongravitational r ApreSN of the BH immediate progenitor can be approximated by PDFs of MHe and 2D joint M 2D joint epreSN−ApreSN PDF kick He−Vkick PDF axis (ApreSN ) and eccentricity radius (epreSN ), the mean anomaly (m), Cygnus X-1 is discussed in Section1D9.2. Given the Vobserved Equations (3) in Fryer & Kalogera (1997), since we assume throu The angle ψ is the polar angle of the position vector of the BH the magnitude (V ), and direction (θ , φ) of the kick velocity position and measured proper motion of Cygnus X-1, we derive k that it is a Helium star. Second, the pre-SN spin of the BH with respect to its pre-SN orbital velocity in the companion’s the b 3σ to be less than the imparted to the BH. θ is the polar angle of the kick with respect the post-SN peculiar velocity of the binary’s center of mass by immediate progenitor and its companion need frame. It is related to the pre-SN parameters as 3 1/2 find breakup angular velocity Ωc just ≈ (GM/R )to . As the calculated to the relative orbital velocity of the BH progenitor prior tracing its orbit in the Galaxy back to the time of BH formation. Figure 6. Orbital evolution of a selected winning binary. Right after the formation of the BH (t = 3.8 Myr), this binary consists of a 14.8 M! BH % & 2 2 2 2 stellar radius R associates with an uncertainty ∆R = 2.5 R" a ran 2 ψ = G(MHethe + MSN 1 − epreSNand . (18) r VHe,preSN explosion, φ is the corresponding azimuthal angle 2 )ApreSN main-sequence star. The top panels show the time evolution of orbital period and eccentricity. The bottom panels show the rates of change of the semim We describe the motion of the binary with respect to a sin right(Valsecchi et al. 2010), max eccentricity e due to tidal effects (solid line), wind mass loss and wind accretion onto the BH (dashed line), and gravitational radiation (dotted line). (see Figure 1 in Kalogera 2000 for a graphical representation). hand Cartesian reference frame, whose originSince coincides with the core collapse is instantaneous, r remains unchanged. ' 2σ ( limit ) The first five parameters are drawn from uniform distributions, 3 ∆R GM2 the Galactic center. The Z-axis points to the northern This gives Galactic a constraint recei 1 + (20) Ωc = 3 while the last two are drawn from isotropic distributions. It is 2 R2 pole, while the X-axis points in the direction from the projected R2 fiden r = ApreSN (1 − eobvious preSN cos Ethat preSN )the progenitor must of course1σ be more massive than position of the Sun onto the Galactic plane to the Galactic center. a no = ApostSN (1 − epostSN cos EpostSN ), (19) for the BH companion. the BH, but there is no absolute upper limit for the progenitor In this reference frame, the Sun is located at (X! , Y! , Z! ) = rentl mass. We adopt provide a discussion on this to be satisfied with | cos E | ! 1. MHe ! 20 M! ,7.and (−8.5, 0, 0.03) kpc (Joshi 2007; Ghez et al. which 2008;needs Gillessen postSN only ORBITAL EVOLUTION AFTER THE SN EXPLOSION mass loss the system a natal kick imparted upperand limit in Section 9.1. et al. 2009; Reid et al. 2009) and has a peculiarThemotion (Ufrom !, amo The orbital evolution of the simulated binaries, which are to the BH can induce a post-SN peculiar velocity (V ) pec,postSN 3σ The relations between pre- and post-SN parameters have been V! , W! ) = (11.1, 12.24, 7.25) km s−1 (Schönrich al. 2010). ∼2 × generated from the Monte Carlo simulations described in at the et binary’s center of mass. Its magnitude is determined by by Hills Cygnus X-1 is currently located at a distance Equations of 1.86+0.12 kpcin Paper Iderived Section 6, is calculated up to the current epoch. After the (28)–(32) and is required to fall(1983): within the BH −0.11 −1 formation of the BH, the orbital parameters of the binary are derived in Section 5, which is 22–32 km s . from the Sun, with a Galactic longitude l = 71.◦range 3, and a Galactic by m subject to secular changes due to the tidal torque exerted by In addition, there are two more restrictions on the properties of 2σ 2 2 plica latitude b = 3.◦ 1 (Lestrade et al. 1999; Reid etpreal.and 2011). This Vk +First, VHe,preSN + 2V VHe,preSNthecos M2due ) to the loss of orbital angular BH + and BHθon= its G(M companion, post-SN binary components. we require thatkboth " and stellar wind. Since ! colla means Cygnus X-1 is currently ∼130 kpc above the Galactic momentum via gravitational radiation components have to fit within their pre- and post-SN Roche lobe the tidal interactions2depend on1 both the orbital and rotational at periapsis. We impose this condition to avoid complications a ma 1σ plane. , − × properties of the MSr companion, arising from MT-induced changes in the stellar structure of the ApostSNthe star’s rotational angular and To model the Galaxy, we adopt the Galactic potential of velocity (Ω) right after SN explosion that formed the BH MS companion that later becomes the BH companion of the ecce Carlberg & Innanen (1987) with updated model parameters (13) enters the problem as an additional unknown quantity. Here, XRB. To calculate the Roche lobe radius of each component Figure 7. Probability distribution functions of the BH immediate (He-rich) ecce we assume the rotational angular velocity of the BH companion in kick eccentric pre-the and we adopt the fitting of Kuijken & Gilmoreprogenitor (1989).mass The(Mequations magnitude (Vk ) post-SN imparted orbits, to the BH. He ) and natal governing at th system’s motion in the Galaxy are integrated backward in time, 7 # $ joint Vjoint Figure Two-dimensional joint ApreSNjoint –epreSN e confidence levels: 68.3% (red), Th Fig 7. 9.The two dimension probability Fig 5. The probability distribution functions of M and V . Figure 7. Probability distribution functions of the BH immediate (He-rich) preSN−ApreSN Figure 8. Two-dimensional confidence 68.3% (red), 95.4% He kick k –MHe Fig 6. The two M probability distribution 2 dimension up to the time corresponding to the current system’s age tsys He−Vkick levels: G(M 1 and − e99.7% 95.4% (yellow), and 99.7% (blue). BH + M2 )ApostSN postSN(blue). (yellow), studi progenitor mass (M ) and natal kick magnitude (V ) imparted to the BH. In Figure 7, we present the probability distribution functions distribution function. He k function. given by the successful sequences. We follow the methodology Figure 7. Probability distribution functions of the BH immediate (He-rich) # color version of this figure is available in the online journal.) 2He ) 2 2(A color2 version of this figure is available in the online journal.) progenitor mass (M ) and natal kick magnitude(A (200 (PDFs) of the BH immediate (He-rich) progenitor mass (M (V ) imparted to the BH. He k V sin θ cos φ + [sin ψ(V + V cos θ ) = r of Gualandris et al. (2005) to initialize the parameters for the He,preSN k k the c Figure 8. Two-dimensional joint Vnatal confidence levels: 68.3% (red), 95.4% and natal kick magnitude (V ). We find M to be in a $ k –MHekick k He We found that the BH immediate progenitor mass (M ) is between 15 to 20 M , and the magnitude (V ) imparted to the BH is ≤ 77 km/s, both at 95% confidence. As shown in the two He ⊙ kick 2 integration, which accounts for the uncertainties in the estimated −1 Figure 8. Two-dimensional joint Vk –MHe confidence levels: 68. − V . (14) cos ψ sin θ sin φ] (yellow), and 99.7% (blue). dyna k In Figure 7, we present the probability distribution functions range of 15.0–20.0 M , and V to be !77 km s , both at ! k a∗event, > 0.95if at 3σ .isTo< and determine whether theMBH dimensional joint velocity Vkick−MHe probability natal coreparameter collapse MHe 1799.7% M⊙(blue). . For small distance and measured components. Wedistribution generate thefunction (see Fig. 6), the BH might probably have received a non Inzero (yellow), He, a Figure 7, we kick presentatthethe probability distribution functions earli 95% confidence. Figure 8 illustrates the two-dimensional joint(A color20version M! . We impose limit based the journal.) physics involved in of this figurethis is available in theon online (PDFs) of the BH immediate (He-rich) progenitor mass (M ) He was born with an extreme spin, we first estimate how much (A color version of this figure is available in the online journal (PDFs) of the BH immediate (He-rich) progenitor mass (MHe ) initial system’sVposition by the Galactic coordinates (l, b) andall the observed properties of Cygnus X-1. For a more detail presentation minimum km/s is necessary to explain of this work, see Wong et al. (2012). kick of ~55 durin Vk –M evolution of massive stars. A discussion limit cankick magnitude (V ). Wemass levels, which shows that ifbe M thanorbital the Here, is the separation between the BH progenitor and on thisand He confidence Heinis rless and natal kick magnitude (V ). We find M to a the BH could have accreted from its companion’s stellar natal find M to be in a k He k He a random distance drawing from a Gaussian distribution. We since −1 −1 found in Section 9.1. Given our understanding ofrange mass of loss ∼17 M!M , the BH might have received asnon-zero natal kickat the be its companion time of SN explosion, 15.0–20.0 M , and V to be !77 km s , both at wind since the time of BH formation. The winning binaries range of 15.0–20.0 , and V to be !77 km , both at ! k ! by drawing k generate initial system’s velocity earli from Helium stars, it seems that the BH has potentially received at the core-collapse event.randomly For small the MHe , a minimum Vk of 95% confidence. Figure 8 illustrates the two-dimensional joint 20 M . We impose this limit based on the physic ! of all successful sequences show that at maximum the BH has 95% confidence. Figure 8 illustrates radial the two-dimensional joint 20 M . We impose this limit based Reference on the −1 physics involved in proper motions (µR.A. , ∼55 µdecl. ) and velocity the current cons −3 natal kick velocity of !77 km s km s−1 heliocentric is necessary for explaining observedr = A!a small (95% confidence) Vk –MHe confidence levels, which shows the evolution of massive stars.the A discussion on t that if M∼2 less than He is accreted × 10 M . Since it is impossible to spin BH (1 − e cos E ), (15) % preSN preSN preSN V the evolution of massive stars. A discussion on this limit can –M confidence levels, which shows that if M is less than (V0 ) from kGaussian distributions. current age isHe He prog Belczynski, Bulik,and T., Fryer, C.,during et al. 2010, 714, 1217 event. Lestrade, J.-F., Preston, Jones, et al. 1999,received A&A, 344,a 1014 be found in Section amount 9.1. Given our understanding ∼17 MR.A., BHD.L., might have non-zero natal by kick properties ofThe Cygnus X-1.system’s Furthermore, both the MHeK.,PDF the ApJ, core-collapse ! , the up to a > 0.95 accreting that negligible of mass, ∗ Belczynski, K., Kalogera, V., Rasio, F. A., et al. 2008, ApJS, 174, 223 McConnell, M. L., Zdziarski, A. A., Bennett, K., et al. 2002, ApJ, 572, 984 uniformly∼17 distributed between 4.8 and 7.6 Myr (see Figure 4). be found in Section 9.1. Given our understanding of mass loss M! , the BH might have received a non-zero natal kick histo from Helium stars, it seems that thespin BH has potenti at the core-collapse event. For small M , a minimum Based on the dynamical model of Orosz et al. (2011), Gou the two-dimensional joint Vk –MHe confidence levels show that HeBH needs to V k of an extreme the have spin at birth. This high Brocksopp, C., E Tarasov, A.eccentric E., Lyuty, V.M, & Roche,and P. 1999, A&A, 343, 861 Orosz, J. A., McClintock, J. E., −1Aufdenberg, J. P, et al. 2011, ApJ, 742, 84 −1 where is the anomaly is related tothe m BH asin has Figureat5 the shows the the possible positions of Cygnus X-1 at a small natal kick velocity of !77 km s ∼55 km s is necessary for explaining the current observed (95% from Helium stars, it seems that potentially received core-collapse event. small M , a minimum V of Fi He k et al. (2011) found that the BH Cygnus X-1 has a spin maximum MFor is constrained by our adopted upper limit of He Cadolle Bel, M., Sizun, P., Goldwurm, A., et al. 2006, A&A, 446, 591 Paxton, B., 2004, PASP, 116, 699 has implications about BH formation and the role of rotation −1 properties of Cygnus X-1. Furthermore, both the MHe PDF and during the core-collapse event. the time ∼55 of BH obtainedfor from integrating our a a small natal kick velocity of !77 km s kmformation, s−1 is necessary explaining the 3000 current observed (95% confidence) Frontera, F., Palazzi, E., Zdziarski, A. A., et al. 2001, ApJ, 546, 1027 Reid, M. J., McClintock, J. E., Narayan, R., et al. 2011, ApJ, 742, 83 in core collapse. Axelsson et al. (2011) also concluded that the Based on the dynamical model of Orosz et al. the two-dimensional joint VV. confidence levels show that k –M He ApJ, trajectories backward in time. As there is no trajectory crossing Gies, D. R., Bolton, C. T., Thomson, J. R., et al. 2003, ApJ, 583, 424 Wong, T.-W., Valsecchi, F., Fragos, T., Kalogera, 2012, 747 111 m = E − e sin E. (16) 9 properties of Cygnus X-1. Furthermore, both the MHe PDF and during the core-collapse event. observed spin connects involved et al. (2011) found in thatcore the collapse BH in Cygnus X-1 the maximum MHe is constrained by our adopted upper limit of to processes Gou, L., McClintock, J. E., Reid, M. J., et al. 2011, ApJ, 742, 85 Based on the dynamical model of Orosz et al. (2011), Gou the two-dimensional joint Vk –MHe confidence levels show that and is not likely to originate from the synchronous rotation of 6 limit of the BH progenitor. et al. (2011) found that the BH in Cygnus X-1 has a spin the maximum MHe is constrained by our adopted upper 9 U