Simulation-calibrated Bayesian inference for progenitor properties of the microquasar SS 433

TL;DR

The paper tackles the problem of inferring the ZAMS progenitor properties of SS 433-like high-mass X-ray binaries by coupling the COSMIC binary population synthesis forward model with a Bayesian inversion framework. It constructs a ten-dimensional progenitor parameter space and a multivariate Gaussian likelihood over six observed SS 433 properties, performing posterior sampling with dynamic nested sampling. To ensure reliable posteriors in a complex, high-dimensional space, the authors implement simulation-based calibration (SBC), iteratively tightening ZAMS priors and validating posterior-conditional calibration, which substantially improves the inference of progenitor parameters. They report 90% credible intervals for key ZAMS quantities and find that the most informative results arise under specific assumptions about mass transfer, natal kicks, and common-envelope evolution, while also noting potential tensions with recent progenitor-mass estimates. The work demonstrates the viability of direct probabilistic inference of X-ray binary progenitors and points to future extensions that incorporate jet properties, alternative population models, and advanced optimization strategies to further constrain high-accretion-rate systems like SS 433.

Abstract

SS$\,$433 is one of the most extreme Galactic X-ray binaries, exhibiting semi-relativistic jets and super-critical accretion, and harboring a compact object, likely a black hole. Despite decades of observation and modeling, the precise nature of its progenitor binary remains uncertain. To estimate the zero-age main sequence (ZAMS) properties of binaries that evolve into SS$\,$433-like systems, we apply simulation-based calibration to Bayesian inference and convolve a multivariate Gaussian likelihood constructed from six measured binary parameters of SS$\,$433 with the isolated binary evolution model COSMIC. Employing the dynamic nested sampler of $\texttt{dynesty}$, we perform posterior inference over a ten-dimensional progenitor parameter space defined by the masses, orbital parameters, mass transfer possibilities, and natal kick velocity. We find that SS$\,$433-like systems arise from specific regions of binary evolution parameter space depending on key assumptions, such as the mass transfer rate and uncertainty taken from observations. Our simulation-based calibration framework, implemented with a suite of machine learning algorithms and scored by a heuristic reliability metric, allows us to iteratively build posterior distributions of the progenitors of SS$\,$433-like systems. This analysis reveals 90% confidence intervals for the ZAMS primary mass (8, 11) M$_\odot$, secondary mass (32, 40) M$_\odot$, orbital period (136, 2259) days, eccentricity (0.26, 0.6), common envelope evolution efficiency (0.44, 0.76), accreted fraction in stable mass transfer (0.22, 0.6), and black hole natal kick velocity magnitude (5, 68) km/s. These results demonstrate the feasibility of direct probabilistic inference of X-ray binary progenitors to offer new insights into the evolution of high-accretion-rate systems such as SS$\,$433.

Figures (4)

• Figure 1: The evolution of a single representative SS 433-like binary generated from our inverse-Bayesian sampling of the COSMIC population synthesis model. The first three panels show the evolution of the binary masses, orbital period, and stellar types, respectively, over the lifetime of the binary. The properties of the initially more (less) massive star are shown with red solid (blue dashed) lines. The bottom panel shows a zoomed-in view of the phase of stable mass transfer where the initially more massive star that formed into a black hole accretes material from the initially less massive star; the corresponding mass transfer rate (green solid line) evolves in time and peaks at $\approx 7.0e-4 \rm \,M_{\odot}/yr$ and exceeds the observed mass transfer rate of SS 433 (gray dashed line) of $10^{-4} \rm \,M_{\odot}/yr$ for about 0.01 Myr.
• Figure 2: Posterior distributions of the ZAMS parameters inferred from our uncalibrated (i.e., without simulation-based calibration) inverse-Bayesian sampling framework: the ZAMS masses, orbital period and eccentricity, the common envelope evolution efficiency $\alpha_{\rm CE}$ and the stable mass transfer accreted fraction $f_{\rm acc}$, and the natal kick velocity magnitude $v_{\rm k}$ and direction in 3-dimensional space. Two datasets are shown corresponding to two numbers of live points, 5,000 in red and 100,000 in blue, with the same fiducial ZAMS prior bounds and all else equal across the two runs that produced these data. The lines in the 2D plots correspond to contours enclosing 50% and 90% of the total probability mass.
• Figure 3: Kernel density estimation of subsets of the dataset from Fig. \ref{['F:CornerConverge']} with 100,000 live points inferred using fiducial priors, where each line color corresponds to a different fraction (10%, green; 25%, blue; 50%, red; 75%, purple) of the full dataset (100%, orange).
• Figure 4: Calibrated output after two SBC iterations from the case of fiducial initial priors, i.e., the 83% of SS 433-like binaries after the second SBC iteration. The ten ZAMS parameters are shown along with the six XRB parameters.