[Paper Review] Constraining Mass of Cygnus X-1 from Analysis of the Hard State Spectral Data using TCAF Solution

Indrani Banerjee, Ayan Bhattacharjee+3 more|arXiv (Cornell University)|Apr 26, 2019

Particle Detector Development and Performance1 references4 citations

TL;DR

This study constrains the mass of Cygnus X-1 using spectral fitting of RXTE/PCA data during its hard state via the Two Component Advective Flow (TCAF) model, an independent method free from distance and companion mass dependencies. The analysis yields a consistent mass of $14.20 \pm 0.36M_{\odot}$, in excellent agreement with dynamical estimates and providing a tighter constraint than previous methods.

ABSTRACT

The galactic black hole candidate Cygnus X-1, one of the brightest sources in the sky, is the first ever black hole candidate to be discovered. Despite being a very well-studied object due to its persistent brightness in X-rays, there has been much difficulty in determining its mass since its discovery. In spite of continuous efforts since the early nineteen seventies, there is yet no concensus on its mass for nearly four decades. The uncertainties in the mass measurements are due to the high degree of error involved in the measurement of its distance. In 2011, Orosz et al. constrained the mass of this object to be M = 14.8 +/- 1.0 M_Sun using dynamical methods. In this paper, we obtained the mass of Cygnus X-1, using a completely independent method, namely, carrying out the spectral analysis using Two Component Advective Flow (TCAF) solution based fits file and the archival data of RXTE PCA instrument. Our result does not require the distance of the source or the information about the companion. Each spectral fit with the TCAF gives one best fitted mass. Averaging fitted masses of Cygnus X-1 over a span of five months of observation during its persistent hard phase, mass of the source comes out to be M_avg = 14.20 +/- 0.36 M_Sun, which is consistent with the dynamically estimated mass.

Motivation & Objective

• To determine the mass of Cygnus X-1 using a method independent of distance and companion star properties.
• To test the applicability of the TCAF model in constraining black hole mass from hard state spectral data.
• To provide a tighter, more robust mass estimate for Cygnus X-1 by averaging over multiple spectral fits during a persistent hard state.
• To validate the TCAF model’s ability to reproduce stable flow parameters in the hard state, supporting its use in mass estimation.

Proposed method

• Spectral fitting of RXTE/PCA archival data in the 2.5–45.0 keV energy band using the TCAF model as a local additive table model in XSPEC.
• Incorporation of a Gaussian component to model the Fe Kα emission line, with energy between 6.2–6.8 keV and width constrained between 0.001–0.8 keV.
• Fitting six key parameters: black hole mass $M_{\text{BH}}$, normalization $N$, disk accretion rate $\dot{m}_d$, halo accretion rate $\dot{m}_h$, shock location $X_s$, and shock strength $R$.
• Using the TCAF framework, which models accretion as a combination of Keplerian disk flow and sub-Keplerian halo flow with shock heating and CENBOL formation.
• Performing spectral fits over a five-month span (MJD 50916–51016) during the persistent hard state to assess parameter stability.
• Averaging the best-fit mass values from individual spectral fits to derive a robust, time-averaged mass estimate.

Experimental results

Research questions

• RQ1 Can the TCAF model provide a reliable, independent estimate of the black hole mass in Cygnus X-1 without relying on distance or companion star parameters?
• RQ2 How stable are the TCAF flow parameters (e.g., $\dot{m}_d$, $\dot{m}_h$, $X_s$, $R$) during the persistent hard state of Cygnus X-1?
• RQ3 Does the spectral fit with TCAF and a Gaussian component accurately reproduce the observed hard state X-ray spectrum without requiring a disc component?
• RQ4 Is the derived mass from TCAF fitting consistent with the dynamically measured mass of $14.8 \pm 1.0M_\odot$?
• RQ5 Can the TCAF model constrain the black hole mass with higher precision than previous spectral methods?

Key findings

• The average black hole mass of Cygnus X-1 derived from TCAF spectral fits is $14.20 \pm 0.36M_{\odot}$, indicating a high-precision, independent mass estimate.
• The disk accretion rate $\dot{m}_d$ remains stable at $0.72 \pm 0.03\dot{M}_{\text{Edd}}$ over the five-month observation period.
• The halo accretion rate $\dot{m}_h$ is consistently high at $1.80 \pm 0.06\dot{M}_{\text{Edd}}$, indicating strong sub-Keplerian accretion.
• The shock location $X_s$ is found to be $72.28r_S$ on average, with minimal variation between $70.38r_S$ and $73.35r_S$, indicating a stable shock position.
• The shock strength $R$ is $1.22 \pm 0.03$, suggesting a moderately strong shock transition in the accretion flow.
• The data/model ratio remains near unity up to 45 keV, confirming the absence of significant disc or reflection components, consistent with the hard state.