On Black Hole Mass Estimation from X-Ray Spectra of Ultraluminous X-Ray Sources

Abstract

We propose a methodology to derive a black-hole mass for super-critical accretion flow. Here, we use the extended disk blackbody (extended DBB) model, a fitting model in which the effective temperature profile obeys the relation $T_{\rm eff} \propto r^{-p}$, with $r$ being the disk radius and $p$ being treated as a fitting parameter. We first numerically calculate the theoretical flow structure and its spectra for a given black-hole mass, $M$, and accretion rate, $\dot{M}$. Through fitting to the theoretical spectra by the extended DBB model, we can estimate the black-hole mass, $M_{\rm x}$, assuming that the innermost disk radius is $r_{\rm in}=3r_{\rm g} (\propto M_{\rm x})$, where $r_{\rm g}$ is the Schwarzschild radius. We find, however, that the estimated mass deviates from that adopted in the spectral calculations, $M$, even for low-$\dot{M}$ cases. We also find that the deviations can be eliminated by introducing a new correction for the innermost radius. Using this correction, we calculate mass correction factors, $M/M_{\rm x}$, in the super-critical regimes for some sets of $M$ and $\dot M$, finding that a mass correction factor ranges between $M/M_{\rm x} \sim$1.2-1.6. The higher is $\dot{M}$, the larger does the mass correction factor tend to be. Since the correction is relatively small, we can safely conclude that the black holes in ULXs, which Vierdayanti et al. (2006, PASJ, 58, 915) analyzed, are stellar-mass black holes with the mass being $<100M_{\odot}$.

Two Eclipsing Ultraluminous X-Ray Sources in M51

“…(confirmed also by the numerical results of Vierdayanti et al 2008), but such a BH would be far below Eddington at the observed luminosity »Ĺ 2 10 39 erg s −1 , contrary to our initial slim disk assumption. Therefore, the slim disk model cannot be self-consistently applied to the spectrum of ULX-1.…”

“…A proper treatment of the observed properties of a slim disk requires additional parameters such as the viewing angle and the BH spin (Vierdayanti et al 2008(Vierdayanti et al , 2013Saḑowski 2009). However, Equations (4) and (5) are already good enough as a first-order approximation to test the consistency of a slim disk model for ULX-1.…”

“…On the other hand, the "disk" in ULX-2 is the dominant continuum emission component. It is probably emitted by a non-standard, geometrically thicker disk with advection, photon trapping and outflows (slim disk model), extending all the way down to the innermost stable circular orbit and possibly even a little further inside it (Vierdayanti et al 2008 (Gladstone et al 2009). It is a classic example of a ULX in the soft ultraluminous regime (Sutton et al 2013).…”

X-Ray Spectral and Optical Properties of a Ulx in NGC 4258 (M106)

“…Also, if we consider the normalization parameter of the DISKPBB model in XM3 data, we may calculate another mass value for the compact object. We derived a true inner disk radius of R in √ cos i ≈ 40 km using the correction factor ξ = 0.353 and a spectral hardening factor κ = 3 (Vierdayanti et al 2008). Assuming a moderate disk inclination i = 60 and taking the mass correction factor as minimum (∼ 1.2) the mass of the compact object in X-6 can be calculated as M ∼ 10M for a non-spining BH.…”

Search for gravitational radiation from intermediate mass black hole binaries in data from the second LIGO-Virgo joint science run

“…Although no conclusive detections have been made to date, intermediate mass black holes are very intriguing astrophysical objects, with growing observational and theoretical evidence for their existence [1,2]. Their discovery would be a major breakthrough in our understanding of massive black hole formation [3,4], stellar-cluster evolution [5][6][7][8][9][10], and hyper/ultraluminous x-ray sources [11][12][13][14][15][16][17]. Coalescing intermediate mass black hole binaries (IMBHBs) are also the strongest candidate gravitational-wave (GW) sources accessible to ground-based interferometric detectors such as LIGO and Virgo [18,19].…”