How Beaming Shapes the Demographics of Ultraluminous X-ray Sources?

TL;DR

ULX demographics hinge on how beaming boosts apparent luminosities. The authors compare a classical beaming law with a log-modified version that includes a $(1+\ln \dot{m})^2$ factor, applying binary population synthesis to generate $X$-ray luminosity functions at $Z=0.016$ and $Z=0.0016$, distinguishing BH and NS accretors. They find that the log-modified prescription reproduces the observed high-luminosity slope and yields more observable NS ULXs, while the classical model overpredicts intrinsic bright sources yet underestimates their observability. These results imply that adopting a realistic beaming law is essential for interpreting ULX demographics and the neutron-star contribution, with metallicity shaping the high-L tail and selection effects constraining the visible population.

Abstract

Ultraluminous X-ray sources (ULXs) are off-nuclear compact objects with apparent luminosities above 10^39 erg/s, often exceeding the Eddington limit for stellar-mass black holes. Beaming is a commonly invoked mechanism to explain their extreme brightness, and the dependence of the beaming factor on accretion rate is a critical parameter. In this work, we investigate how different beaming prescriptions affect the predicted properties of ULX populations. Using binary population synthesis, we construct synthetic X-ray luminosity functions (XLFs) for both classical and log-modified beaming models at solar and sub-solar metallicities. The classical model predicts a larger intrinsic number of bright ULXs, but strong beaming reduces their observable fraction, resulting in fewer visible ULXs compared to the log-modified model. The log-modified prescription yields a shallower slope at high-luminosity, aligning better with observed XLFs, and increases the fraction of observable neutron star ULXs above 10^39 erg/s. These results underscore the significant role of the beaming law in shaping ULX statistical distributions and assessing neutron star contributions to the population.

Figures (5)

• Figure 1: Comparison of the classical and log-modified beaming prescriptions as a function of the dimensionless mass accretion rate $\dot{m}$. The gray, blue, and orange shaded regions correspond to $\dot{m}<1$, $1<\dot{m}<38.5$, and $\dot{m}>38.5$, respectively. Top panel: Intrinsic luminosity $L_{\rm int}$ as a function of $\dot{m}$. A clear transition from linear scaling at $\dot{m} \leq 1$ to logarithmic scaling at $\dot{m} > 1$ is visible. The solid and dashed lines represent BH and NS of the same mass. Middle panel: Beaming factor $b(\dot{m})$ for the classical (dashed line) and log-modified (solid line) prescriptions. The classical model adopts a constant $x=1$ with $\dot{m}_{\rm crit}=8.5$, whereas in the log-modified model $x$ varies with $\dot{m}$, giving $\dot{m}_{\rm crit}=38.5$. Bottom panel: Observed isotropic luminosity $L_{\rm iso} = L_{\rm int} / b$ for the two models. At low accretion rates ($\dot{m} < 8.5$), the two curves coincide. At high accretion rates, the log-modified model predicts a significantly lower $L_{\rm iso}$ than the classical model, with the divergence between the two curves increasing rapidly with $\dot{m}$, consistent with the behavior of $b(\dot{m})$. Solid and dash-dotted lines denote BH and NS in the classical model; dashed and dotted lines show the log-modified case. All curves are shown for a central compact object mass of $M_1 = 1.4~{\rm M_{\odot}}$, taken as a typical NS mass and serving as a reference case.
• Figure 2: Simulated and observed XLFs at two metallicities: $Z = 0.0016$ (top) and $Z = 0.016$ (bottom). Simulated results are shown as gray curves for the total population and blue curves for the cumulative visible population, with dashed and solid lines indicating the classical and log-modified models, respectively. Observed XLFs are overplotted as thin magenta ($Z = 0.0016$) and green ($Z = 0.016$) lines.
• Figure 3: Probability distribution histograms of ULX counts as a function of isotropic luminosity. The vertical axis shows time-weighted counts per luminosity bin. Gray bars denote the total number of systems, while blue (classical model) and salmon (log-modified model) bars indicate the visible subsets. The top panels correspond to $Z = 0.0016$, and the bottom panels correspond to $Z = 0.016$, respectively.
• Figure 4: Observable fraction of ULXs as a function of luminosity. The horizontal axis shows $\log_{10}L_{\rm iso}$, and the vertical axis shows the visible fraction, defined as the ratio of visible systems to the total number of systems in each luminosity bin. The red circles connected by a solid line represent the classical model, while the blue squares connected by a dashed line represent the log-modified model. Since metallicity has only a minor impact on this ratio, only the $Z = 0.016$ case is shown.
• Figure 5: Visible XLFs of BH and NS ULXs at two metallicities. The upper and lower panels correspond to $Z = 0.0016$ and $Z = 0.016$, respectively. Dashed and solid lines denote the classical and log-modified models, while blue and gray curves represent NSs and BHs. For comparison, the thin magenta dotted line shows the observed XLF at $Z = 0.0016$, and the thin green dashed line shows the observed XLF at $Z = 0.016$.