Mass Ratio Distribution Research Articles

Determination of the yield, mass and structure of silver patches on colloidal silica using multiwavelength analytical ultracentrifugation

Anisotropic nanoparticles offer considerable promise for applications but also present significant challenges in terms of their characterization. Recent developments in the electroless deposition of silver patches directly onto colloidal silica particles have opened up a simple and scalable synthesis method for patchy particles with tunable optical properties. Due to the reliance on patch nucleation and growth, however, the resulting coatings are distributed in coverage and thickness and some core particles remain uncoated. To support process optimization, new methods are required to rapidly determine patch yield, thickness and coverage. Here we present a novel approach based on multiwavelength analytical ultracentrifugation (MWL-AUC) which permits simultaneous hydrodynamic and spectroscopic characterization. The patchy particle colloids are produced in a continuous flow mixing process that makes use of a KM-type micromixer. By varying the process flow rate or metal precursor concentration we show how the silver to silica mass ratio distribution derived from the AUC-measured sedimentation coefficient distribution can be influenced. Moreover, through reasoned assumptions we arrive at an estimation of the patch yield that is close to that determined by arduous analysis of scanning electron microscopy (SEM) images. Finally, combining MWL-AUC, electrodynamic simulations and SEM image analysis we establish a procedure to estimate the patch thickness and coverage.

Detailed evolutionary models of massive contact binaries – I. Model grids and synthetic populations for the Magellanic Clouds

ABSTRACT The majority of close massive binary stars with initial periods of a few days experience a contact phase, in which both stars overflow their Roche lobes simultaneously. We perform the first dedicated study of the evolution of massive contact binaries and provide a comprehensive prediction of their observed properties. We compute 2790 detailed binary models for the Large and Small Magellanic Clouds each, assuming mass transfer to be conservative. The initial parameter space for both grids span total masses from 20 to 80$\, \mathrm{M}_\odot$ , orbital periods of 0.6–2 d and mass ratios of 0.6–1.0. We find that models that remain in contact over nuclear time-scales evolve towards equal masses, echoing the mass ratios of their observed counterparts. Ultimately, the fate of our nuclear-time-scale models is to merge on the main sequence. Our predicted period–mass ratio distributions of O-type contact binaries are similar for both galaxies, and we expect 10 such systems together in both Magellanic Clouds. While we can largely reproduce the observed distribution, we overestimate the population of equal-mass contact binaries. This situation is somewhat remedied if we also account for binaries that are nearly in contact. Our theoretical distributions work particularly well for contact binaries with periods <2 d and total masses $\lessapprox 45\, \mathrm{M}_\odot \,$. We expect stellar winds, non-conservative mass transfer, and envelope inflation to have played a role in the formation of the more massive and longer-period contact binaries.

A Flexible Gaussian Process Reconstruction Method and the Mass Function of the Coalescing Binary Black Hole Systems

Abstract We develop a new method based on Gaussian processes to reconstruct the mass distribution of binary black holes (BBHs). Instead of prespecifying the formalisms of mass distribution, we introduce a more flexible and nonparametric model with which the distribution can be mainly determined by the observed data. We first test our method with simulated data and find that it can well recover the injected distribution. Then, we apply this method to analyze the data of BBHs’ observations from LIGO-Virgo Gravitational-Wave Transient Catalog 2. By reconstructing the chirp mass distribution, we find that there is a peak or a platform located at 20–30 M ⊙ rather than a single-power-law-like decrease from low mass to high mass. Moreover, one or two peaks in the chirp mass range of  < 20 M ⊙ may be favored by the data. Assuming a mass-independent mass ratio distribution of p(q) ∝ q 1.4, we further obtain a distribution of primary mass and find that there is a feature located in the range of (30, 40) M ⊙, which can be related to Broken Power Law and Power Law + Peak distributions described in Collaboration et al. Besides, the merger rate of BBHs is estimated to be  = 26.29 − 8.96 + 14.21 Gpc − 3 yr − 1 , supposing that there is no redshift evolution.

OGLE-2018-BLG-1185b: A Low-mass Microlensing Planet Orbiting a Low-mass Dwarf

Abstract We report an analysis of the planetary microlensing event OGLE-2018-BLG-1185, which was observed by a large number of ground-based telescopes and by the Spitzer Space Telescope. The ground-based light curve indicates a low planet–host star mass ratio of q = (6.9 ± 0.2) × 10−5, which is near the peak of the wide-orbit exoplanet mass-ratio distribution. We estimate the host star and planet masses with a Bayesian analysis using the measured angular Einstein radius under the assumption that stars of all masses have an equal probability of hosting the planet. The flux variation observed by Spitzer is marginal, but still places a constraint on the microlens parallax. Imposing a conservative constraint that this flux variation should be Δf Spz < 4 instrumental flux units yields a host mass of M host = 0.37 − 0.21 + 0.35 M ⊙ and a planet mass of m p = 8.4 − 4.7 + 7.9 M ⊕ . A Bayesian analysis including the full parallax constraint from Spitzer suggests smaller host star and planet masses of M host = 0.091 − 0.018 + 0.064 M ⊙ and m p = 2.1 − 0.4 + 1.5 M ⊕ , respectively. Future high-resolution imaging observations with the Hubble Space Telescope or Extremely Large Telescope could distinguish between these two scenarios and help reveal the planetary system properties in more detail.

VAMANA: modeling binary black hole population with minimal assumptions

The population analysis of compact binaries involves the reconstruction of some of the gravitational wave (GW) signal parameters, such as, the mass and the spin distribution, that gave rise to the observed data. This article introduces VAMANA, which reconstructs the binary black hole population using a mixture model and facilitates excellent density measurement as informed by the data. VAMANA uses a mixture of weighted Gaussians to reconstruct the chirp mass distribution. We expect Gaussian mixtures to provide flexibility in modeling complex distributions and enable us in capturing details in the astrophysical chirp mass distribution. Each of the Gaussian in the mixture is combined with another Gaussian and a power-law to simultaneously model the spin component aligned with the orbital angular momentum and the mass ratio distribution, thus also wing us to capture their variation with the chirp mass. Additionally, we can also introduce broadband smoothing by restricting the Gaussian mixture to lie within a threshold distance of a predefined reference chirp mass distribution. Using simulated data we show the robustness of our method in reconstructing complex populations for a large number of observations. We also apply our method to the publicly available catalog of GW observations made during LIGO’s and Virgo’s first and second observation runs and present the reconstructed mass, spin distribution, and the estimated merger rate of binary black holes.

Impact of common envelope development criteria on the formation of LIGO/Virgo sources

The treatment and criteria for development of unstable Roche lobe overflow (RLOF) that leads to the common envelope (CE) phase have hindered the area of evolutionary predictions for decades. In particular, the formation of black hole-black hole (BH-BH), black hole-neutron star (BH-NS), and neutron star-neutron star (NS-NS) merging binaries depends sensitively on the CE phase in classical isolated binary evolution model. All these mergers are now reported as LIGO/Virgo sources or source candidates. CE is even considered by some as a mandatory phase in the formation of BH-BH, BH-NS, or NS-NS mergers in binary evolution models. At the moment, there is no full first-principles model for the development of the CE. We employed the StarTrack population synthesis code to test the current advancements in studies on the stability of RLOF for massive donors to assess their effect on the LIGO/Virgo source population. In particular, we allowed for more restrictive CE development criteria for massive donors (M > 18 M⊙). We also tested a modified condition for switching between different types of stable mass transfer and between the thermal or nuclear timescale. The implemented modifications significantly influence the basic properties of merging double compact objects, sometimes in non-intuitive ways. For one of the tested models, with restricted CE development criteria, the local merger rate density for BH-BH systems increased by a factor of 2–3 due to the emergence of a new dominant formation scenario without any CE phase. We find that the changes in highly uncertain assumptions on RLOF physics may significantly affect: (i) the local merger rate density; (ii) shape of the mass and mass ratio distributions; and (iii) dominant evolutionary formation (with and without CE) scenarios of LIGO/Virgo sources. Our results demonstrate that without sufficiently strong constraints on RLOF physics, it is not possible to draw fully reliable conclusions about the population of double compact object systems based on population synthesis studies.

The mass-ratio distribution of tertiary-induced binary black hole mergers

ABSTRACT Many proposed scenarios for black hole (BH) mergers involve a tertiary companion that induces von Zeipel–Lidov–Kozai (ZLK) eccentricity cycles in the inner binary. An attractive feature of such mechanisms is the enhanced merger probability when the octupole-order effects, also known as the eccentric Kozai mechanism, are important. This can be the case when the tertiary is of comparable mass to the binary components. Since the octupole strength [∝(1 − q)/(1 + q)] increases with decreasing binary mass ratio q, such ZLK-induced mergers favour binaries with smaller mass ratios. We use a combination of numerical and analytical approaches to fully characterize the octupole-enhanced binary BH mergers and provide semi-analytical criteria for efficiently calculating the strength of this enhancement. We show that for hierarchical triples with semimajor axial ratio a/aout ≳ 0.01–0.02, the binary merger fraction can increase by a large factor (up to ∼20) as q decreases from unity to 0.2. The resulting mass-ratio distribution for merging binary BHs produced in this scenario is in tension with the observed distribution obtained by the LIGO/VIRGO collaboration, although significant uncertainties remain about the initial distribution of binary BH masses and mass ratios.

Undetected Binary Stars Cause an Observed Mass-dependent Age Gradient in Upper Scorpius

Abstract Young stellar associations represent a key site for the study of star formation, but to accurately compare observations to models of stellar evolution, the age of an association must be determined. The Upper Scorpius region is the youngest section of the Scorpius–Centaurus OB association, which is the largest collection of nearby, young, low-mass stars. The true age of Upper Scorpius is not clear, and an observed mass-dependent age gradient in Upper Scorpius, as well as in other star-forming regions, complicates age measurements. The age gradient may indicate a genuine astrophysical feature or may be an artifact of unrecognized systematic effects in stellar age measurements. We have conducted a synthetic red-optical low-resolution spectroscopic survey of a simulated analog to the Upper Scorpius star-forming region to investigate the effects of unresolved binary stars (which have mass-dependent demographics) on age measurements of a stellar population. We found that the observed mass-dependent age gradient in Upper Scorpius can be explained by a population of undetected binary stars. For a simulated population with an age of 10 (rms = 2) Myr, we measured an age of 10.5 (rms = 3.5) Myr for F stars and of 7.5 (rms = 5.8) Myr for M stars. This discrepancy is caused by the mass-dependent mass ratio distribution and the variable steepness of the mass–luminosity relation. Our results support the previously suggested 10 Myr age for Upper Scorpius, with a small intrinsic age spread.

Electro-optical properties and frequency response of polymer-dispersed liquid crystal gratings doped with multi-walled carbon nanotubes

In this paper, a model developed for establishing the lowest possible driving voltage of Holographic polymer-dispersed liquid crystal (H-PDLC) based on the Maxwell–Wagner effect doped with specific nanotubes. The dependences of the turning point frequency on the doping concentration and the distribution of multi-walled carbon nanotubes (MWCNTs) are discussed. The MWCNTs can effectively adjust the relaxation frequency of grating, increase its LC droplet size, and decrease its dielectric permittivity. The diffraction efficiency and threshold voltage of the prepared switchable Bragg grating were found to reach 91% and 0.77 V μm−1, respectively, at 5 kHz. By comparing the results of experiments and simulations, we obtained the mass distribution ratio of MWCNTs in the LC and polymer regions. It was found that the MWCNTs diffuse into the LC regions during the polymerization process, and the use of a suitable driving frequency and doping concentration can minimize the driving voltage. This work provides a reference for studying the dynamic range optical tuning and polymerization kinetics of nanoparticles in switchable gratings.

On the Modeling of Algol-Type Binaries

In earlier papers, we presented a binary evolutionary code for the purpose of reproducing the orbital parameters, masses, radii, and location in the Hertzsprung Russell diagram (abbreviated as HRD) of well-observed Algol systems. In subsequent versions, the effects of mass and angular momentum losses and tidal coupling were included in order to produce the observed distributions of orbital periods and mass ratios of Algol-type binaries. The mass loss includes stellar wind and possible liberal evolution, when the gainer star is not capable to absorb all of the matter during mass transfer from the donor star. We added magnetic braking to our code to better reproduce the observed equatorial velocities. Large equatorial velocities of mass-gaining stars are now lowered by tidal interaction and magnetic braking. Tides are mainly at work at short orbital periods, leaving magnetic braking alone at work during longer orbital periods. The observed values of the equatorial velocities of mass gainers in Algol-type binaries are mostly well reproduced by our code. According to our models, Algols have short periods with a strong magnetic field.

Stellar Radial Velocities in the Old Open Cluster M67 (NGC 2682). II. The Spectroscopic Binary Population

Abstract We present and analyze 120 spectroscopic binary and triple cluster members of the old (4 Gyr) open cluster M67 (NGC 2682). As a cornerstone of stellar astrophysics, M67 is a key cluster in the WIYN Open Cluster Study (WOCS); radial-velocity (RV) observations of M67 are ongoing and extend back over 45 yr, incorporating data from seven different telescopes, and allowing us to detect binaries with orbital periods ≲104 days. Our sample contains 1296 stars (604 cluster members) with magnitudes of 10 ≤ V ≤ 16.5 (about 1.3–0.7 M ⊙), from the giants down to ∼4 mag below the main-sequence turnoff, and extends in radius to 30′ (7.4 pc at a distance of 850 pc, or ∼7 core radii). This paper focuses primarily on the main-sequence binaries, but orbital solutions are also presented for red giants, yellow giants, and sub-subgiants. Out to our period detection limit and within our magnitude and spatial domain, we find a global main-sequence incompleteness-corrected binary fraction of 34% ± 3%, which rises to 70% ± 17% in the cluster center. We derive a tidal circularization period of . We also analyze the incompleteness-corrected distributions of binary orbital elements and masses. The period distribution rises toward longer periods. The eccentricity distribution, beyond P circ, is consistent with a uniform distribution. The mass-ratio distribution is also consistent with a uniform distribution. Overall, these M67 binaries are closely consistent with similar binaries in the galactic field, as well as with the old (7 Gyr) open cluster NGC 188. WOCS. 83.

Speckle Observations of TESS Exoplanet Host Stars: Understanding the Binary Exoplanet Host Star Orbital Period Distribution

Abstract We present high-resolution speckle interferometric imaging observations of TESS exoplanet host stars using the NN-EXPLORE Exoplanet and Stellar Speckle Imager instrument at the 3.5 m WIYN telescope. Eight TESS objects of interest that were originally discovered by Kepler were previously observed using the Differential Speckle Survey Instrument. Speckle observations of 186 TESS stars were carried out, and 45 (24%) likely bound companions were detected. This is approximately the number of companions we would expect to observe given the established 46% binarity rate in exoplanet host stars. For the detected binaries, the distribution of stellar mass ratio is consistent with that of the standard Raghavan distribution and may show a decrease in high-q systems as the binary separation increases. The distribution of binary orbital periods, however, is not consistent with the standard Ragahavan model, and our observations support the premise that exoplanet-hosting stars with binary companions have, in general, wider orbital separations than field binaries. We find that exoplanet-hosting binary star systems show a distribution peaking near 100 au, higher than the 40–50 au peak that is observed for field binaries. This fact led to earlier suggestions that planet formation is suppressed in close binaries.

Building and Calibrating the Binary Star Population Using Kepler Data

Abstract Modeling binary star populations is critical to linking the theories of star formation and stellar evolution with observations. In order to test these theories, we need accurate models of observable binary populations. The Kepler Eclipsing Binary Catalog (KEBC), with its estimated >90% completeness, provides an observational anchor on binary population models. In this work we present the results of a new forward model of the binary star population in the Kepler field. The forward model takes a single star population from a model of the Galaxy and pairs the stars into binaries by applying the constraints on the population from the results of observational binary population surveys such as Raghavan et al. and Duchêne & Kraus. A synthetic binary population is constructed from the initial distributions of orbital parameters. We identify the eclipsing binary sample from the generated binary star population and compare this with the observed sample of eclipsing binaries contained in the KEBC. Finally, we update the distributions of the synthetic population and repeat the process until the synthetic eclipsing binary sample agrees with the KEBC. The end result of this process is a model of the underlying binary star population that has been fit to observations. We find that for fixed flat mass ratio and eccentricity input distributions, the binary period distribution is logarithmically flat above ∼3.2 days. With additional constraints on distributions from observations, we can further adjust the synthetic binary population by relaxing other input constraints, such as mass ratio and eccentricity.

The Long-term Evolution of Main-sequence Binaries in DRAGON Simulations

Abstract We present a comprehensive investigation of main-sequence binaries in the DRAGON simulations, which are the first one-million-particle direct N-body simulations of globular clusters. We analyze the orbital parameters of the binary samples in two of the DRAGON simulations, D1-R7-IMF93 and D2-R7-IMF01, focusing on their secular evolution and correlations up to 12 Gyr. These two models have different initial stellar mass functions: Kroupa 1993 (D1-R7-IMF93) and Kroupa 2001 (D2-R7-IMF01); and different initial mass-ratio distributions: random paring (D1-R7-IMF93) and a power law (D1-R7-IMF93). In general, the mass ratio of a population of binaries increases over time due to stellar evolution, which is less significant in D2-R7-IMF01. In D1-R7-IMF93, primordial binaries with a mass ratio q ≈ 0.2 are most common, and the frequency linearly declines with increasing q at all times. Dynamical binaries of both models have higher eccentricities and larger semimajor axes than primordial binaries. They are preferentially located in the inner part of the star cluster. Secular evolution of binary orbital parameters does not depend on the initial mass-ratio distribution, but is sensitive to the initial binary distribution of the system. At t = 12 Gyr, the binary fraction decreases radially outwards, and mass segregation is present. A color difference of 0.1 mag in F330W − F814W and 0.2 mag in NUV − y between the core and the outskirts of both clusters is seen, which is a reflection of the binary radial distribution and the mass segregation in the cluster. The complete set of data for primordial and dynamical binary systems at all snapshot intervals is made publicly available.

Binary black hole mergers from LIGO/Virgo O1 and O2: Population inference combining confident and marginal events

We perform a statistical inference of the astrophysical population of binary black hole (BBH) mergers observed during the first two observing runs of Advanced LIGO and Advanced Virgo, including events reported in the GWTC-1 and IAS catalogs. We derive a novel formalism to fully and consistently account for events of arbitrary significance. We carry out a software injection campaign to obtain a set of mock astrophysical events subject to our selection effects, and use the search background to compute the astrophysical probabilities $p_{\rm astro}$ of candidate events for several phenomenological models of the BBH population. We emphasize that the values of $p_{\rm astro}$ depend on both the astrophysical and background models. Finally, we combine the information from individual events to infer the rate, spin, mass, mass-ratio and redshift distributions of the mergers. The existing population does not discriminate between random spins with a spread in the effective spin parameter, and a small but nonzero fraction of events from tidally-torqued stellar progenitors. The mass distribution is consistent with one having a cutoff at $m_{\rm max} = 41^{+10}_{-5}\,\rm M_\odot$, while the mass ratio favors equal masses; the mean mass ratio $\bar q> 0.67$. The rate shows no significant evolution with redshift. We show that the merger rate restricted to BBHs with a primary mass between 20 and $30\, \rm M_\odot$, and a mass ratio $q > 0.5$, and at $z \sim 0.2$, is 1.5 to $5.3\,{\rm Gpc^{-3} yr^{-1}}$ (90\% c.l.); these bounds are model independent and a factor of $\sim 3$ tighter than that on the local rate of all BBH mergers, and hence are a robust constraint on all progenitor models. Including the events in our catalog increases the Fisher information about the BBH population by $\sim 47\%$, and tightens the constraints on population parameters.

Pairing function of visual binary stars

ABSTRACTAn all-sky sample of 1227 visual binaries based on Washington Double Star catalogue is constructed to infer the initial mass function (IMF), mass ratio, and projected distance distribution with a dedicated population synthesis model. Parallaxes from Gaia DR2 and Hipparcos are used to verify the distance distribution. The model is validated on the single-star Tycho-2 sample and successfully reproduces the observed magnitudes and angular separations. The projected separation distribution follows f(s) ∼ s−1.2 in 102–2 × 103 au range for 1–4.5 m⊙ primary stars. Several algorithms are explored as pairing functions. Random pairing is confidently rejected. Primary-constrained pairing (PCP) and split-core pairing (SCP), the scenarios adopting primary component’s or total system’s mass as fundamental, are considered. The preferred IMF slope is α ∼ 2.8 either way. A simple power-law mass ratio distribution is unlikely, but the introduction of a twin excess provides a favourable result. PCP with f(q) ∼ q−1 is preferred with a tiny twin fraction, models with f(q) ∼ q−1.5 are acceptable when a larger twin excess is allowed. SCP is similar to PCP when a larger slope of the power law is adopted: f(q) ∼ qβ + 0.7.

Radiation Stability of Sorbents in Medical 225Ac/213Bi Generators

ABSTRACT The radiation resistance of sorbents AG 50 W-X8, Ac resin, and UTEVA resin in contact with hydrochloric solutions under irradiation with 10 MeV electrons in a wide absorbed-dose range up to 2 MGy was studied. Mass distribution ratios and total sorption capacity were used as quantitative measures to determine the changes in ion exchange or extractive abilities of the resins. Based on the data obtained and on a review of the performance of the 225Ac/213Bi generators reported in the literature, the applicability of investigated resins for high activity loaded onto different generator systems was concluded.

Binary Fraction Estimation of Main-sequence Stars in 12 Open Clusters: Based on the Homogeneous Data of LAMOST Survey and Gaia DR2

Abstract Based on the homogeneous low-resolution spectra data observed by Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) and the Gaia Data Release 2 (Gaia DR2) photometric data, we chose 12 open clusters (OCs) to study their fundamental parameters and binary fractions (BFs). For each OC, more than 20 cluster members were observed by LAMOST, and some of them were observed multiple times. We used these data to derive precise metallicities of OCs. Combining the metallicities and the Gaia DR2 photometric data, we used the isochrone fitting method to obtain fundamental parameters of these OCs. By fixing metallicity during the fitting, we avoided the effect of degeneracy between the metallicity and reddening. Based on the fundamental parameters, we utilized the synthetic color–magnitude diagram (CMD) method to derive the BFs of main-sequence (MS) stars, the mass functions (MFs) with correction of binaries, and the companion mass ratio distributions. The BFs of observed MS stars in OCs in this paper vary from 29% to 55%. Extrapolating the synthetic CMD to the hydrogen-burning limit, the BFs vary from 13% to 47%. Combining MFs published by previous literature with our results, we found that MFs with correction of binaries were steeper than those without correction by Δα = 0.6 ± 0.16 in the mass range of m ∈ [1M ⊙, 2.67M ⊙]. We found that the companion mass ratio distribution in OCs of our sample was flat. We also found a significant positive correlation between the BF and density.

The absolute magnitudes MJ, the binary fraction, and the binary mass ratios of M7–M9.5 dwarfs

ABSTRACT We use the large homogeneous sample of late M dwarfs, M7–M9.5, of Ahmed & Warren (2019) matched to Gaia DR2, to measure the relation between absolute magnitude and spectral type, and to infer the multiplicity fraction of the population, and the distribution of mass ratios in the binary systems. Binaries are identified photometrically as overluminous sources. In order to define a sample that is unbiased with respect to multiplicity we use distance limits that are a function of G − J colour to define a volume-complete sample of 2706 systems. The G − J colours are very precise, with random errors all less than 0.02. We measure absolute magnitudes MJ that are on average 0.5 mag brighter than previous determinations. We find evidence that the discrepancies arise from differences in spectral types in different samples. The measured binary fraction is $16.5\pm 0.8{{\ \rm per\ cent}}$, of which $98{{\ \rm per\ cent}}$ are unresolved: both values are consistent with results of previous studies. The distribution of excess flux in the binaries, compared to the singles, is used to infer the mass ratio distribution f(q)∝qγ, where q = Ms/Mp. We infer a very steep distribution over this spectral range, with γ > 10 ($99{{\ \rm per\ cent}}$ probability). This says that unresolved ultracool M dwarf binaries reside almost exclusively in equal mass systems, and implies that the spectral types of the unresolved binaries match to with 0.5 spectral subtypes. The intrinsic scatter in absolute magnitude MJ for ultracool M dwarfs at fixed G − J colour is measured to be 0.21 mag.

Extracting the Possible Intrinsic Relation between the Radiative Efficiency and Mass of QSOs: A Maximum Likelihood Method and its Application to the SDSS DR7 QSOs

Abstract Radiative efficiencies of QSOs and their distribution encode rich information on the evolution of both masses and spins of massive black holes (MBHs) across cosmic time. In this paper, we develop a maximum likelihood method to statistically extract the intrinsic relation between radiative efficiency (ϵ) and mass (M •) of QSOs from their distribution on the luminosity–(empirically estimated virial) mass plane. By using mock samples, we find that strong constraint can be put on the ϵ–M • relation at redshift z ≲ 0.4 from uniform QSO samples similar to those in the Sloan Digital Sky Survey (SDSS) and from QSO samples at z ∼ 0.6 (or ≲1.0) if the magnitude limit of the survey can be ∼1–2 (or 2–3) magnitude deeper. Applying this method to the SDSS DR7 QSOs with z ≲ 0.7, we find (or ) correlation for QSOs with the masses obtained according to the H β (or Mg ii) empirical mass estimator. These contradictory results may be due to the unknown systematic errors in the two mass estimators, preventing an accurate constraint on the ϵ–M • relation by using current available QSO samples. We find that both the estimates of MBH mass and Eddington ratio distribution functions can be affected by the ϵ– relation, suggesting that the determination of this relation is important for understanding the accretion and growth history of MBHs. In the future, the intrinsic ϵ–M • relation is expected to be strongly constrained by using QSO samples obtained from surveys deeper than the SDSS if the host galaxy contamination and systematic errors of the mass estimator(s) can be well modeled or removed.