High-velocity stars as a result of encounters between stars and massive binary black holes in galactic nuclei

Numerical simulations of the motions of stars in the gravitational fields of binary black holes with various component mass ratios have been carried out. Two models are considered: (1) the two-body problem with two fixed centers; (2) the general three-body problem. The first model is applicable only over short times Δt ? T, where T is the period of the binary system. The second model is applicable at all times except for during close encounters of stars with one of the binary components, r ≤ 0.00002 pc, where r is the distance from the star to the nearer black hole. In very close passages, relativistic corrections must be taken into account. Estimates of the probability of formation of high-velocity stars as a result of such interactions are obtained. It is shown that this mechanism is not suitable for the nucleus of our Galaxy due to the probable absence of a second massive black hole in the central region of the Galaxy.     

Stars with relativistic speeds in the Hills scenario

The dynamical capture of a binary system consisting of a supermassive black hole (SMBH) and an ordinary star in the gravitational field of a central (more massive) SMBH is considered in the three-body problem in the framework of a modified Hills scenario. The results of numerical simulations predict the existence of objects whose spatial speeds are comparable to the speed of light. The conditions for and constraints imposed on the ejection speeds realized in a classical scenario and the modified Hills scenario are analyzed. The star is modeled using an N-body approach, making it possible to treat it as a structured object, enabling estimation of the probability that the object survives when it is ejected with relativistic speed as a function of the mass of the star, the masses of both SMBHs, and the pericenter distance. It is possible that the modern kinematic classification for stars with anomalously high spatial velocities will be augmented with a new class—stars with relativistic speeds.     

The disruption of close binaries in the gravitational field of a supermassive black hole and the formation of hypervelocity stars

The formation of hypervelocity stars due to the dynamical capture of one component of a closebinary system by the gravitational field of a supermassive black hole (SMBH) is modeled. The mass of the black hole was varied between 10⁶ and 10⁹ M _⊙. In the model, the problem was considered first as a three-body problem (stage I) and then as an N-body problem (stage II). In the first stage, the effect of the inclination of the internal close-binary orbit (the motion of the components about the center of mass of the binary system) relative to the plane of the external orbit (the motion of the close binary around the SMBH) on the velocity with which one of the binary components is ejected was assessed. The initial binary orbits were generated randomly, with 10 000 orbits considered for each external orbit with a fixed pericenter distance r _p . Analysis of the results obtained in the first stage of the modeling enables determination of the binary-orbit orientations that are the most favorable for high-velocity ejection, and estimation of the largest possible ejection velocities V _max. The boundaries of the region of stellar disruption derived from the balance of tidal forces and self-gravitation are discussed using V _max-r _p plots, which generalize the results of the first stage of the modeling. Since a point-mass representation does not enable predictions about the survival of stars during close passages by a SMBH, there is the need for a second stage of the modeling, in which the tidal influence of the SMBH is considered. An approach treating a star like a structured finite object containing N bodies (N = 4000) enables the derivation of more accurate limits for the zone of efficient acceleration of hypervelocity stars and the formulation of conditions for the tidal disruption of stars.     

Close binaries containing Supermassive Black Holes

We consider the evolution of binary systems formed by a Supermassive Black Hole (SMBH) residing in the center of a galaxy or a globular cluster and a star in its immediate vicinity. The star is assumed to fill its Roche lobe, and the SMBH accretes primarily the matter of this star. The evolution of such a system is mainly determined by the same processes as for an ordinary binary. The main differences are that the donor star is irradiated by hard radiation emitted during accretion onto the SMBH; in a detached system, nearly all the donor wind is captured by the black hole, which strongly affects the evolution of the semi-major axis; it is not possible for companions of the most massive SMBHs to fill their Roche lobes, since the corresponding orbital separations are smaller than the radius of the last stable orbit in the gravitational field of the SMBH. Moreover, there may not be efficient exchange between the orbital angular momentum and the angular momentum of the overflowing matter in such systems. Our computations assumed that, if the characteristic timescale for mass transfer is smaller than the thermal timescale of the star, no momentum exchange occurs. Absorption of incident external radiation in the stellar envelope was treated using the same formalism that was used when computing the radiative transfer in the stellar atmosphere. Numerical simulations show that Roche-lobe overflow is possible for a broad range of initial system parameters. The evolution of semi-detached systems containing a star and a SMBH nearly always ends with the dynamical disruption of the star. Stars with masses close to the solar mass are destroyed immediately after they fill their Roche lobes. During the accretion of matter of disrupted stars, the SMBH can achieve quasar luminosities. If the SMBH accretes ambient gas as well as gas stripped from stars, the star is subject to additional radiation in the detached phase of its evolution, strengthening its stellar wind. This leads to an increase of the semi-major axis and subsequent decrease of the probability of Roche-lobe overflow during the subsequent evolution of the system.     

The evolution of stars paired with supermassive black holes

A star located in the close vicinity of a supermassive black hole (SMBH) in a galactic nucleus or a globular-cluster core could form a close binary with the SMBH, with the star possibly filling its Roche lobe. The evolution of such binary systems is studied assuming that the SMBH mainly accretes matter from the companion star and that the presence of gas in the vicinity of the SMBH does not appreciably influence variations in the star’s orbit. The evolution of the star–SMBH system is mainly determined by the same processes as those determining the evolution of ordinary binaries. The main differences are that the star is subject to an incident flux of hard radiation arising during the accretion of matter by the SMBH, and, in detached systems, the SMBH captures virtually all the wind emitted by its stellar companion, which appreciably influences the evolution of the major axis of the orbit. Moreover, the exchange between the orbital angular momentum and the angular momentum of the overflowing matter may not be entirely standard in such systems. The computations assume that there will be no such exchange of angular momentum if the characteristic timescale for mass transfer is shorter than the thermal time scale of the star. The absorption of external radiation in the stellar envelope was computed using the same formalism applied when computing the opacity of the stellar matter. The numerical simulations show that, with the adopted assumptions, three types of evolution are possible for such a binary system, depending on the masses and the initial separation of the SMBH and star. Type I evolution leads to the complete destruction of the star. Only this type of evolution is realized for low-mass main-sequence (MS) stars, even those with large initial separations from their SMBHs. Massive MS stars will also be destroyed if the initial separation is sufficiently small. However, two other types of evolution are possible for massive stars, with a determining role in the time variations of the parameters of the star–SMBH system being played by the possible growth of the massive star into a red giant during the time it is located in the close vicinity of the SMBH. Type II evolution can be realized for massive MS stars that are initially farther from the SMBH than in the case of disruption. In this case, the massive star fills its Roche lobe during its expansion, but is not fully destroyed; the star retreats inside its Roche lobe after a period of intense mass loss. This type of evolution is characterized by an increase in the orbital period of the system with time. As a result, the remnant of the star (its former core) is preserved as a white dwarf, and can end up at a fairly large distance from the SMBH. Type III evolution can be realized formassiveMSstars that are initially located still farther from their SMBHs, and also for massive stars that are already evolved at the initial time. In these cases, the star moves away from the SMBH without filling its Roche lobe, due to its intense stellar wind. The remnants of such stars can also end up at a fairly large distances from their SMBHs.     

The formation and evolution of the most massive stars

We consider the formation of massive stars under the assumption that a young star accretes material from the protostellar cloud through its accretion disk while losing gas in the polar directions via its stellar wind. The mass of the star reaches its maximum when the intensity of the gradually strengthening stellar wind of the young star becomes equal to the accretion rate. We show that the maximum mass of the forming stars increases with the temperature of gas in the protostellar cloud T ₀, since the rate at which the protostellar matter is accreted increases with T ₀. Numerical modeling indicates that the maximum mass of the forming stars increases to ～900 M _⊙ for T ₀ ～ 300 K. Such high temperatures of the protostellar gas can be reached either in dense star-formation regions or in the vicinity of bright active galactic nuclei. It is also shown that, the lower the abundance of heavy elements in the initial stellar material Z, the larger the maximum mass of the star, since the mass-loss rate due to the stellar wind decreases with decreasing Z. This suggests that supermassive stars with masses up to 10⁶ M _⊙ could be formed at early stages in the evolution of the Universe, in young galaxies that are almost devoid of heavy elements. Under the current conditions, for T ₀ = (30–100) K, the maximum mass of a star can reach ～100M _⊙, as is confirmed by observations. Another opportunity for the most massive stars to increase their masses emerges in connection with the formation and early stages of evolution of the most massive close binary systems: the most massive stars can be produced either by coalescence of the binary components or via mass transfer in such systems.     

Late stages of the evolution of close compact binaries: Type I supernovae,gamma-ray bursts,and supersoft X-ray sources

We consider the evolution of close binaries resulting in the most intensive explosive phenomena in the stellar Universe—Type Ia supernovae and gamma-ray bursts. For Type Ia supernovae, which represent thermonuclear explosions of carbon-oxygen dwarfs whose masses reach the Chandrasekhar limit during the accretion of matter from the donor star, we derive the conditions for the accumulation of the limiting mass by the degenerate dwarf in the close binary. Accretion onto the degenerate dwarf can be accompanied by supersoft X-ray radiation with luminosity 1–10⁴ L _⊙. Gamma-ray bursts are believe to accompany the formation and rapid evolution of compact accretion-decretion disks during the formation of relativistic objects—black holes and neutron stars. The rapid (～1 M _⊙/s) accretion of matter from these disks onto the central compact relativistic star results in an energy release of ～0.1 M _⊙ c ² ～ 10⁵³ erg in the form of gamma-rays and neutrinos over a time of 0.1–1000 s. Such disks can form via the collapse of the rapidly rotating cores of Type Ib, Ic supernovae, which are components in extremely close binaries, or alternately due to the collapse of accreting oxygen-neon degenerate dwarfs with the Chandrasekhar mass into neutron stars, or the merging of neutron stars with neutron stars or black holes in close binaries. We present numerical models of the evolution of some close binaries that result in Type Ia supernovae, and also estimate the rates of these supernovae (～0.003/year) and of gamma-ray bursts (～10^?4/year) in our Galaxy for various evolutionary scenarios. The collimation of the gamma-ray burst radiation within an opening angle of several degrees “matches” the latter estimate with the observed rate of these events, ～10^?7–10^?8/year calculated for a galaxy with the mass of our Galaxy.     

Spots and activity of Pleiades stars from observations with the Kepler Space Telescope (K2)

Observations of the K2 continuation of Kepler Space Telescope program are used to estimate the spot coverage S (the fractional spotted area on the surface of an active star) for stars of the Pleiades cluster. The analysis is based on data on photometric variations of 759 confirmed clustermembers, together with their atmospheric parameters, masses, and rotation periods. The relationship between the activity (S) of these Pleiades stars and their effective temperatures shows considerable change in S for stars with temperatures T _eff less than 6100 K (this can be considered the limiting value for which spot formation activity begins) and a monotonic increase in S for cooler objects (a change in the slope for stars with Teff ~ 3700 K). The scatter in this parameter ΔS about its mean dependence on the (V ?Ks)0 color index remains approximately the same over the entire (V?K _s )0 range, including cool, fully convective dwarfs. The computated S values do not indicate differences between slowly rotating and rapidly rotating stars with color indices 1.1 < (V?K _s )0 < 3.7. The main results of this study include measurements of the activity of a large number of stars having the same age (759 members of the Pleiades cluster), resulting in the first determination of the relationship between the spot-forming activity and masses of stars. For 27 stars with masses differing from the solarmass by nomore than 0.1M⊙, themean spot coverage is S = 0.031±0.003, suggesting that the activity of candidate young Suns is more pronounced than that of the present-day Sun. These stars rotate considerably faster than the Sun, with an average rotation period of 4.3^d. The results of this study of cool, low-mass dwarfs of the Pleiades cluster are compared to results from an earlier study of 1570 M stars.     

Wolf-Rayet stars,black holes,and gamma-ray bursters in close binaries

We consider the evolutionary status of observed close binary systems containing black holes and Wolf-Rayet (WR) stars. When the component masses and the orbital period of a system are known, the reason for the formation of a WR star in an initial massive system of two main-sequence stars can be established. Such WR stars can form due to the action of the stellar wind from a massive OB star (M_OB≥50M_⊙), conservative mass transfer between components with close initial masses, or the loss of the common envelope in a system with a large (up to ～25) initial component mass ratio. The strong impact of observational selection effects on the creation of samples of close binaries with black holes and WR stars is demonstrated. We estimate theoretical mass-loss rates for WR stars, which are essential for our understanding the observed ratio of the numbers of carbon and nitrogen WR stars in the Galaxy \(\dot M_{WR} (M_ \odot yr^{ - 1} ) = 5 \times 10^{ - 7} (M_{WR} /M_ \odot )^{1.3} \). We also estimate the minimum initial masses of the components in close binaries producing black holes and WR stars to be ～25M_⊙. The spatial velocities of systems with black holes indicate that, during the formation of a black hole from a WR star, the mass loss reaches at least several solar masses. The rate of formation of rapidly rotating Kerr black holes in close binaries in the Galaxy is ～3×10^?6 yr^?1. Their formation may be accompanied by a burst of gamma radiation, possibly providing clues to the nature of gamma-ray bursts. The initial distribution of the component mass ratios for close binaries is dN～dq=dM₂/M₁ in the interval 0.04?q₀≤1, suggesting a single mechanism for their formation.     

Analysis of the dynamical instability of several multiple stars with weak hierarchy

The dynamical stability of 16 multiple stars is analyzed using Monte Carlo simulations with allowance for the errors in the observational data. The analysis was carried out by varying the uncertainties in the initial observational data. Six different stability criteria were considered, and the dynamical evolution was studied using numerical simulations. Eleven of the systems are probably stable, whereas five systems (HD 40887, HD 136176, HD 150680, HD 217675, and HD 222326) are probably unstable (the probability that they are unstable is 0.94 or more accordingt o the results of forward and backward simulations over intervals of 10⁶ yr). The results of the simulations were most consistent with the criteria of Mardling-Aarseth (the correlation coefficient between the probabilities of disruption inferred from the stability criterion and numerical integration was r = 0.998), Valtonen-Karttunen (r = 0.998), and Eggleton-Kiseleva (r = 0.997). In about 92–93% of all cases, these criteria yield results that are consistent with the numerical simulations. These criteria also yield high disruption probabilities for the unstable systems. Scenarios for the formation of such systems are discussed: temporary capture of a field star by a close binary, perturbation of a stable multiple system by a massive field object, and disruption of small stellar groups or clusters. The probabilities that these scenarios are realized are analyzed.     

Observational manifestations of the common envelope in a close binary

Three-dimensional (3D) numerical models of close binaries are used to study the structure and dynamics of common envelopes formed due to periodic ejections of matter from the accretion disk through the vicinity of the Lagrange point L₃. The results are used to estimate the physical parameters of the envelope, including its 3D matter distribution, and the matter-flow configuration and dynamics. Possible observational manifestations of such envelopes are estimated. We present the envelope’s radialvelocity distributions at various phases and times, as well as model light curves taking into account extinction in the envelope. The envelope becomes optically thick for systems with high mass-exchange rates, ? > 10^?8 M _⊙/year, and has a significant influence on the binary’s observed features. The uneven phase distributions of the matter and density variations due to periodic injections of matter into the envelope are important for interpretations of observations of close binary stars.     

Estimation of the accuracy of methods for determining component masses for low-mass X-ray binary systems

Modern modeling of the population of low-mass X-ray binary systems containing black holes applying standard assumptions leads to a lack of agreement between the modeled and observed mass distributions for the optical components, with the observed masses being lower. This makes the task of estimating the systematic errors in the derived component masses due to imperfect models relevant. To estimate the influence of systematic errors in the derived masses of stars in X-ray binary systems, we considered two approximations for the tidally deformed star in a Roche model. Approximating the star as a sphere with a volume equal to that of the Roche lobe leads to slight overestimation of the equatorial rotational velocity V _rot sin i, and hence to slight underestimation of the mass ratio q = M _x /M _v . Approximating the star as a flat, circular disk with constant local line profiles and a linear limb-darkening law (a classical rotational broadeningmodel) is an appreciably cruder approach, and leads to overestimation of V _rot sin i by about 20%. In the case of high values of q = M _x /M _v , this approximation leads to substantial underestimation of the mass ratio q, which can reach several tens of percent. The mass of the optical star is overestimated by a factor of 1.5 in this case, while the mass of the black hole is changed only slightly. Since most estimates of component mass ratios for X-ray binary systems are carried out using a classical rotational broadening model for the lines, this leads to the need for appreciable corrections to (reductions of) previously published masses for the optical stars, which enhances the contradiction with the standard evolutionary scenario for low-mass X-ray binaries containing black holes.     

The impact of reflection effects on the parameters of the old pre-cataclysmic variables MS Peg and LM Com

We have determined the main parameters of the old precataclysmic variable stars MS Peg and LM Com. The radial velocities of the components, reflection effects in the spectra, and light curves of the systems are studied based on model stellar atmospheres subject to external irradiation. Forty-seven moderate-resolution spectra for MS Peg and 57 for LM Com obtained with the 6-m telescope of the Special Astrophysical Observatory are used to derive the refined orbital periods of 0.1736660 days and 0.2586873 days, respectively; the orbital eccentricities do not exceed e=0.04. The mass (M _w =0.49e_⊙) and radius (e _w =0.015R_⊙) of the MS Peg primary calculated using the gravitational redshift correspond to those for a cooling carbon white dwarf with a thin hydrogen envelope. The parameters of the red dwarf (M _r =0.19M_⊙, T_eff=3560 K, R _r =0.18R_⊙) are close to those derived from evolutionary tracks for main-sequence M stars with solar chemical composition. The radius (R _r =0.22R_⊙) and temperature (T_eff=3650 K) of the LM Com secondary exceed theoretical estimates for main-sequence stars with masses of M _r =0.17M_⊙. The luminosity excess of the red dwarf in LM Com can be explained by a prolonged (T>5×10⁶ yrs) relaxation of the M star to its normal state after the binary leaves the common-envelope stage. For both systems, theoretical U, B, V, and R light curves and spectra calculated using the adopted sets of parameters are generally consistent with the observations. This confirms the radiative origin of the hot spots, the unimportance of horizontal radiative transport, and the absence of large-scale velocity fields with high values (V_trans>50 km/s) at the surfaces of the secondaries. Most of the emission lines in the spectra of these objects are formed under conditions close to thermalization, enabling modeling of their pro files in an LTE approximation. A strong λ3905 Å emission line has been identified as the 3s²3p4s 1P⁰-3s²3p² 1S SiI λ3905.52 Å line formed in the atmosphere of the hot spot. The observed intensity can be explained by non-LTE “superionization” of SiI atoms by soft UV radiation from the white dwarf. We suggest a technique for identifying binaries whose cool components are subject to UV irradiation based on observations of λ3905 Å emission in their spectra.     

Photometric and polarimetric activity of the Herbig Ae star VX Cas

We present the results of our simultaneous photometric and polarimetric observations of the Herbig Ae/Be star VX Cas acquired in 1987–2001. The star belongs to the UX Ori subtype of young variable stars and exhibits a rather low level of photometric activity: only six Algol-like minima with amplitudes ΔV>1^m were recorded in 15 years of observations. Two of these minima, in 1998 and 2001, were the deepest in the history of the star’s photometric studies, with V amplitudes of about 2^m. In each case, the dimming was accompanied by an increase in the linear polarization in agreement with the law expected for variable circumstellar extinction. The highest V polarization was about 5%. Observations of VX Cas in the deep minima revealed a turnover of the color tracks, typical of stars of this type and due to an increased contribution from radiation scattered in the circumstellar disk. We separated the observed polarization of VX Cas into interstellar (P_is) and intrinsic (P_in) components. Their position angles differ by approximately 60°, with P_is dominating in the bright state and P_in dominating during the deep minima. The competition of these two polarization components leads to changes in both the degree and position angle of the polarization during the star’s brightness variations. Generally speaking, in terms of the behavior of the brightness, color indices, and linear polarization, VX Cas is similar to other UX Ori stars studied by us earlier. A number of episodes of photometric and polarimetric activity suggest that, in their motion along highly eccentric orbits, circumstellar gas and dust clouds can enter the close vicinity of the star (and be disrupted there).     

Signs of magnetic accretion in X-ray pulsars

The spin-down mechanism of accreting neutron stars is discussedwith an application to one of the best studied X-ray pulsars GX301-2. We show that the maximum possible spin-down torque applied to a neutron star from the accretion flow can be evaluated as K _sd ^(t) = ??²/(r _m r _cor)^3/2. The spin-down rate of the neutron star in GX301-2 can be explained provided the magnetospheric radius of the neutron star is smaller than its canonical value. We calculate the magnetospheric radius considering the mass-transfer in the binary system in the frame of the magnetic accretion scenario suggested by V.F. Shvartsman. The spin-down rate of the neutron star expected within this approach is in a good agreement with that derived from observations of GX301-2.     

The evolutionary status of ultraluminous X-ray sources

We analyze models for quasi-stationary, ultraluminous X-ray sources (ULXs) with luminosities 10³⁸–10⁴⁰ erg/s exceeding the Eddington limit for a ～1.4M_⊙ neutron star. With the exception of relatively rare stationary ULXs that are associated with supernova remnants or background quasars, most ULXs are close binary systems containing a massive stellar black hole (BH) that accretes matter donated by a stellar companion. To explain the observed luminosities of ～10⁴⁰ erg/s, the mass of the BH must be ～40M_⊙ if the accreted matter is helium and ～60M_⊙ if the accreted matter has the solar chemical composition. We consider donors in the form of main-sequence stars, red giants, red supergiants, degenerate helium dwarfs, heavy disks that are the remnants of disrupted degenerate dwarfs, helium nondegenerate stars, and Wolf-Rayet stars. The most common ULXs in galaxies with active star formation are BHs with Roche-lobe-filling main-sequence companions with masses ～7M_⊙ or close Wolf-Rayet companions, which support the required mass-exchange rate via their strong stellar winds. The most probable candidate ULXs in old galaxies are BHs surrounded by massive disks and close binaries containing a BH and degenerate helium-dwarf, red-giant, or red-supergiant donor.     

The mass of the compact object in the X-ray binary 4U 1700-37

The results of a systematic analysis of master radial-velocity curves for the X-ray binary 4U 1700-37 are presented. The dependence of the mass of the X-ray component on the mass of the optical component is derived in a Roche model based on a fit of the master radial-velocity curve. The parameters of the optical star are used to estimate the mass of the compact object in three ways. The masses derived based on information about the surface gravity of the optical companion and various observational data are 2.25 _?0.24 ^+0.23 M_⊙ and 2.14 _?0.56 ^+0.50 M_⊙. The masses based on the radius of the optical star, 21.9R_⊙, are 1.76 _?0.21 ^+0.20 M_⊙ and 1.65 _?0.56 ^+0.78 M_⊙. The mass of the optical component derived from the mass-luminosity relation for X-ray binaries, 27.4M_⊙, yields masses for the compact object of 1.41 _?0.08 ⁺ M_⊙ and 1.35 _?0.18 ^+0.18 M_⊙.     

Physical parameters of the eclipsing binary V798 Cep

The first high-accuracy CCD UBV RI(RI)_C light curves for the recently discovered eclipsing binary V798 Cep (P = 16 ^d .08, V = 11 ^m . 8) are presented; this star is included in our program of eclipsing systems with considerable eccentricities. A photometric solution for the light curves and physical characteristics of the component stars are derived. The orbital eccentricity is quite high, e = 0.437. The longitude of periastron is close to 180°, making studies of the apsidal motion difficult. V798 Cep may be a hierarchical system.     

Acceleration of spatial motions of stars by close-binary supermassive black holes in galactic nuclei

The conditions for the acceleration of the spatial motions of stars by close-binary supermassive black holes (SMBHs) in galactic nuclei are analyzed in order to derive the velocity distribution for stars ejected from galaxies by such black holes. A close binary system consisting of two SMBHs in circular orbits was subject to a spherically symmetrical “barrage” of solar-mass stars with various initial velocities. The SMBHs were treated as point objects with Newtonian gravitational fields. Models with binary component-mass ratios of 1, 0.1, 0.01, and 0.001 were studied. The results demonstrate the possibility of accelerating neutron stars, stellar-mass black holes, and degenerate dwarfs to velocities comparable to the relative orbital velocities of the binary-SMBH components. In the stage when the binary components are merging due to the action of gravitational-wave radiation, this velocity can approach the speed of light. The most massive binary black-holes (M ? 10⁹M_⊙) can also accelerate main-sequence stars with solar or subsolar masses to such velocities.     

IR photometry and models for the dust envelopes of two carbon stars

The results of JHKLM photometry of two carbon stars are presented: the irregular variable NQ Cas and the Mira star BD Vul. Data on the mean fluxes supplemented with mid-IR observations with the IRAS, AKARI, andWISE satellites are used to compute spherically symmetrical model dust envelopes for the stars, consisting of particles of amorphous carbon and silicon carbide. The optical depth in the visible for the comparatively cool dust envelope of BD Vul, with a dust temperature at its inner boundary T₁ = 610 K, is fairly low: τ_V = 0.13. The dust envelope of NQ Cas is appreciably hotter (T₁ = 1550 K), and has τ_V = 0.32. The estimated mass-loss rates are 1.5 × 10^?7M_⊙/yr for NQ Cas and 5.9 × 10^?7M_⊙/yr for BD Vul.