Two-temperature Accretion Flows Research Articles

Line profile of nuclear de-excitation gamma-ray emission from very hot plasma

ABSTRACT De-excitation gamma-ray lines, produced by nuclei colliding with protons, provide information about astrophysical environments where particles have kinetic energies of 10–100 MeV per nucleon. In general, such environments can be categorized into two types: the interaction between non-thermal MeV cosmic rays and ambient gas, and the other is thermal plasma with a temperature above a few MeV. In this paper, we focus on the latter type and investigate the production of de-excitation gamma-ray lines in very hot thermal plasma, especially the dependence of the line profile on the plasma temperature. We have calculated the line profile of prompt gamma rays from 12C and 16O and found that when nuclei have a higher temperature than protons, gamma-ray line profiles can have a complex shape unique to each nucleus species. This is caused by anisotropic gamma-ray emission in the nucleus rest frame. We propose that the spectroscopy of nuclear de-excitation gamma-ray lines may enable to probe energy distribution in very hot astrophysical plasmas. This diagnostics can be a new and powerful technique to investigate the physical state of a two-temperature accretion flows onto a black hole, especially the energy distributions of the protons and nuclei, which are difficult to access for any other diagnostics.

Two-temperature accretion flows around strongly magnetized stars and their spectral analysis

ABSTRACT We investigate two-temperature accretion flows onto strongly magnetized compact stars. Matter is accreted in the form of an accretion disc upto the disc radius (rd), where, the magnetic pressure exceeds both the gas and ram pressure and thereafter the matter is channelled along the field lines onto the poles. We solve the equations of motion self-consistently along the field lines, incorporating radiative processes like bremsstrahlung, synchrotron, and inverse-comptonization. For a given set of constants of motion, the equations of motion do not produce unique transonic solution. Following the second law of thermodynamics, the solution with the highest entropy is selected and thereby eliminating the degeneracy in solution. We study the properties of these solutions and obtain corresponding spectra as a function of the magnetic field (B*), spin period (P) and accretion rate of the star ($\dot{M}$ ). A primary shock is always formed just near the surface. The enhanced radiative processes in this post-shock region slows down the matter and it finally settles on the surface of the star. This post-shock region contributes to ${\gtrsim}99.99~{{\ \rm per\ cent}}$ of the total luminosity obtained from the accretion flow. It is still important to study the full accretion flow because secondary shocks may be present for some combination of B*, P, and $\dot{M}$ in addition to primary shocks. We find that secondary shocks, if present, produce an extended emission at higher energies in the spectra.

Temperature properties in magnetized and radiatively cooled two-temperature accretion flows on to a black hole

ABSTRACT Simplified assumptions about the thermodynamics of the electrons are normally employed in general-relativistic magnetohydrodynamic (GRMHD) simulations of accretion on to black holes. To counter this, we have developed a self-consistent approach to study magnetized and radiatively cooled two-temperature accretion flows around a Kerr black hole in two spatial dimensions. The approach includes several heating processes, radiative cooling, and a coupling between the electrons and the ions via Coulomb interaction. We test our approach by performing axisymmetric GRMHD simulations of magnetized tori accreting on to a Kerr black hole under various astrophysical scenarios. In this way, we find that the inclusion of the Coulomb interaction and the radiative cooling impacts the thermodynamical properties of both the ions and electrons, changing significantly the temperature distribution of the latter, and underlining the importance of a two-temperature approach when imaging these flows. In addition, we find that the accretion rate influences the bulk properties of the flow as well as the thermodynamics of the electrons and ions. Interestingly, we observe qualitatively distinct temperature properties for SANE and MAD accretion modes while maintaining the same accretion rates, which could help distinguishing MAD and SANE accretion flows via observations. Finally, we propose two new relations for the temperature ratios of the electrons, ions, and of the gas in terms of the plasma-β parameter. The new relations represent a simple and effective approach to treat two-temperature accretion flows on supermassive black holes such as Sgr A* and M 87*.

Viscous dissipative two-temperature accretion flows around black holes

Viscous dissipative two-temperature accretion flows around black holes

Improved Model of X-Ray Emission from Hot Accretion Flows

We have developed an improved model of X-ray emission from optically thin, two-temperature accretion flows, kerrflow, using an exact Monte Carlo treatment of global Comptonization as well as with a fully general relativistic description of both the radiative and hydrodynamic processes. It also includes pion-decay electrons, whose synchrotron emission dominates the seed photon yield at in flows around supermassive black holes. We consider in detail the dependence of the model spectra on the black hole spin, the electron heating efficiency, the plasma magnetization, and the accretion rate, and we discuss the feasibility of constraining these parameters by analyzing X-ray spectra of nearby low-luminosity active galactic nuclei. We note some degeneracies that hinder precise estimations of these parameters when individual X-ray spectra are analyzed. These degeneracies are eliminated when several spectra from a given source are fitted jointly, which then allows us to reliably measure the model parameters. We find significant differences with previous spectral models of hot-flow emission, related with the computational methods for Comptonization. Finally, we briefly consider and discuss the dependence on the viscosity parameter and on the outflow strength.

Two-temperature solutions and emergent spectra from relativistic accretion discs around black holes

Aims. We investigate a two-temperature advective transonic accretion disc around a black hole and analyse its spectrum in the presence of radiative processes such as bremsstrahlung, synchrotron, and inverse-Comptonisation. The aim is to link the emergent spectra with constants of motion of the accretion disc fluid, however, the number of unknowns in two-temperature theory exceeds the number of equations for a given set of constants of motion. We intend to remove the degeneracy using a general methodology and obtain a unique solution, along with its spectrum. Methods. We used hydrodynamic equations (continuity, momentum, and energy conservation equation) to obtain sonic points and solutions. To solve these equations of motion we used the 4th order Runge-Kutta method. For the spectral analysis, general and special relativistic effects were taken into consideration. The system is, nonetheless, degenerate and we remove the degeneracy by choosing the solution with maximum entropy, as dictated by the second law of thermodynamics. Results. We obtained a unique transonic solution for a given set of constants of motion. The entropy expression is a tool used to make a selection between the degenerate solutions. We found that Coulomb coupling is a weak energy exchange term, which allows protons and electrons to settle down into two different temperatures, justifying, hence, our study of two-temperature flows. The information of the electron flow allows us to model the spectra. We show that the spectra of accretion solutions depend on the associated constants of motion. At low accretion rates, bremsstrahlung is important. A fraction of the bremsstrahlung photons may be of higher energy than the neighbouring electrons, energising them through the process of Compton scattering. Synchrotron emission, on the other hand, provides soft photons, which can be inverse-Comptonised to produce a hard power law part in the spectrum. Luminosity increases with the increase in the accretion rate of the system, as well as with the increase in BH mass. However, the radiative efficiency of the flow has almost no dependence on the BH mass, but it sharply rises with the increase in the accretion rate. The spectral index, however, hardens with the increase in the accretion rate, while it does not change much with the variation in BH mass. In addition to the constants of motion, the value of the plasma beta parameter and magnitude of magnetic dissipation in the system also helps in shaping the spectrum. A shocked solution exists in two-temperature accretion flows in a limited region of the parameter space. We find that a shocked solution is always brighter than a solution without a shock. Conclusions. An accreting system in two-temperature regime admits multiple solutions for the same set of constants of motion, producing widely different spectra. Comparing the observed spectrum with that derived from a randomly chosen accretion solution would give us a wrong estimation of the accretion parameters of the system. The form of entropy measurement we obtained helped us to remove the degeneracy of the solutions and allowed us to understand the physics of the system, shorn of arbitrary assumptions. In this work, we show how the spectra and luminosities of an accreting system depend on the constants of motion, producing solutions ranging from radiatively inefficient flows to luminous flows. An increase in BH mass quantitatively changes the system, making the system more luminous, while the spectral bandwidth also increases. A higher BH mass system spans the range from radio to gamma-rays. However, increasing the accretion rate around a BH of certain mass has little influence on the frequency range of the spectra.

A comparative study of single-temperature and two-temperature accretion flows around black holes

We study the properties of sub-Keplerian accretion disk around a stationary black hole, considering bremsstrahlung, synchrotron and Comptonization of synchrotron photons as radiative cooling mechanisms active in the disk. We obtain the solutions of two-temperature global accretion flow (TTAF) and compare it with the results obtained from single-temperature (STAF) model. We observe that flow properties, in particular, the radial profile of electron and ion temperatures differ noticeably in the adopted models for flows with identical boundary conditions fixed at the outer edge of the disk. Since the electron temperature is one of the key factors to regulate the radiative processes, we argue that physically motivated description of electron temperature needs to be considered in studying the astrophysical phenomena around black holes.

Properties of two-temperature dissipative accretion flow around black holes

We study the properties of two-temperature accretion flow around a non-rotating black hole in presence of various dissipative processes where pseudo-Newtonian potential is adopted to mimic the effect of general relativity. The flow encounters energy loss by means of radiative processes acted on the electrons and at the same time, flow heats up as a consequence of viscous heating effective on ions. We assumed that the flow is exposed with the stochastic magnetic fields that leads to Synchrotron emission of electrons and these emissions are further strengthen by Compton scattering. We obtain the two-temperature global accretion solutions in terms of dissipation parameters, namely, viscosity (α) and accretion rate (⁠|${\dot{m}}$|⁠), and find for the first time in the literature that such solutions may contain standing shock waves. Solutions of this kind are multitransonic in nature, as they simultaneously pass through both inner critical point (xin) and outer critical point (xout) before crossing the black hole horizon. We calculate the properties of shock-induced global accretion solutions in terms of the flow parameters. We further show that two-temperature shocked accretion flow is not a discrete solution, instead such solution exists for wide range of flow parameters. We identify the effective domain of the parameter space for standing shock and observe that parameter space shrinks as the dissipation is increased. Since the post-shock region is hotter due to the effect of shock compression, it naturally emits hard X-rays, and therefore, the two-temperature shocked accretion solution has the potential to explain the spectral properties of the black hole sources.

General relativistic magnetohydrodynamical simulations of the jet in M 87

(abridged) The connection between black hole, accretion disk, and radio jet can be best constrained by fitting models to observations of nearby low luminosity galactic nuclei, in particular the well studied sources Sgr~A* and M87. There has been considerable progress in modeling the central engine of active galactic nuclei by an accreting supermassive black hole coupled to a relativistic plasma jet. However, can a single model be applied to a range of black hole masses and accretion rates? Here we want to compare the latest three-dimensional numerical model, originally developed for Sgr A* in the center of the Milky Way, to radio observations of the much more powerful and more massive black hole in M87. We postprocess three-dimensional GRMHD models of a jet-producing radiatively inefficient accretion flow around a spinning black hole using relativistic radiative transfer and ray-tracing to produce model spectra and images. As a key new ingredient to these models, we allow the proton-electron coupling in these simulations depend on the magnetic properties of the plasma. We find that the radio emission in M87 is well described by a combination of a two-temperature accretion flow and a hot single-temperature jet. The model fits the basic observed characteristics of the M87 radio core. The best fit model has a mass-accretion rate of Mdot approx 9x10^{-3} MSUN/YR and a total jet power of P_j \sim 10^{43} erg/s. Emission at 1.3mm is produced by the counter jet close to the event horizon. Its characteristic crescent shape surrounding the black hole shadow could be resolved by future millimeter-wave VLBI experiments. The model was successfully derived from one for the supermassive black hole in center of the Milky Way by appropriately scaling mass and accretion rate. This suggests the possibility that this model could also apply to a larger range of low-luminosity black holes.

ELECTRON HEATING BY THE ION CYCLOTRON INSTABILITY IN COLLISIONLESS ACCRETION FLOWS. I. COMPRESSION-DRIVEN INSTABILITIES AND THE ELECTRON HEATING MECHANISM

In systems accreting well below the Eddington rate, the plasma in the innermost regions of the disk is collisionless and two-temperature, with the ions hotter than the electrons. Yet, whether a collisionless faster-than-Coulomb energy transfer mechanism exists in two-temperature accretion flows is still an open question. We study the physics of electron heating during the growth of ion velocity-space instabilities, by means of multi-dimensional particle-in-cell (PIC) simulations. A large-scale compression - embedded in a novel form of the PIC equations - continuously amplifies the field. This constantly drives a pressure anisotropy P_perp > P_parallel, due to the adiabatic invariance of the particle magnetic moments. We find that, for ion plasma beta values beta_i ~ 5-30 appropriate for the midplane of low-luminosity accretion flows, mirror modes dominate if the electron-to-proton temperature ratio is > 0.2, whereas if it is < 0.2 the ion cyclotron instability triggers the growth of strong Alfven-like waves, that pitch-angle scatter the ions to maintain marginal stability. We develop an analytical model of electron heating during the growth of the ion cyclotron instability, which we validate with PIC simulations. We find that for cold electrons (beta_e < m_e/m_i), the electron energy gain is controlled by the magnitude of the E-cross-B velocity induced by the ion cyclotron waves. This term is independent of the initial electron temperature, so it provides a solid energy floor even for electrons starting with extremely low temperatures. On the other hand, the electron energy gain for beta_e > m_e/m_i - governed by the conservation of the magnetic moment in the growing fields of the instability - is proportional to the initial electron temperature. Our results have implications for two-temperature accretion flows as well as the solar wind and intracluster plasmas. [abridged]

ON THE ROLE AND ORIGIN OF NONTHERMAL ELECTRONS IN HOT ACCRETION FLOWS

We study the X-ray spectra of tenuous, two-temperature accretion flows using a model involving an exact, Monte Carlo computation of the global Comptonization effect as well as general relativistic description of both the flow structure and radiative processes. In our previous work we found that in flows surrounding supermassive black holes, thermal synchrotron radiation is not capable of providing a sufficient seed photons flux to explain the X-ray spectral indices as well as the cut-off energies measured in several best-studied AGNs. In this work we complete the model by including seed photons provided by nonthermal synchrotron radiation and we find that it allows to reconcile the hot flow model with the AGN data. We take into account two possible sources of nonthermal electrons. First, we consider e+- produced by charged-pions decay, which should be always present in the innermost part of a two-temperature flow due to proton-proton interactions. We find that for a weak heating of thermal electrons (small delta) the synchrotron emission of pion-decay e+- is much stronger than the thermal synchrotron emission in the considered range of bolometric luminosities, L Ledd. The small-delta model including hadronic effects in general agrees with the AGN data, except for the case of a slowly rotating black hole and a thermal distribution of protons. For large-delta, the pion-decay e+- have a negligible effect and then in this model we consider nonthermal electrons produced by direct acceleration. We find an approximate agreement with the AGN data for the fraction of the heating power of electrons which is used for the nonthermal acceleration eta~0.1.

Low angular momentum flow model II for Sgr A

We examine 1D two-temperature accretion flows around a supermassive black hole, adopting the specific angular momentum \lambda, the total specific energy \epsilon and the input accretion rate \dot M_{input} = 4.0x10^{-6} solar mass/yr estimated in the recent analysis of stellar wind of nearby stars around Sgr A*. The two-temperature flow is almost adiabatic even if we take account of the heating of electrons by ions, the bremsstrahlung cooling and the synchrotron cooling, as long as the ratio \beta of the magnetic energy density to the thermal energy density is taken to be as \beta < 1. The different temperatures of ions and electrons are caused by the different adiabatic indices of ions and electrons which depend on their temperature states under the relativistic regime. The total luminosity increases with increasing \beta and results in - 10^{35} - 10^{36} erg/s for \beta=10^{-3} - 1. Furthermore, from 2D time-dependent hydrodynamical calculations of the above flow, we find that the irregularly oscillating shocks are formed in the inner region and that the luminosity and the mass-outflow rate vary by a factor of 2 -- 3 and 1.5 -- 4, respectively. The time variability may be relevant to the flare activity of Sgr A*.

Coupled jet-disk model for Sagittarius A*: explaining the flat-spectrum radio core with GRMHD simulations of jets

The supermassive black hole in the center of the Milky Way, Sgr A*, displays a nearly flat radio spectrum that is typical for jets in active galactic nuclei. Indeed, time-dependent magnetized models of radiatively inefficient accretion flows (RIAFs), which are commonly used to explain the millimeter, near-infrared, and X-ray emission of Sgr A*, often also produce jet-like outflows. However, the emission from these models has so far failed to reproduce the flat radio spectrum. Aims: We investigate whether current accretion simulations can produce the compact flat spectrum emission by simply using a different prescription for the heating of the radiating particles in the jet. Methods: We studied the radiative properties of accretion flows onto a black hole produced in time-dependent general-relativistic magnetohydrodynamic (GRMHD) simulations. A crucial free parameter in these models has always been the electron temperature, and here we allowed for variations in the proton-to-electron temperature ratios in the jet and disk. Results: We found that the flat spectrum is readily reproduced by a standard GRMHD model if one has an almost isothermal jet coupled to a two-temperature accretion flow. The low-frequency radio mission comes from the outflowing sheath of matter surrounding the strongly magnetized nearly empty jet. The model is consistent with the radio sizes and spectrum of Sgr A*. Conclusions: Hence, GRMHD models of accreting black holes can in principle naturally reproduce jets that match observed characteristics. For Sgr A* the model fit to the spectrum predicts higher mass-accretion rates when a jet is included than without a jet. Hence, the impact of the recently discovered G2 cloud that is expected to be accreted onto Sgr A* might be less severe than currently thought.

Gamma-ray emission from proton–proton interactions in hot accretion flows

We present a model of gamma-ray emission through neutral pion production and decay in two-temperature accretion flows around supermassive black holes. We refine previous studies of such a hadronic gamma-ray emission by taking into account (1) relativistic effects in the photon transfer and (2) absorption of gamma-ray photons in the radiation field of the flow. We use a fully general relativistic description of both the radiative and hydrodynamic processes, which allows us to study the dependence on the black hole spin. The spin value strongly affects the gamma-ray emissivity within ~10 gravitational radii. The central regions of flows with the total luminosities L < 0.001 of the Eddington luminosity are mostly transparent to photons with energies below 10 GeV, permitting investigation of the effects of space-time metric. For such L, an observational upper limit on the gamma-ray (0.1-10 GeV) to X-ray (2-10 keV) luminosity ratio of L_{0.1-10 GeV}/L_{2-10 keV} << 0.1 can rule out rapid rotation of the black hole; on the other hand, a measurement of L_{0.1-10 GeV}/L_{2-10 keV} ~ 0.1 cannot be regarded as the evidence of rapid rotation, as such a ratio can also result from a flat radial profile of gamma-ray emissivity (which would occur for nonthermal acceleration of protons in the whole body of the flow). At L > 0.01 of the Eddington luminosity, the gamma-ray emission from the innermost region is strongly absorbed and the observed gamma-rays do not carry information on the value of the black hole spin. We note that if the X-ray emission observed in Centaurus A comes from an accretion flow, the hadronic gamma-ray emission from the flow should contribute significantly to the MeV/GeV emission observed from the core of this object, unless it contains a slowly rotating black hole and protons in the flow are thermal.

Unbeamed γ-rays from low luminosity AGNs

Proton-proton collisions in two-temperature accretion flows may lead to substantial γ-ray emission through neutral pion production and decay. We present preliminary results of our study of such a hadronic γ-ray emission. We find that radiatively inefficient flows around rapidly rotating black holes can produce the MeV/GeV luminosities up to ∼ 1044 erg s−1. The hot-flow model predictions for the hadronic and the Comptonization components can put interesting constraints on the properties of the source of high-energy radiation in AGNs. As an example, we compare our model with X/γ-ray observations of Centaurus A and we find that the model with a maximally rotating black hole strongly overpredicts the γ-ray flux.

Black Hole Accretion in Low States: Electron Heating

Plasmas in an accretion flow are heated by magneto-hydrodynamic turbulence generated through the magneto- rotational instability (MRI). The viscous stress driving the accretion is intimately connected to the microscopic processes of turbulence dissipation. We show that, in a few well-observed black hole accretion systems, there is compelling observational evidence of efficient electron heating by turbulence or collective plasma effects in low accretion states, when Coulomb collisions are not efficient enough to establish a thermal equilibrium between electrons and ions at small radii. We consider a Keplerian two-temperature accretion flow with a constant mass accretion rate in the pseudo- Newtonian gravitational potential and take into account the bremsstrahlung, synchrotron, and inverse Comptonization cooling processes. The balance of gravitational energy dissipation and turbulence energy cascade requires that the viscous stress be proportional to the product of the turbulence kinetic energy density and the total turbulence energy density, which may contradict the result of some shearing box simulations that the viscous stress is proportional to the magnetic field energy density. The critical mass accretion rate, below which the two-temperature solution may exist, is determined by the cooling processes and the collisional energy exchanges between electrons and ions and has very weak dependence on the collision-less heating of electrons by turbulence, which becomes more important at lower accretion rates. If the collision-less heating is dominated by the transit-time damping processes, small scale waves propagating obliquely with respect to the large scale magnetic field are prohibited, which may affect the saturate state of the MRI driven turbulence significantly. The plasma also needs to be strongly magnetized with the magnetic field and proton energy densities comparable so that electrons can share more of the dissipated gravitational energy. The heating of relativistic electrons is efficient since the heating rate is proportional to the mean momentum of the particles, and the electron heating may also be enhanced by their resonant scattering with small scale nearly parallel propagating waves.

Energy transfer and magnetic field generation via ion-beam driven instabilities in an electron-ion plasma

Whether an efficient collisionless temperature equilibration mechanism exists for a two-temperature ion-electron plasma, with Ti&amp;gt;Te, is important for understanding astrophysical phenomena such as two-temperature accretion flows and collisionless shocks in supernova remnants or gamma-ray bursts. In this paper, counter-streaming ion beam-driven two-stream, Weibel (or filamentation), and oblique instabilities are studied using two-dimensional (2D) particle-in-cell (PIC) simulations as a possible plasma instability that could operate in such astrophysical objects. The PIC simulations show interplay among these instabilities and that distinct stages with different dominant modes occur during the nonlinear evolution period. Although the 2D results show stronger electron-ion coupling than the one-dimensional (1D) instabilities, it is still too weak to rule out existing two-temperature accretion solutions. The nonrelativistic quasilinear equations for the 1D Weibel plus 1D two-stream modes are numerically solved to compare the results with the 2D PIC simulations and qualitative similarities were found. The equations also show that the magnetic fields generated by the Weibel instability decay to zero in the end.

A possible disk mechanism for the 23-day QPO in Mkn 501

Optically, thin two-temperature accretion flows may be thermally and viscously stable, but acoustically unstable. Here, we propose that the O-mode instability of a cooling-dominated optically thin two-temperature inner disk may explain the 23-day quasi-periodic oscillation (QPO) period observed in the TeV and X-ray light curves of Mkn 501 during its 1997 high state. In our model the relativistic jet electrons Compton upscatter the disk soft X-ray photons to TeV energies, so that the instability-driven X-ray periodicity will lead to a corresponding quasi-periodicity in the TeV light curve and produce correlated variability. We analyse the dependence of the instability-driven quasi-periodicity on the mass ( M) of the central black hole, the accretion rate ( M ˙ ) and the viscous parameter ( α) of the inner disk. We show that in the case of Mkn 501 the first two parameters are constrained by various observational results, so that for the instability occurring within a two-temperature disk where α = 0.05–1.0, the quasi-period is expected to lie within the range of 8–100 days, as indeed the case. In particular, for the observed 23-day QPO period our model implies a viscosity coefficient α ⩽ 0.28, a sub-Eddington accretion rate M ˙ ≃ 0.02 M ˙ Edd and a transition radius to the outer standard disk of r 0 ∼ 60 r g , and predicts a period variation δ P/ P ∼ 0.23 due to the motion of the instability region.

X-ray emissions from two-temperature accretion flows within a dipole magnetic funnel

We investigate the hydrodynamics of accretion channelled by a dipolar magnetic field (funnel flows). We consider situations in which the electrons and ions in the flow cannot maintain thermal equilibrium [two-temperature (2T) effects] due to strong radiative loss, and determine the effects on the keV X-ray properties of the systems. We apply this model to investigate the accretion shocks of white dwarfs in magnetic cataclysmic variables (mCVs). We have found that the incorporation of 2T effects could harden the keV X-rays. Also, the dipolar model yields harder X-ray spectra than the standard planar model if white dwarf is sufficiently massive (≳1 M⊙). When fitting observed keV X-ray spectra of mCVs, the inclusion of 2T hydrodynamics and a dipolar accretion geometry lowers estimates for white dwarf masses when compared with masses inferred from models excluding these effects. We find mass reductions ≲9 per cent in the most massive cases.

Hot One‐Temperature Accretion Flows Revisited

The effectiveness of the thermal coupling of ions and electrons in optically thin, hot accretion flows is investigated in a phenomenological approach. In the limit of complete coupling, we focus on the one-temperature accretion flows around black holes. Based on a global analysis, the results are compared with two-temperature accretion flow models and with the observations of black hole sources. Many features of one-and two-temperature solutions are quite similar. That is, hot one-temperature solutions are found to exist for mass flow rates less than a critical value, i.e.,. M less than or similar to 10 alpha(2)M(Edd), where M-Edd = L-Edd/c(2) is the Eddington accretion rate. When. M less than or similar to 10(-3)alpha(2)M(Edd), the viscous energy is mainly balanced by the advective cooling, i.e., the solution is in the advection-dominated accretion flow ( ADAF) regime. On the other hand, when 10(-3)alpha(2)M(Edd) less than or similar to M less than or similar to 10 alpha(2)M(Edd), radiative cooling is effective and is mainly balanced by advective heating, placing the solution in the regime of luminous hot accretion flow ( LHAF). When. M greater than or similar to 10 alpha(2)M(Edd), the accretion flow collapses at a transition radius with only the standard optically thick and geometrically thin disk solution existing in the innermost regions. We have fitted the spectra of the two black hole sources with the one-temperature models, Sgr A* and XTE J1118+ 480, which have been examined successfully with two-temperature models. It is found that the one-temperature models do not provide acceptable fits to the multi-wavelength spectra of Sgr A* nor to XTE J1118+ 480 as a result of the higher temperatures characteristic of the one-temperature models. It is concluded that the thermal coupling of ions and electrons cannot be fully effective and that a two-temperature description is required.