Jets In X-ray Binaries Research Articles

Swift J1727.8–1613 Has the Largest Resolved Continuous Jet Ever Seen in an X-Ray Binary

Multiwavelength polarimetry and radio observations of Swift J1727.8–1613 at the beginning of its recent 2023 outburst suggested the presence of a bright compact jet aligned in the north–south direction, which could not be confirmed without high-angular-resolution images. Using the Very Long Baseline Array and the Long Baseline Array, we imaged Swift J1727.8–1613 during the hard/hard-intermediate state, revealing a bright core and a large, two-sided, asymmetrical, resolved jet. The jet extends in the north–south direction, at a position angle of −0.60° ± 0.07° east of north. At 8.4 GHz, the entire resolved jet structure is ∼110(d/2.7kpc)/sini au long, with the southern approaching jet extending ∼80(d/2.7kpc)/sini au from the core, where d is the distance to the source and i is the inclination of the jet axis to the line of sight. These images reveal the most resolved continuous X-ray binary jet, and possibly the most physically extended continuous X-ray binary jet ever observed. Based on the brightness ratio of the approaching and receding jets, we put a lower limit on the intrinsic jet speed of β ≥ 0.27 and an upper limit on the jet inclination of i ≤ 74°. In our first observation we also detected a rapidly fading discrete jet knot 66.89 ± 0.04 mas south of the core, with a proper motion of 0.66 ± 0.05 mas hr−1, which we interpret as the result of a downstream internal shock or a jet–interstellar medium interaction, as opposed to a transient relativistic jet launched at the beginning of the outburst.

The Cause of the Difference in the Propagation Distances between Compact and Transient Jets in Black Hole X-Ray Binaries

Accreting black hole binaries change their properties during evolution, passing through two main luminous states, dominated by either hard or soft X-rays. In the hard state, steady compact jets emitting multiwavelength radiation are present. Those jets are usually observed in radio, and when resolved, their extent is ≲1015 cm. Then, during hard-to-soft transitions, powerful ejecta in the form of blobs appear. They are observed up to distances of ∼1018 cm, which are ≳1000 times larger than the extent of hard-state jets. On the other hand, estimates of the accretion rates during most luminous hard states and the hard-to-soft transitions are very similar, implying that maximum achievable powers of both types of jets are similar and cannot cause a huge difference in their propagation. Instead, we explain the difference in the propagation length by postulating that the ejecta consist of electron-ion plasmas, whereas the hard-state jets consist mostly of electron–positron pairs. The inertia of the ejecta are then much higher than those of compact jets, and the former are not readily stopped by ambient media. A related result is that the accretion flow during the hard state is of standard and normal evolution, while it is a magnetically arrested disk during transient ejections. The pairs in hard-state jets can be produced by collisions of photons of the hard spectrum emitted by hot accretion flows within the jet base. On the other hand, the X-ray spectra during the state transitions are relatively soft, and the same process produces much fewer pairs.

Numerical simulations of relativistic jets

In this paper, we review recent and ongoing work by our group on numerical simulations of relativistic jets. Relativistic outflows in astrophysics are related to dilute, high energy plasmas, with physical conditions out of the reach of current laboratory capabilities. Simulations are thus imperative for the study of these objects. We present a number of such scenarios that have been studied by our group at the Universitat de València. In particular, we have focused on the evolution of extragalactic outflows through galactic and intergalactic environments, deceleration by interaction with stars or clouds or the propagation of jets in X-ray binaries and interaction with stellar winds from massive companions. All also share their role as particle acceleration sites and production of non-thermal radiation throughout the electromagnetic spectrum. Therefore, our work is not only aimed at understanding the impact of outflows on their environments and thus their role in galaxy and cluster evolution, but also the nature and capabilities of these sites as generators of high- and very-high-energy radiation and cosmic rays.

Formation and Evolution of Transient Jets and Their Cavities in Black Hole X-Ray Binaries

We propose a model explaining the origin of transient/episodic jets in black hole X-ray binaries, in which they are caused by transitions from a collimated, strongly magnetized jet to a wide, uncollimated outflow. The change occurs when the accretion flow leaves the magnetically choked state due to an increase of the accretion rate for a weakly varying magnetic flux. The formed powerful jet then detaches from its base, and propagates as a discrete ejection. The uncollimated outflow then produces a relativistic plasma that fills the surroundings of the black hole, contributing to the formation of a low-density cavity. While the pressure in the cavity is in equilibrium with the surrounding interstellar medium (ISM), its inertia is orders of magnitude lower than that of the ISM. This implies that the plasma cannot efficiently decelerate the ejecta, explaining most of the observations. The modest deceleration within the cavities observed in some cases can then be due to the presence of clouds and/or filaments, forming a wide transition zone between the cavity and the ISM.

Possible contribution of X-ray binary jets to the Galactic cosmic ray and neutrino flux

ABSTRACT For over a century, the identification of high-energy cosmic ray (CR) sources remains an open question. For Galactic CRs with energy up to 1015 eV, supernova remnants (SNRs) have traditionally been thought the main candidate source. However, recent TeV γ-ray observations have questioned the SNR paradigm. Propagating CRs are deflected by the Galactic magnetic field, hence, γ-rays and neutrinos produced via inelastic hadronic interactions are the only means for unveiling the CR sources. In this work, we study the γ-ray and neutrino emission produced by CRs accelerated inside Galactic jets of stellar-mass black holes in X-ray binaries (BHXBs). We calculate the intrinsic neutrino emission of two prototypical BHXBs , Cygnus X–1 and GX 339–4, for which we have high-quality, quasi-simultaneous multiwavelength spectra. Based on these prototypical sources, we discuss the likelihood of the 35 known Galactic BHXBs to be efficient CR accelerators. Moreover, we estimate the potential contribution to the CR spectrum of a viable population of BHXBs that reside in the Galactic plane. When these BHXBs go into outburst, they may accelerate particles up to hundreds of TeV that contribute to the diffuse γ-ray and neutrino spectra while propagating in the Galactic medium. Using HERMES, an open-source code that calculates the hadronic processes along the line of sight, we discuss the contribution of BHXBs to the diffuse γ-ray and neutrino fluxes, and compare these to their intrinsic γ-ray and neutrino emissions. Finally, we discuss the contribution of BHXBs to the observed spectrum of Galactic CRs.

Bursts from Space: MeerKAT – the first citizen science project dedicated to commensal radio transients

ABSTRACT The newest generation of radio telescopes is able to survey large areas with high sensitivity and cadence, producing data volumes that require new methods to better understand the transient sky. Here, we describe the results from the first citizen science project dedicated to commensal radio transients, using data from the MeerKAT telescope with weekly cadence. Bursts from Space: MeerKAT was launched late in 2021 and received ∼89 000 classifications from over 1000 volunteers in 3 months. Our volunteers discovered 142 new variable sources which, along with the known transients in our fields, allowed us to estimate that at least 2.1 per cent of radio sources are varying at 1.28 GHz at the sampled cadence and sensitivity, in line with previous work. We provide the full catalogue of these sources, the largest of candidate radio variables to date. Transient sources found with archival counterparts include a pulsar (B1845-01) and an OH maser star (OH 30.1–0.7), in addition to the recovery of known stellar flares and X-ray binary jets in our observations. Data from the MeerLICHT optical telescope, along with estimates of long time-scale variability induced by scintillation, imply that the majority of the new variables are active galactic nuclei. This tells us that citizen scientists can discover phenomena varying on time-scales from weeks to several years. The success both in terms of volunteer engagement and scientific merit warrants the continued development of the project, while we use the classifications from volunteers to develop machine learning techniques for finding transients.

Compact jets dominate the continuum emission in low-luminosity active galactic nuclei

Low-luminosity active galactic nuclei (LLAGN) are special among their kind due to the profound structural changes that the central engine experiences at low accretion rates (≲ 10−3 in Eddington units). The disappearance of the accretion disc – the blue bump – leaves behind a faint optical nuclear continuum whose nature has been largely debated. This is mainly due to serious limitations on the observational side imposed by the starlight contamination from the host galaxy and the absorption by hydrogen, preventing the detection of these weak nuclei in the infrared (IR) to ultraviolet (UV) range. We addressed these challenges by combining multi-wavelength sub-arcsecond resolution observations – able to isolate the genuine nuclear continuum – with nebular lines in the mid-IR, which allowed us to indirectly probe the shape of the extreme UV continuum. We found that eight of the nearest prototype LLAGN are compatible with pure compact jet emission over more than ten orders of magnitude in frequency. This consists of self-absorbed synchrotron emission from radio to the UV plus the associated synchrotron self-Compton component dominating the emission in the UV to X-ray range. Additionally, the LLAGN continua show two particular characteristics when compared with the typical jet spectrum seen in radio galaxies: (i) a very steep spectral slope in the IR-to-optical/UV range (−3.7 < α0 < −1.3; Fν ∝ να0); and (ii) a very high turnover frequency (0.2–30 THz; 1.3 mm–10 μm) that separates the optically thick radio emission from the optically thin continuum in the IR-to-optical/UV range. These attributes can be explained if the synchrotron continuum is mainly dominated by thermalised particles at the jet base or the corona with considerably high temperatures, whereas only a small fraction of the energy (∼20%) would be distributed along the high-energy power-law tail of accelerated particles. On the other hand, the nebular gas excitation in LLAGN is in agreement with photo-ionisation from inverse Compton radiation (αx ∼ −0.7), which would dominate the nuclear continuum shortwards of ∼3000 Å, albeit a possible contribution from low-velocity shocks (< 500 km s−1) to the line excitation cannot be discarded. No sign of a standard hot accretion disc is seen in our sample of LLAGN, nevertheless, a weak cold disc (< 3000 K) is detected at the nucleus of the Sombrero galaxy, though its contribution to the nebular gas excitation is negligible. Our results suggest that the continuum emission in LLAGN is dominated at all wavelengths by undeveloped jets, powered by a thermalised particle distribution with high energies, on average. This is in agreement with their compact morphology and their high turnover frequencies. This behaviour is similar to that observed in peaked-spectrum radio sources and also compact jets in quiescent black hole X-ray binaries. Nevertheless, the presence of extended jet emission at kiloparsec scales for some of the objects in the sample is indicative of past jet activity, suggesting that these nuclei may undergo a rejuvenation event after a more active phase that produced their extended jets. These results imply that the dominant channel for energy release in LLAGN is mainly kinetic via the jet, rather than the radiative one. This has important implications in the context of galaxy evolution, since LLAGN probably represent a major but underestimated source of kinetic feedback in galaxies.

The simultaneity of emission from approaching and receding jets

ABSTRACT We show that the standard Blandford–Königl model for compact conical relativistic jets has a peculiar feature: At a given observed frequency of radiation, the emission from the approaching jet arrives at the location of a distant observer at the same time as the emission from the counterjet for all finite inclination angles. We show that this result can be used to determine whether jets are genuinely symmetric, if the cross-coherence between radio and X-ray time series can be measured at high Fourier frequency for a sample of neutron star X-ray binaries with a range of inclination angles. We also discuss echo mapping techniques that can be used to look for deviations from the standard model in high cadence time series data on X-ray binary jets, and conclude that these can plausibly be applied to some systems.

A broadband radio view of transient jet ejecta in the black hole candidate X-ray binary MAXI J1535–571

AbstractWe present a broadband radio study of the transient jets ejected from the black hole candidate X-ray binary MAXI J1535–571, which underwent a prolonged outburst beginning on 2017 September 2. We monitored MAXI J1535–571 with the Murchison Widefield Array (MWA) at frequencies from 119 to 186 MHz over six epochs from 2017 September 20 to 2017 October 14. The source was quasi-simultaneously observed over the frequency range 0.84–19 GHz by UTMOST (the Upgraded Molonglo Observatory Synthesis Telescope) the Australian Square Kilometre Array Pathfinder (ASKAP), the Australia Telescope Compact Array (ATCA), and the Australian Long Baseline Array (LBA). Using the LBA observations from 2017 September 23, we measured the source size to be$34\pm1$mas. During the brightest radio flare on 2017 September 21, the source was detected down to 119 MHz by the MWA, and the radio spectrum indicates a turnover between 250 and 500 MHz, which is most likely due to synchrotron self-absorption (SSA). By fitting the radio spectrum with a SSA model and using the LBA size measurement, we determined various physical parameters of the jet knot (identified in ATCA data), including the jet opening angle ($\phi_{\rm op} = 4.5\pm1.2^{\circ}$) and the magnetic field strength ($B_{\rm s} = 104^{+80}_{-78}$mG). Our fitted magnetic field strength agrees reasonably well with that inferred from the standard equipartition approach, suggesting the jet knot to be close to equipartition. Our study highlights the capabilities of the Australian suite of radio telescopes to jointly probe radio jets in black hole X-ray binaries via simultaneous observations over a broad frequency range, and with differing angular resolutions. This suite allows us to determine the physical properties of X-ray binary jets. Finally, our study emphasises the potential contributions that can be made by the low-frequency part of the Square Kilometre Array (SKA-Low) in the study of black hole X-ray binaries.

Probing Jet Launching in Neutron Star X-Ray Binaries: The Variable and Polarized Jet of SAX J1808.4–3658

Abstract We report on an optical photometric and polarimetric campaign on the accreting millisecond X-ray pulsar (AMXP) SAX J1808.4–3658 during its 2019 outburst. The emergence of a low-frequency excess in the spectral energy distribution in the form of a red excess above the disk spectrum (seen most prominently in the z, i, and R bands) is observed as the outburst evolves. This is indicative of optically thin synchrotron emission due to a jet, as seen previously in this source and in other AMXPs during outburst. At the end of the outburst decay, the source enters a reflaring state. The low-frequency excess is still observed during the reflares. Our optical (BVRI) polarimetric campaign shows variable linear polarization (LP) throughout the outburst. We show that this is intrinsic to the source, with low-level but significant detections (0.2%–2%) in all bands. The LP spectrum is red during both the main outburst and the reflaring state, favoring a jet origin for this variable polarization over other interpretations, such as Thomson scattering with free electrons from the disk or the propelled matter. During the reflaring state, a few episodes with stronger LP levels (1%–2%) are observed. The low-level, variable LP is suggestive of strongly tangled magnetic fields near the base of the jet. These results clearly demonstrate how polarimetry is a powerful tool for probing the magnetic field structure in X-ray binary jets, as for active galactic nuclei jets.

Eclipses of jets and discs of X-ray binaries as a powerful tool for understanding jet physics and binary parameters

ABSTRACT We calculate the expected effects on the spectral energy distributions and light curves in X-ray binary jets from eclipses by the donor stars. Jets will be eclipsed for all inclination angles, with just the height along the jet where the eclipse takes place being set by the orbital parameters. Typically, eclipses will lead to 5–10 per cent reductions in the jet emission over a range of a factor of few in wavelength with a periodic modulation. In ideal systems with high inclination angles, relatively even mass ratios, and modest jet speeds, the eclipses may be deeper. We discuss how eclipses can be used to measure binary system parameters, as well as the height of the bases of the jets. We also discuss how, with data sets that will likely require future facilities, more detailed tests of models of jet physics could be made by establishing deviations from the standard recipes for compact conical flat spectrum jets and by determining the ingress and egress durations of the eclipses and measuring the transverse size of the jets. We provide representative calculations of expectations for different classes of systems, demonstrating that the most promising target for showing this effect in the radio band is the longer period ‘atoll’-class neutron star X-ray binaries, while in the optical and infrared bands, the best candidates are likely to be the most edge-on black hole X-ray binaries. We also discuss the effects of the outer accretion disc eclipsing the inner jet.

Infrared interferometry to spatially and spectrally resolve jets in X-ray binaries

ABSTRACT Infrared interferometry is a new frontier for precision ground-based observing, with new instrumentation achieving milliarcsecond (mas) spatial resolutions for faint sources, along with astrometry on the order of 10 microarcseconds (μas). This technique has already led to breakthroughs in the observations of the supermassive black hole at the Galactic centre and its orbiting stars, active galactic nucleus, and exo-planets, and can be employed for studying X-ray binaries (XRBs), microquasars in particular. Beyond constraining the orbital parameters of the system using the centroid wobble and spatially resolving jet discrete ejections on mas scales, we also propose a novel method to discern between the various components contributing to the infrared bands: accretion disc, jets, and companion star. We demonstrate that the GRAVITY instrument on the Very Large Telescope Interferometer should be able to detect a centroid shift in a number of sources, opening a new avenue of exploration for the myriad of transients expected to be discovered in the coming decade of radio all-sky surveys. We also present the first proof-of-concept GRAVITY observation of a low-mass XRB transient, MAXI J1820+070, to search for extended jets on mas scales. We place the tightest constraints yet via direct imaging on the size of the infrared emitting region of the compact jet in a hard state XRB.

Collisionless tangential discontinuity between pair plasma and electron–proton plasma

We study with a one-dimensional particle-in-cell simulation the expansion of a pair cloud into a magnetized electron–proton plasma as well as the formation and subsequent propagation of a tangential discontinuity that separates both plasmas. Its propagation speed takes the value that balances the magnetic pressure of the discontinuity against the thermal pressure of the pair cloud and the ram pressure of the protons. Protons are accelerated by the discontinuity to a speed that exceeds the fast magnetosonic speed by the factor of 10. A supercritical fast magnetosonic shock forms at the front of this beam. An increasing proton temperature downstream of the shock and ahead of the discontinuity leaves the latter intact. We create the discontinuity by injecting a pair cloud at a simulation boundary into a uniform electron–proton plasma, which is permeated by a perpendicular magnetic field. Collisionless tangential discontinuities in the relativistic pair jets of x-ray binaries (microquasars) are in permanent contact with the relativistic leptons of their inner cocoon, and they become the sources of radio synchrotron emissions.

High-energy cosmic ray production in X-ray binary jets

ABSTRACT As smaller analogues of active galactic nuclei, X-ray binaries (XRBs) are also capable of launching jets that accelerate particles to high energies. In this work, we re-examine XRB jets as potential sources of high-energy cosmic rays (CRs) and explore whether they could provide a significant second Galactic component to the CR spectrum. In the most intriguing scenario, XRB-CRs could dominate the observed spectrum above the so-called knee feature at ∼3 × 1015 eV, offering an explanation for several key issues in this transition zone from Galactic to extragalactic CRs. We discuss how such a scenario could be probed in the near future via multimessenger observations of XRB jets, as well as diffuse Galactic neutrino flux measurements.

Lorentz Factors of Compact Jets in Black Hole X-Ray Binaries

Abstract Compact, continuously launched jets in black hole X-ray binaries (BHXBs) produce radio to optical/IR synchrotron emission. In most BHXBs, an IR excess (above the disk component) is observed when the jet is present in the hard spectral state. We investigate why some BHXBs have prominent IR excesses and some do not, quantified by the amplitude of the IR quenching or recovery over the transition from/to the hard state. We find that the amplitude of the IR excess can be explained by inclination-dependent beaming of the jet synchrotron emission and the projected area of the accretion disk. Furthermore, we see no correlation between the expected and the observed IR excess for Lorentz factor 1, which is strongly supportive of relativistic beaming of the IR emission, confirming that the IR excess is produced by synchrotron emission in a relativistic outflow. Using the amplitude of the jet fade and recovery over state transitions and the known orbital parameters, we constrain for the first time the bulk Lorentz factor range of compact jets in several BHXBs (with all the well-constrained Lorentz factors lying in the range of Γ = 1.3–3.5). Under the assumption that the Lorentz factor distribution of BHXB jets is a power law, we find that . We also find that the very high amplitude IR fade/recovery seen repeatedly in the BHXB GX 339–4 favors a low inclination angle ( ) of the jet.

ALMA observations of A0620–00: fresh clues on the nature of quiescent black hole X-ray binary jets

ABSTRACT We report on Atacama Large Millimeter Array (ALMA) continuum observations of the black hole X-ray binary A0620–00 at an X-ray luminosity nine orders of magnitude sub-Eddington. The system was significantly detected at 98 GHz (at 44 ± 7 $\mu$Jy) and only marginally at 233 GHz (20 ± 8 $\mu$Jy), about 40 d later. These results suggest either an optically thin sub-mm synchrotron spectrum, or highly variable sub-mm jet emission on month time-scales. Although the latter appears more likely, we note that, at the time of the ALMA observations, A0620–00 was in a somewhat less active optical-IR state than during all published multiwavelength campaigns when a flat-spectrum, partially self-absorbed jet has been suggested to extend from the radio to the mid-IR regime. Either interpretation is viable in the context of an internal shock model, where the jet’s spectral shape and variability are set by the power density spectrum of the shells’ Lorentz factor fluctuations. While strictly simultaneous radio–mm-IR observations are necessary to draw definitive conclusions for A0620–00, the data presented here, in combination with recent radio and sub-mm results from higher luminosity systems, demonstrate that jets from black hole X-ray binaries exhibit a high level of variability – either in flux density or intrinsic spectral shape, or both – across a wide spectrum of Eddington ratios. This is not in contrast with expectations from an internal shock model, where lower jet power systems can be expected to exhibit larger fractional variability owing to an overall decrease in synchrotron absorption.

Polarization and QPOs from jets in black hole systems

The historical observations of polarized jet emission for Blazars are reviewed and previous models are discussed. Motivated by this, a model for polarization of both steady and transient behavior using a helical magnetic field is presented. The variety of observed correlations and anti-correlations between the electric polarization angle, the degree of polarization and optical flux can be explained by this model. In addition, the phenomena of quasi-periodic oscillations (QPO) behavior seen in jets in X-ray Binaries (XRBs) is also explained by a model based on helical trajectories of emitting blobs and the resulting time scales and harmonics of the QPO are derived. In both the models, the input parameters are the inclination angle, the Lorentz factor of the jet and pitch angle of the magnetic helix.

Characterization of the infrared/X-ray subsecond variability for the black hole transient GX 339-4

We present a detailed analysis of the X-ray/IR fast variability of the Black-Hole Transient GX 339-4 during its low/hard state in August 2008. Thanks to simultaneous high time-resolution observations made with the VLT and RXTE, we performed the first characterisation of the sub-second variability in the near-infrared band - and of its correlation with the X-rays - for a low-mass X-ray binary, using both time and frequency-domain techniques. We found a power-law correlation between the X-ray and infrared fluxes when measured on timescales of 16 seconds, with a marginally variable slope, steeper than the one found on timescales of days at similar flux levels. We suggest the variable slope - if confirmed - could be due to the infrared flux being a non-constant combination of both optically thin and optically thick synchrotron emission from the jet, as a result of a variable self-absorption break. From cross spectral analysis we found an approximately constant infrared time lag of $\approx$ 0.1s, and a very high coherence of $\approx$ 90 per cent on timescales of tens of seconds, slowly decreasing toward higher frequencies. Finally, we report on the first detection of a linear rms-flux relation in the emission from a low-mass X-ray binary jet, on timescales where little correlation is found between the X-rays and the jet emission itself. This suggests that either the inflow variations and jet IR emission are coupled by a non-linear or time-variable transform, or that the IR rms-flux relation is not transferred from the inflow to the jet, but is an intrinsic property of emission processes in the jet.

Observational Signatures of Mass-loading in Jets Launched by Rotating Black Holes

Abstract It is widely believed that relativistic jets in X-ray binaries (XRBs) and active-galactic nuclei are powered by the rotational energy of black holes. This idea is supported by general-relativistic magnetohydrodynamic (GRMHD) simulations of accreting black holes, which demonstrate efficient energy extraction via the Blandford–Znajek mechanism. However, due to uncertainties in the physics of mass loading, and the failure of GRMHD numerical schemes in the highly magnetized funnel region, the matter content of the jet remains poorly constrained. We investigate the observational signatures of mass loading in the funnel by performing general-relativistic radiative transfer calculations on a range of 3D GRMHD simulations of accreting black holes. We find significant observational differences between cases in which the funnel is empty and cases where the funnel is filled with plasma, particularly in the optical and X-ray bands. In the context of Sgr A*, current spectral data constrains the jet filling only if the black hole is rapidly rotating with a ≳ 0.9. In this case, the limits on the infrared flux disfavor a strong contribution from material in the funnel. We comment on the implications of our models for interpreting future Event Horizon Telescope observations. We also scale our models to stellar-mass black holes, and discuss their applicability to the low-luminosity state in XRBs.

Electron transport with re-acceleration and radiation in the jets of X-ray binaries

This paper studies acceleration processes of background thermal electrons in X-ray binary jets via turbulent stochastic interactions and shock collisions. By considering turbulent magnetized jets mixed with fluctuation magnetic fields and ordered, large-scale one, and numerically solving the transport equation along the jet axis, we explore the influence of such as magnetic turbulence, electron injections, location of an acceleration region, and various cooling rates on acceleration efficiency. The results show that (1) the existence of the dominant turbulent magnetic fields in the jets is necessary to accelerate background thermal electrons to relativistic energies. (2) Acceleration rates of electrons depend on magnetohydrodynamic turbulence types, from which the turbulence type with a hard slope can accelerate electrons more effectively. (3) An effective acceleration region should be located at the distance $>10^3R_{\rm g}$ away from the central black hole ($R_{\rm g}$ being a gravitational radius). As a result of acceleration rates competing with various cooling rates, background thermal electrons obtain not only an increase in their energies but also their spectra are broadened beyond the given initial distribution to form a thermal-like distribution. (4) The acceleration mechanisms explored in this work can reasonably provide the electron maximum energy required for interpreting high-energy $\gamma$-ray observations from microquasars, but it needs to adopt some extreme parameters in order to predict a possible very high-energy $\gamma$-ray signal.