Probing Jet Compositions with Extreme Mass Ratio Binary Black Holes

Abstract

Determining whether black hole jets are dominated by leptonic or baryonic matter remains an open question in high-energy astrophysics. We propose that extreme mass ratio binary (EMRB) black holes, where an intermediate mass secondary black hole (a “miniquasar”) periodically interacts with the accretion flow of a supermassive black hole (SMBH), offer a natural laboratory to probe jet composition. In an EMRB, the miniquasar jet is launched episodically after each disk-crossing event, triggered by the onset of super-Eddington accretion. The resulting emissions exhibit temporal evolution as the jet interacts with the SMBH accretion disk. Depending on whether the jet is leptonic or hadronic in composition, the radiative signatures differ substantially. Notably, a baryonic jet produces a more pronounced gamma-ray output than a purely leptonic jet. By modeling the evolution of the multifrequency characteristic features, it is suggested that the gamma-ray-to-UV emissions may serve as a diagnostic tool capable of distinguishing between leptonic and baryonic scenarios. The resulting electromagnetic signals, when combined with multi-messenger observations, offer a powerful means to constrain the physical nature of relativistic jets from black holes.

1. Introduction

Relativistic jets from black hole systems are observed across a wide range of mass scales, from stellar-mass black holes, known as microquasars e.g., [1,2,3], to supermassive black holes that power active galactic nuclei (AGNs) e.g., [4,5,6,7]. AGN jets can extend over vast distances, sometimes spanning thousands to millions of light years, and have significant energy and momentum, capable of shaping their surrounding environments [5,8,9]. Together with complicated jet acceleration e.g., [10] and particle acceleration within the jet e.g., [11], the jet composition, whether dominated by leptons or hadrons, plays a crucial role in determining how energy is transported, dissipated and transferred. Due to their differing masses and interaction cross sections with ambient media such as intracluster medium, leptons and hadrons affect the jet’s energy dissipation and acceleration mechanisms in fundamentally different ways [12,13,14]. In addition, the composition of the cosmic ray can also be related to the particles available within the jet and experience further acceleration [15,16,17].

The spectral properties of AGNs, particularly blazars e.g., [6,15,18], provide information on the composition of the jet. A typical blazar spectrum displays two distinct peaks. It is widely accepted that leptons produce the lower-frequency peak through synchrotron radiation [6]. However, the source of the higher-frequency peak at gamma-ray range remains unclear. This ambiguity is due to the possibility that gamma-ray emissions could either result from inverse Compton (IC) scattering by leptons or through specific hadronic processes, such as the decay of pions leading to gamma-ray production.

Compared to the jets from steady-state, isolated sources, jets originating from black hole systems might offer valuable insights into jet composition, for example, time-varying systems with episodic, transient jets from binary systems, such as OJ 287 [19,20], GRO 1655-40 [21], XTE J1550-564 [22], or systems where jets interact with the surrounding matter [13,14]. Motivated by this paradigm, we investigate the emission characteristics of periodic jets in extreme mass ratio binary black holes (EMRBs). We focus on EMRBs with a mass ratio $q=M_{\mathrm{secondary}}/M_{\mathrm{primary}}<{10}^{-4}\mathrm{to}{10}^{-6}$ consisting of an SMBH with $M_{\mathrm{SMBH}}={10}^6\mathrm{to}{10}^9{\mathrm{M}}_{\odot }$ as the primary and a secondary black hole with $M_{\mathrm{mq}}={10}^2\mathrm{to}{10}^5{\mathrm{M}}_{\odot }$ (hereafter, we refer to such intermediate mass black holes as “miniquasars”).

These EMRB systems provide a promising avenue for probing the composition of relativistic jets through their associated radiative signatures, since transient jet-launching and jet–ambient interactions naturally occur periodically. Figure 1 displays a schematic plot for the EMRB of interest, in which the semi-major axis extends beyond the boundary of the SMBH disk (∼${10}^3{R}_{\mathrm{g}}$, where $R_{\mathrm{g}}\equiv G{M}_{\mathrm{SMBH}}/c^2$). The configuration enables periodic interactions between the miniquasar and the thin accretion disk around the SMBH, and the lauching and quenching of the miniquasar jet. As the miniquasar traverses the SMBH accretion flow (between phases (a) and (b) in Figure 1), it can capture material from the SMBH accretion disk. This captured mass is subsequently accreted by the miniquasar, thus initiating the miniquasar jet (phase (b) in Figure 1). Before the miniquasar reaches the apocenter, the mass is already depleted, leading to the cessation of the jet (phase (c) in Figure 1). The transient accretion considered here is different from the continuous accretion when the secondary is embedded in the SMBH accretion disk due to capture by dynamical interactions of nuclear star clusters within the AGN disk e.g., [23,24], or because the secondary orbits the AGN with an untilted circular orbit during the inspiral phase [25]. In addition, the emission features due to the encounter between the accretion disk of an SMBH with another SMBH are studied in [26,27].

In this work, we specifically focus on emissions resulting from the interaction between the miniquasar jet and the SMBH accretion disk (as illustrated by phase (c) in Figure 1) and its evolution. By examining the orbital-phase-dependent multi-wavelength emissions during the jet–disk collision, we demonstrate how the light curves of the UV and gamma-ray power would vary with different jet compositions. The inherent orbital modulation in these binaries offers a unique advantage over jets from isolated AGNs or standalone miniquasars, breaking the spectral degeneracies that typically hinder efforts to constrain jet composition in steady-state observations.

For EMRBs with SMBH $M_{\mathrm{SMBH}}$ mass and semi-major axis a, the orbital period is

$$T=2\pi \sqrt{{\displaystyle \frac{a^3}{G{M}_{\mathrm{SMBH}}}}}.$$

(1)

The orbital period for EMRBs with different $M_{\mathrm{SMBH}}$ and a is presented in Figure 2. The orbital frequencies of the EMRBs of interest here, with its apocenter outside the edge of the SMBH disk and periods typically on the order of years, lie within the sensitivity band of the pulsar timing arrays (PTAs) [28,29,30,31], as indicated by the blue-shaded region in Figure 2. The EMRBs considered here are the progenitors of extreme mass ratio inspiral (EMRI) systems. With AGN-assist dynamics and the decay of orbital energy by gravitational waves, the systems gradually spiral and transition into EMRIs [32], eventually reaching higher frequencies detectable by the LISA [33,34], as indicated by the red-shaded region.

The paper is organized as follows. In Section 2, we provide an overview of the leptonic and hadronic composition of jets, and related radiation mechanisms. In Section 3, we present our method and results for modeling the phased resolved jet emission features, including the computation of orbits and estimation of leptonic and hadronic emissions from the jet. The summary and final remarks are provided in Section 4.

2. Jet Composition and Radiation Mechanism

2.1. Jet Composition

Based on the observed synchrotron emissions below gamma-ray [6,15], electrons are identified as an essential component of the jet composition. The number density ratio between different species, electrons ${\mathrm{e}}^{-}$, positron ${\mathrm{e}}^{+}$, and protons $\mathrm{p}$ are denoted by

$$\begin{array}{cc}{\xi }_{{\mathrm{e}}^{+}}& \equiv {\displaystyle \frac{n_{{\mathrm{e}}^{+}}}{n_{{\mathrm{e}}^{-}}}},\\ {\xi }_{\mathrm{p}}& \equiv {\displaystyle \frac{n_{\mathrm{p}}}{n_{{\mathrm{e}}^{-}}}},\end{array}$$

(2)

and charge neutrality require

$${\xi }_{{\mathrm{e}}^{+}}+{\xi }_{\mathrm{p}}=1.$$

(3)

The total intrinsic jet power $P_{\mathrm{jet}}$ can therefore be decomposed into powers carried by electrons $P_{{\mathrm{e}}^{-}}$, positron $P_{{\mathrm{e}}^{+}}$, magnetic field $P_{\mathrm{B}}$, and protons $P_{\mathrm{p}}$:

$$P_{\mathrm{jet}}=P_{{\mathrm{e}}^{-}}+P_{{\mathrm{e}}^{+}}+P_{\mathrm{p}}+P_{\mathrm{B}}.$$

(4)

After normalizing the decomposed power with $P_{{\mathrm{e}}^{-}}$ by

$$\begin{array}{cc}{P}_{{\mathrm{e}}^{+}}& ={\xi }_{{\mathrm{e}}^{+}}{P}_{{\mathrm{e}}^{-}},\hfill \\ P_{\mathrm{p}}& ={\displaystyle \frac{{\mathrm{m}}_p\langle {\gamma }_{\mathrm{p}}\rangle }{{\mathrm{m}}_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}{P}_{{\mathrm{e}}^{-}},\hfill \\ P_{\mathrm{B}}& =P_{{\mathrm{e}}^{-}}+P_{{\mathrm{e}}^{+}},\hfill \end{array}$$

(5)

The total jet power, Equation (4), can therefore be rewritten as

$$P_{\mathrm{jet}}=P_{{\mathrm{e}}^{-}}\left(2+2{\xi }_{{\mathrm{e}}^{+}}+{\displaystyle \frac{{\mathrm{m}}_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{{\mathrm{m}}_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}\right).$$

(6)

In the above formula, $m_{\mathrm{p}}/m_{\mathrm{e}}(\approx$1836) is the mass ratio between a proton and an electron, $\langle {\gamma }_{\mathrm{e}}\rangle$ and $\langle {\gamma }_{\mathrm{p}}\rangle$ are the average particle Lorentz factors for electrons and positrons and protons in the jet comoving frame, respectively, and the equipartition between magnetic energy and lepton energy is adopted. Assuming the emissions of the jet iarecharacterized by a power-law spectrum with power-law index $\overline{\alpha }$, with the view angle $\theta$ and jet bulk Lorentz factor $\mathrm{\Gamma }={(1-{\beta }^2)}^{-1/2}$, where $\beta$ is the ratio of the speed of the jet and the speed of light, the observed jet power $P_{\mathrm{jet}}^{\mathrm{obs}}$ is related to the intrinsic jet power by

$$P_{\mathrm{jet}}^{\mathrm{obs}}={\delta }^{3+\overline{\alpha }}{P}_{\mathrm{jet}},$$

(7)

with the Doppler factor

$$\delta ={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\beta cos\theta )}}.$$

(8)

Furthermore, the power ratio of different species can be defined by

$$\begin{array}{cc}{\eta }_{{\mathrm{e}}^{-}}& \equiv {\displaystyle \frac{{\mathrm{P}}_{{\mathrm{e}}^{-}}}{{\mathrm{P}}_{\mathrm{jet}}}},\hfill \\ {\eta }_{{\mathrm{e}}^{+}}& \equiv {\displaystyle \frac{{\mathrm{P}}_{{\mathrm{e}}^{+}}}{{\mathrm{P}}_{\mathrm{jet}}}}={\xi }_{{\mathrm{e}}^{+}}{\eta }_{{\mathrm{e}}^{-}},\hfill \\ {\eta }_{\mathrm{p}}& \equiv {\displaystyle \frac{{\mathrm{P}}_{\mathrm{p}}}{{\mathrm{P}}_{\mathrm{jet}}}}={\displaystyle \frac{{\mathrm{m}}_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{{\mathrm{m}}_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}{\eta }_{{\mathrm{e}}^{-}}.\hfill \end{array}$$

(9)

Therefore, the energy content of the jet depends on the kinetic energy ratio between baryons and leptons $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$. Figure 3 displays the relative contribution of the hardronic components of the jet power, ${\eta }_{\mathrm{p}}$, with different ratios of $\langle {\gamma }_p\rangle /\langle {\gamma }_e\rangle$. The higher $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$ or ${\xi }_p$, the more jet power is carried by the baryons.

Based on the mass content of the species within the jet, three potential configurations can be classified:

• Pair-Dominated/Leptonic Jet: The mass of a pair-dominated jet is dominated by electrons and positrons, with a negligible contribution from protons ${\xi }_{\mathrm{p}}\simeq 0$. The efficiency of leptonic loading into jets can be related to pair loading $\gamma +\gamma \to e^{-}+e^{+}$ with MeV photons [35,36,37].
• Baryonic Jet: A baryonic jet’s mass is dominated by its baryonic (hadronic) component: protons and/or heavier nuclei. For example, the shaded region in Figure 3 indicates a case when the barynoic component contributes $>99.9\%$ of the total mass, and a representative baryonic jet. A commonly adopted pure electron–proton jet with ${\xi }_{\mathrm{p}}=1$ belongs to this model. In a baryonic jet, although the mass is predominantly composed of baryonic matter, the energy content may not be similarly dominated. This occurs when leptons have significantly higher energy $\langle {\gamma }_{\mathrm{e}}\rangle \gg \langle {\gamma }_{\mathrm{p}}\rangle$, as illustrated in Figure 3. The efficient acceleration of the jet’s hadronic components is crucial for generating ultra-high-energy cosmic rays (UHECR) [38] and UHE neutrinos [39]. Neutrino-producing blazars (such as TXS 0506+056 [40,41]) offer corroborative evidence of a hadronic component in the jet.
• Hybrid Jet: A hybrid lepton–hadronic jet has a mass composition between a pure pair-dominated/leptonic jet and a hadronic jet, encompassing the most general jet composition, which includes electrons, positrons, and baryons. Multifrequency and multi-messenger modeling efforts for the analysis of blazar observations usually include all these components [18].

2.2. Leptonic Interactions and Associated Radiative Processes

The radiative processes associated with leptons include synchrotron, bremsstrahlung, and IC emission. The synchrotron radiation typically peaks at a frequency of ${\nu }_{\mathrm{syn}}\sim {10}^6{\gamma }^{2}B\mathrm{Hz}$, where B is the magnetic field strength in Gauss, and $\gamma$ is the Lorentz factor of the leptons; see, e.g., [42]. Accurate modeling of the spectral energy distribution (SED) requires not only the underlying radiative mechanisms but also the energy distribution of the lepton population. Rather than assuming monoenergetic leptons, it is essential to specify the number density as a function of energy for the ensemble of leptons.

In the IC process, relativistic leptons scatter incoming soft photons, possibly through single or multiple scattering events. A single scattering can boost the photon energy by a factor of up to ∼${\gamma }^2$ relative to the frequency of seed photons [43]. Synchrotron radiation is inherently highly polarized, and Compton scattering can further enhance the polarization of the upscattered high-energy photons [42,43], providing potential diagnostics for both the lepton population and the magnetic field configuration [44].

Bremsstrahlung emission occurs when leptons are accelerated in the Coulomb fields of ambient charged particles. Although bremsstrahlung may contribute significantly to the SED of hot and dense accretion flows [45], it is generally negligible in jet environments. This is due to the low particle densities in jets and the suppression of bremsstrahlung cross sections at relativistic lepton energies [46].

2.3. Hadronic Interactions and Associated Radiative and Particle Processes

The hadron can interact with the photon (photo–hadronic, $\mathrm{p}\gamma$, process) and other hadrons (proton–proton, pp, interaction). The interactions of the hadronic process facilitate the formation of pions (${\pi }^0,{\pi }^{+},{\pi }^{-}$), whose decay subsequently results in the production of gamma rays from ${\pi }^0$ and leptons and neutrinos from ${\pi }^{+}$ and ${\pi }^{-}$. Direct production channels related to pion production include [42,47]

$$\mathrm{p}+\gamma \to {\Delta }^{+}\to \left\{\begin{array}{c}\mathrm{p}+{\pi }^0\hfill \\ \mathrm{n}+{\pi }^{+}\hfill \end{array}\right.,$$

(10)

and

$$\begin{array}{c}\hfill \mathrm{p}+\mathrm{p}\to \left\{\begin{array}{c}\mathrm{p}+\mathrm{p}+{\pi }^0\hfill \\ \mathrm{p}+\mathrm{p}+{\pi }^{+}+{\pi }^{-}\hfill \\ \mathrm{p}+\mathrm{n}+{\pi }^{+}\hfill \end{array}\right..\end{array}$$

(11)

The decay of the neutral pion produced in the above mechanism then results in strong gamma-ray emissions

$$\begin{array}{c}\hfill {\pi }^0\to \gamma +\gamma ,\end{array}$$

(12)

and charged pions produce secondary leptons and neutrinos:

$$\begin{array}{cc}\hfill {\pi }^{+}\to & {\mu }^{+}+{\nu }_{\mu }\hfill \\ \hfill & \downarrow \hfill \\ \hfill & {\mathrm{e}}^{+}+{\nu }_{\mathrm{e}}+{\overline{\nu }}_{\mu }\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\pi }^{-}\to & {\mu }^{-}+{\overline{\nu }}_{\mu }\hfill \\ \hfill & \downarrow \hfill \\ \hfill & {\mathrm{e}}^{-}+{\overline{\nu }}_{\mathrm{e}}+{\nu }_{\mu }\hfill \end{array}$$

(14)

The relative importance of these hadronic processes differs across black hole mass scales. The neutral pion production threshold in p$\gamma$ intereactions is set by the photon energy at approximately 140 MeV in the proton rest frame. In the lab frame, assuming head-on collisions, a criterion of $2{\gamma }_{\mathrm{p}}{E}_{\gamma }\ge 140$ MeV is required, which implies the need for a high-energy photon field at approximately hundreds of MeV. Both the p$\gamma$ and the pp channels are expected for AGN jets, within which UHECR can be produced e.g., [48,49]. In comparison, miniquasar jets can accelerate protons less efficiently, and the gamma-ray emissions from the hadronic process would mainly be contributed by the pp process.

3. EMRB as a Probe to Jet Composition

The periodic interactions between the accretion flow of the primary SMBH and the secondary miniquasar present a distinctive opportunity to examine the composition of the miniquasar jet. As illustrated in phase (b) of Figure 1, the collision between the miniquasar jet and the accretion flow of the SMBH can lead to significant variations in emission characteristics, with specific details associated with the jet’s composition. For instance, as opposed to a pair-dominated jet, the collision can result in proton–proton interactions within a baryonic jet.

In the following, after systematically investigating the parameter space for EMRBs of interest, we model the orbital dynamics and the UV and gamma-ray light curves dependent on orbital movement for a representative scenario, and perform a comparative analysis of the emission signatures that correspond to different jet composition.

3.1. Orbital Parameter Space

In addition to the semi-major axis a (see Figure 2), the orbital characteristics of the miniquasar are also governed by its eccentricity e. Figure 4 presents the pericenter $r_{\mathrm{peri}}=a(1-e)$ and apocenter $r_{\mathrm{apo}}=a(1+e)$ within the parameter space. As demonstrated in Figure 1, our focus is on EMRBs within which the miniquasar undergoes repeated interactions with SMBH accretion by entering and leaving the SMBH accretion disk. With an outer boundary of the SMBH disk of ∼${10}^3{R}_g$, the shaded region in Figure 1 satisfies these criteria.

With a given semi-major axis a, eccentricity e, and the mass of SMBH $M_{\mathrm{SMBH}}$, the orbit of the miniquasar can be computed by Kepler’s equation [50,51], which relates the mean anomaly $M\left(t\right)$ to the eccentric anomaly $E\left(t\right)$ via

$$\begin{array}{cc}\hfill M\left(t\right)& =E\left(t\right)-esinE\left(t\right).\hfill \end{array}$$

(15)

The mean anomaly evolves linearly in time:

$$\begin{array}{cc}\hfill M\left(t\right)& ={\displaystyle \frac{2\pi }{T}}(t-t_0),\hfill \end{array}$$

(16)

with T being the orbital period and $t_0$ the time of pericenter passage, which is set to $t_0=0$ without loss of generality. Once $E\left(t\right)$ is determined numerically, the radius to the SMBH can be recovered from

$$\begin{array}{cc}\hfill r\left(t\right)& =a(1-ecosE(t\left)\right).\hfill \end{array}$$

(17)

Together, these equations provide a complete parametric description of orbital motion, linking geometrical shape with temporal evolution.

3.2. Modeling Multi-Frequency Emissions of the Miniquasar Jet

The emission of jets at different frequencies can be quantified by evaluating the transformation of jet power into emission power, guided by the radiation mechanisms described in Section 2.2 and Section 2.3. We focus on the energetic event during the miniquasar jet’s interaction with the SMBH accretion (phase (b) in Figure 1). In such cases, the magnetic field strength and the photon field are set by the local conditions of the accretion disk.

For a miniquasar jet interacting with the thin disk of a SMBH, without losing generality, we consider a bulk Lorentz factor of $\mathrm{\Gamma }\simeq 5$ cf. [1]. The average particle Lorentz factors have values in the range $\langle {\gamma }_{\mathrm{e}}\rangle \sim {10}^{2-3}$ and $\langle {\gamma }_{\mathrm{p}}\rangle \sim {10}^{0-1}$ (cold protons). We estimate the lepton emissions as follows (see also Section 3.2.1). For leptons with a Lorentz factor $\gamma \sim \mathrm{\Gamma }\langle {\gamma }_{\mathrm{e}}\rangle \sim {10}^3$ and a magnetic field strength of $B\sim {10}^{2-3}\mathrm{G}$ in the accretion disk, the synchrotron frequency ${\nu }_{\mathrm{syn}}\sim {10}^6{\gamma }^{2}B\left(\mathrm{Gauss}\right)$ would be ∼${10}^{14-17}$ Hz, which is in the optical/ultraviolet (UV) bands in the co-moving frame moving of the accretion flow. The UV radiation emitted by the disk can be enhanced through IC scattering. As a rough estimate, the frequency of Compton-scattered radiation (with a single scattering event) is ${\nu }_{\mathrm{IC}}\sim {10}^{20-24}\mathrm{Hz}$. This frequency corresponds to energies in the MeV/GeV range.

For the hadronic process, the energy of the gamma rays from the decay of ${\pi }^0$ is around $0.1E_{\mathrm{p}}$, where $E_{\mathrm{p}}\sim \mathrm{\Gamma }\langle {\gamma }_{\mathrm{p}}\rangle m_{\mathrm{p}}{c}^2$. The output gamma-ray also contributes to the GeV photonic radiation. Secondary leptons from the decay of charged pion possess an energy similar to that of $0.1E_{\mathrm{p}}$, corresponding to $\langle {\gamma }_{\mathrm{e}}^{\mathrm{secondary}}\rangle \sim {10}^3$, which is thus similar to that of the primary leptons.

With the above-mentioned potential output radiation ranges in mind, the relative power between optical/UV and MeV/GeV emissions indicates the relative amounts of leptons and hadrons in the jet. For the purpose of examining the general multi-frequency emission characteristics, we do not further distinguish between the optical/UV and MeV/GeV bands in the subsequent discussion, and treat the synchrotron emissions as UV radiation, and both the IC and neutral pion decay emissions as GeV gamma-ray radiation. A comprehensive SED modeling, which requires detailed input energy spectra of the relevant particles and photon fields, is deferred to future work.

The related notations for the later estimation of UV and gamma-ray power are summarized in

Table 1. Summary of notations related to composition of jet content, jet power, and radiation power.

Notation	Definition	Reference
${\xi }_{{\mathrm{e}}^{+}}$	positron to electron ratio	Equation (2)
${\xi }_{\mathrm{p}}$	proton to electron ratio	Equation (2)
${\eta }_{{\mathrm{e}}^{-}}$	jet power carried by electron	Equation (9)
${\eta }_{{\mathrm{e}}^{+}}$	jet power carried by positron	Equation (9)
${\eta }_{\mathrm{p}}$	jet power carried by proton	Equation (9)
$f_{\mathrm{IC}}$	lepton power to $\gamma$-ray power (via IC process) conversion rate	Equations (18) and (27)
$f_{\mathrm{syn}}$	lepton power to UV power (via synchrotron process) conversion rate	Equations (19) and (28)
$f_{pp}$	efficiency of pp interaction	Equation (25)
$f_{{\pi }^0\to \gamma -\mathrm{ray}}$	neutral pion power to $\gamma$-ray power conversion rate	Equation (26)
$f_{{\pi }^{\pm }\to e^{\pm }\to \gamma -\mathrm{ray}}$	charged pion power to $\gamma$-ray power (via IC process of secondary leptons) conversion rate	Equation (27)
$f_{{\pi }^{\pm }\to e^{\pm }\to \mathrm{UV}}$	charged pion power to UV power (via synchrotron process of secondary leptons) conversion rate	Equation (28)

: the electromagnetic properties of the jet emissions from miniquasars are influenced by three main factors. These are the jet composition, symbolized by ${\xi }_{(\dots )}$ (see Equation (2)); the portion of power attributed to different types of particles, represented by ${\eta }_{(\dots )}$ (see Equation (9)); and the efficiency of the conversion of particle energy, indicated by $f_{(\dots )}$. The subscript _(…) specifies the associated species or radiative processes.

3.2.1. Leptonic Emission

The gamma-ray emissions contributed from the lepton composition (both electron and positron) of the jet are generally associated with the IC process. The estimated gamma-ray power is

$$\begin{array}{c}\hfill P_{{\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}\approx f_{\mathrm{IC}}(P_{{\mathrm{e}}^{-}}+P_{{\mathrm{e}}^{+}})=f_{\mathrm{IC}}(1+{\xi }_{{\mathrm{e}}^{+}}){\eta }_{{\mathrm{e}}^{-}}{P}_{\mathrm{jet}}.\end{array}$$

(18)

The energy conversion rate, denoted as $f_{\mathrm{IC}}$, refers to the process by which electrons and positrons transfer their energy to photons through the IC process. While the leptonic energy can be converted into scattered, gamma-ray photons, part of the energy can also be converted to synchrotron, UV photons, with the power

$$\begin{array}{c}\hfill P_{{\mathrm{e}}^{\pm }\to \mathrm{UV}}\approx f_{\mathrm{syn}}(1+{\xi }_{{\mathrm{e}}^{+}}){\eta }_{{\mathrm{e}}^{-}}{P}_{\mathrm{jet}}.\end{array}$$

(19)

The ratio between UV and gamma-ray power can be estimated as the ratio between the energy of the ambient radiation field $U_{\mathrm{ph}}$ and the local magnetic field energy $U_{\mathrm{B}}$:

$$\begin{array}{c}\hfill {\displaystyle \frac{P_{{\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}}{P_{{\mathrm{e}}^{\pm }\to \mathrm{UV}}}}={\displaystyle \frac{f_{\mathrm{IC}}}{f_{\mathrm{syn}}}}\approx {\displaystyle \frac{U_{\mathrm{ph}}}{U_{\mathrm{B}}}}.\end{array}$$

(20)

The conversion of lepton energy to emissions via the IC or synchrotron processes is not entirely efficient. Assuming that the sum of the energy conversion from the processes is approximately constant, as denoted by $\mathcal{Q}$, we impose the following constraint:

$$\begin{array}{c}\hfill f_{\mathrm{IC}}+f_{\mathrm{syn}}=\mathcal{Q}<1.\end{array}$$

(21)

This condition reflects the fact that only a fraction of the available particle energy is radiated away via synchrotron and IC processes, with the remainder either being retained by particles or lost through non-radiative channels such as adiabatic expansion or escape.

3.2.2. Hadronic Emission

The gamma-ray emissions that emanate from the baryonic jet are not only derived from primary electrons but also arise through hadronic processes, such as pp or p$\gamma$ interactions. The interactions of the hadronic process facilitate the formation of pions ${\pi }^0,{\pi }^{+},{\pi }^{-}$, whose decay subsequently results in the production of gamma rays from ${\pi }^0$ and electrons/positrons from ${\pi }^{+},{\pi }^{-}$. As discussed in Section 2.3, the pp interaction between the miniquasar jet and the SMBH accretion flow emerges as a more plausible mechanism for the production of pion. The produced gamma-ray power from ${\pi }^0$ decay (with energy conversion rate $f_{{\pi }^0\to \gamma -\mathrm{ray}}$) and secondary leptons (with energy conversion rate $f_{{\pi }^{\pm }\to e^{\pm }\to \gamma -\mathrm{ray}}$) can be estimated by

$$\begin{array}{cc}{P}_{\mathrm{p}\to \gamma -\mathrm{ray}}\hfill & \approx f_{\mathrm{pp}}({\displaystyle \frac{1}{3}}{f}_{{\pi }^0\to \gamma -\mathrm{ray}}+{\displaystyle \frac{2}{3}}{f}_{{\pi }^{\pm }\to e^{\pm }\to \gamma -\mathrm{ray}})P_{\mathrm{p}}\hfill \\ \hfill & =f_{\mathrm{pp}}({\displaystyle \frac{1}{3}}{f}_{{\pi }^0\to \gamma -\mathrm{ray}}+{\displaystyle \frac{2}{3}}{f}_{{\pi }^{\pm }\to e^{\pm }\to \gamma -\mathrm{ray}}){\displaystyle \frac{{\mathrm{m}}_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{{\mathrm{m}}_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}{\eta }_{{\mathrm{e}}^{-}}{P}_{\mathrm{jet}},\hfill \end{array}$$

(22)

where $f_{\mathrm{pp}}$ is the efficiency of the pp interactions. The coefficients $1/3$ and $2/3$ in Equation (22) are estimated by the cross section of the pion production in the pp collision [42,47]. The secondary leptons from pion decay can also contribute to synchrotron emissions at the UV band:

$$P_{\mathrm{p}\to \mathrm{UV}}\approx f_{\mathrm{pp}}\left({\displaystyle \frac{2}{3}}{f}_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \mathrm{UV}}\right){\displaystyle \frac{{\mathrm{m}}_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{{\mathrm{m}}_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}{\eta }_{{\mathrm{e}}^{-}}{P}_{\mathrm{jet}},$$

(23)

where $f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \mathrm{UV}}$ represents the energy conversion rate to UV emissions. The various energy conversion rates, $f_{(\dots )}$, are estimated in the following.

To approximate $f_{\mathrm{pp}}$, one can examine the correlation between the dynamical timescale $t_{\mathrm{dyn}}$ and the interaction timescale $t_{\mathrm{pp}}$:

$$f_{\mathrm{pp}}=\min (1,{\displaystyle \frac{t_{\mathrm{dyn}}}{t_{\mathrm{pp}}}})=\min (1,{\displaystyle \frac{H/\left(\mathrm{\Gamma }c\right)}{1/\left(n_{\mathrm{p}}{\sigma }_{\mathrm{pp}}c\right)}}).$$

(24)

The estimated dynamical timescale is roughly $H/\left(\mathrm{\Gamma }c\right)\sim \left(0.01R\right)/\left(\mathrm{\Gamma }c\right)\sim {10}^2\mathrm{s}$, assuming that a jet with $\mathrm{\Gamma }=5$ passes through the SMBH accretion disk at $R\sim 100G{M}_{\mathrm{SMBH}}/c^2\sim {10}^{15}\mathrm{cm}$ for $M_{\mathrm{SMBH}}={10}^8{\mathrm{M}}_{\odot }$, and that the disk height is $H\sim 0.01R$. The estimated interaction timescale is $t_{\mathrm{pp}}<1\mathrm{s}$, by adopting the cross section ${\sigma }_{\mathrm{pp}}\sim 30$ mb ($1\mathrm{mb}={10}^{-27}$ cm⁻²) [52], and $n_{\mathrm{p}}\sim {10}^{16-18}{\mathrm{cm}}^{-3}$ (as estimated by the number density of the thin disk solution for the primary SMBH [53,54]). Therefore, we estimate

$$\begin{array}{c}\hfill f_{\mathrm{pp}}\approx 1.\end{array}$$

(25)

In addition, the decay of neutral pions into gamma rays is highly efficient:

$$\begin{array}{c}\hfill f_{{\pi }^0\to \gamma \text{-ray}}\approx 1.\end{array}$$

(26)

Equations (25) and (26) suggest that the interaction between the jet and disk allows for a highly efficient transformation of the proton power into gamma-ray emissions, highlighting a key characteristic of the EMRBs under study.

In comparison, the efficiency of converting charged pion energy to radiation includes the transformation of charged pion energy into lepton energy, denoted as $f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }}(\approx$$0.25$ [47,55]), followed by the conversion of lepton energy into IC or synchrotron radiation energy. We estimate the following efficiency:

$$\begin{array}{c}\hfill f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}=f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }}{f}_{\mathrm{IC}}\approx 0.25f_{\mathrm{IC}}\end{array}$$

(27)

and

$$\begin{array}{c}\hfill f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \mathrm{UV}}=f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }}{f}_{\mathrm{syn}}\approx 0.25f_{\mathrm{syn}}.\end{array}$$

(28)

The secondary leptons therefore follow a similar relation as in Equation (20):

$$\begin{array}{c}\hfill {\displaystyle \frac{f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}}{f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \mathrm{UV}}}}={\displaystyle \frac{f_{\mathrm{IC}}}{f_{\mathrm{syn}}}}\approx {\displaystyle \frac{U_{\mathrm{ph}}}{U_{\mathrm{B}}}}.\end{array}$$

(29)

It is assumed that the constraint for $f_{\mathrm{IC}}$ and $f_{\mathrm{syn}}$, Equation (21), also applies to the secondary leptons, and therefore

$$\begin{array}{c}\hfill f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}+f_{{\pi }^{\pm }\to {\mathrm{e}}^{\pm }\to \mathrm{UV}}\approx 0.25\mathcal{Q}.\end{array}$$

(30)

3.2.3. Estimation of Emission Efficiency and Multi-Frequency Emission Ratio

With the conversion rates from the particle energy to the radiation, $f_{(\dots )}$, we can estimate the radiation power of jets with a different composition. The multi-wavelength radiation power from the jet power carried by the leptonic and hadronic components is, respectively,

$$\begin{array}{c}\hfill {\displaystyle \frac{P_{{\mathrm{e}}^{-}\to \gamma -\mathrm{ray}}+P_{{\mathrm{e}}^{-}\to \mathrm{UV}}}{P_{{\mathrm{e}}^{-}}}}={\displaystyle \frac{P_{{\mathrm{e}}^{+}\to \gamma -\mathrm{ray}}+P_{{\mathrm{e}}^{+}\to \mathrm{UV}}}{P_{{\mathrm{e}}^{+}}}}=f_{\mathrm{IC}}+f_{\mathrm{syn}}=\mathcal{Q}<1,\end{array}$$

(31)

and

$${\displaystyle \frac{P_{\mathrm{p}\to \gamma \text{-ray}}+P_{\mathrm{p}\to \mathrm{UV}}}{P_{\mathrm{p}}}}\approx f_{pp}\left[{\displaystyle \frac{1}{3}}{f}_{{\pi }^0\to \gamma \text{-ray}}+{\displaystyle \frac{2}{3}}\left(f_{{\pi }^{\pm }\to e^{\pm }\to \gamma \text{-ray}}+f_{{\pi }^{\pm }\to e^{\pm }\to \mathrm{UV}}\right)\right]=\left({\displaystyle \frac{1}{3}}+{\displaystyle \frac{\mathcal{Q}}{6}}\right)<1.$$

(32)

The ratios are constant if $\mathcal{Q}$ is constant over time.

For a hybrid jet, the estimated gamma-ray-to-UV emission ratio is

$$\begin{array}{c}\hfill {\displaystyle \frac{P_{\gamma -\mathrm{ray}}}{P_{\mathrm{UV}}}}={\displaystyle \frac{P_{\mathrm{p}\to \gamma -\mathrm{ray}}+P_{{\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}}{P_{\mathrm{p}\to \mathrm{UV}}+P_{{\mathrm{e}}^{\pm }\to \mathrm{UV}}}}\approx {\displaystyle \frac{({\displaystyle \frac{1}{3}}+{\displaystyle \frac{f_{\mathrm{IC}}}{6}}){\displaystyle \frac{m_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{m_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}+(1+{\xi }_{{\mathrm{e}}^{+}})f_{\mathrm{IC}}}{{\displaystyle \frac{\mathcal{Q}-f_{\mathrm{IC}}}{6}}{\displaystyle \frac{m_{\mathrm{p}}\langle {\gamma }_{\mathrm{p}}\rangle }{m_{\mathrm{e}}\langle {\gamma }_{\mathrm{e}}\rangle }}{\xi }_{\mathrm{p}}+(1+{\xi }_{{\mathrm{e}}^{+}})(\mathcal{Q}-f_{\mathrm{IC}})}},\end{array}$$

(33)

which is related to the mass content, ${\xi }_{\mathrm{p}}$ and ${\xi }_{{\mathrm{e}}^{+}}(=1-{\xi }_{\mathrm{p}})$, as well as the energy content, $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$, within the jet. In Figure 5 we present the ratio of gamma-ray-to-UV power with different mass and energy content, assuming $\mathcal{Q}=0.5$ (50% leptonic power within the jet goes to radiation power) and $f_{\mathrm{IC}}=0.8\mathcal{Q}$ (the IC process dominates the leptonic radiation). When ${\xi }_{\mathrm{p}}\to 0$, the jet is pair-dominated and all cases reduced to

$$\begin{array}{c}\hfill {\displaystyle \frac{P_{{\mathrm{e}}^{\pm }\to \gamma -\mathrm{ray}}}{P_{{\mathrm{e}}^{\pm }\to \mathrm{UV}}}}={\displaystyle \frac{f_{\mathrm{IC}}}{f_{\mathrm{syn}}}}\approx {\displaystyle \frac{f_{\mathrm{IC}}}{\mathcal{Q}-f_{\mathrm{IC}}}},\end{array}$$

(34)

which has a value $f_{\mathrm{IC}}/(\mathcal{Q}-f_{\mathrm{IC}})=4$ for this setup. As the ratio $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$ increases, protons account for a greater portion of the jet’s power (refer to Figure 3), which also results in a higher gamma-ray-to-UV power ratio. This is due to the effective gamma-ray production by pp collisions.

3.3. Two-Stage Accretion and Jet Power

As the miniquasar undergoes recurrent orbital phases, episodic accretion and jet-launching are expected to occur when it passes through the dense region of the SMBH accretion flow. An estimate of the passing time for the miniquasar with orbital velocity $v_{\mathrm{orb}}$ to pass through the SMBH disk with disk height H, $t_{\mathrm{pass}}=H/v_{\mathrm{orb}}$, is multiple hours1. Assuming the viscous time scale $t_{\mathrm{visc}}$ is much longer than the passing time (e.g., months), the accretion of the miniquasar can be considered using a two-stage process. The first stage is mass capture through Bondi–Hoyle accretion [56,57] while it crosses the SMBH accretion flow with a supersonic velocity. The captured mass $\Delta M_{\mathrm{cap}}$ is approximately given by

$$\Delta M_{\mathrm{cap}}\approx {\rho }_{\mathrm{disk}}\times \pi {\left({\displaystyle \frac{2G{M}_{\mathrm{mq}}}{v_{\mathrm{rel}}^2}}\right)}^2\times v_{\mathrm{rel}}\times t_{\mathrm{pass}}$$

(35)

where ${\rho }_{\mathrm{disk}}$ is the local gas density, $M_{\mathrm{mq}}$ is the miniquasar mass, $v_{\mathrm{rel}}\sim v_{\mathrm{orb}}$ is the relative velocity between the miniquasar and the disk gas, and $t_{\mathrm{pass}}$ is the disk crossing timescale. The captured mass is related to ${\rho }_{\mathrm{disk}}$, $M_{\mathrm{mq}}$, and $H(\simeq$$v_{\mathrm{rel}}\times t_{\mathrm{pass}})$. The capture radius, $2G{M}_{\mathrm{mq}}/v_{\mathrm{rel}}^2$, is roughly ten times smaller than the disk height. In general, for a fixed mass and accretion rate of SMBH, $\Delta M_{\mathrm{cap}}$ increases with increasing pericenter radius due to the combined effects of orbital velocity, disk density, and disk height as functions of radius (see Appendix A for estimates of $\Delta M_{\mathrm{cap}}$).

The second stage is that in which the miniquasar accretes the captured materials until all available accreting material is exhausted. The evolution of the accretion rate is modeled as a viscous decay [58,59]:

$$\begin{array}{c}\hfill {\dot{M}}_{\mathrm{mq}}\left(t\right)={\dot{M}}_{\mathrm{mq}}(t=0)\times {\left(1+{\displaystyle \frac{t}{t_{\mathrm{visc}}}}\right)}^{-\gamma },\end{array}$$

(36)

with normalization

$${\dot{M}}_{\mathrm{mq}}(t=0)={\displaystyle \frac{\Delta M_{\mathrm{cap}}}{2t_{\mathrm{visc}}}},$$

(37)

where the viscous timescale is $t_{\mathrm{visc}}$ and $\gamma =3/2$ is adopted for demonstrative purposes. Here, we treat $t_{\mathrm{visc}}$ as a free parameter2. Assuming $t_{\mathrm{visc}}$ is of the order of months, the resulting accretion for the miniquasar is super-Eddington. That is, ${\dot{M}}_{\mathrm{mq}}/{\dot{M}}_{\mathrm{Edd}}>1$, with the Eddington accretion rate ${\dot{M}}_{\mathrm{Edd}}\equiv L_{\mathrm{Edd}}/\left(0.1c^2\right)$ and $L_{\mathrm{Edd}}\sim {10}^{38}(M_{\mathrm{BH}}/{\mathrm{M}}_{\odot })\mathrm{erg}{\mathrm{s}}^{-1}$ is the Eddington luminosity, as demonstrated in Appendix A. In our demonstrative example, we assume the mass of the miniquasar is $M_{\mathrm{mq}}=500M_{\odot }$ (see also Figure A1 and

Table 2. Orbital and emission parameters of the demonstrative EMRB system.

Orbital-Related Parameters	Value	Reference
$M_{\mathrm{SMBH}}$	${10}^8{\mathrm{M}}_{\odot }$	Section 3.1
a	1500 $R_g$	Section 3.1
e	0.7	Section 3.1
emissions-related parameters	value	note
$\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$	${10}^{-2}$	Equation (9)
$\mathcal{Q}$	0.5	Equations (21) and (30)
$f_{\mathrm{IC}}$	0.8 $\mathcal{Q}$	Equations (20) and (29)
$f_{\mathrm{pp}}$	1	Equation (25)
$f_{{\pi }^0\to \gamma -\mathrm{ray}}$	1	Equation (26)
miniquasar accretion parameters	value	note
$M_{\mathrm{mq}}$	500 $M_{\odot }$	related to Equation (35)
$t_{\mathrm{visc}}$	2 weeks	related to Equations (36) and (37)

).

We are especially interested in jet-launching together with transient accretion. The miniquasar accretion flow with ${\dot{M}}_{\mathrm{mq}}/{\dot{M}}_{\mathrm{Edd}}>1$ is of the slim disk type, e.g., [60,61], and radiation pressure dominated jet can be launched [62,63,64]. Applying such empirical disk–jet coupling to the miniquasar provides a useful constraint on the jet power and on the viscous timescale $t_{\mathrm{visc}}$ so that the miniquasar jet can be triggered or quenched. In order to maintain the miniquasar accretion within a regime that supports jet activity, it is required that3

$${\displaystyle \frac{{\dot{M}}_{\mathrm{mq}}}{{\dot{M}}_{\mathrm{Edd}}}}\ge 1.$$

(38)

Assuming a typical super-Eddington accretion with $\dot{M_{\mathrm{mq}}}/{\dot{M}}_{\mathrm{Edd}}\sim 10$, a typical miniquasar intrinsic jet power $P_{\mathrm{jet}}$ can be estimated by

$$\begin{array}{c}\hfill P_{\mathrm{jet}}=\eta {\dot{M}}_{\mathrm{mq}}{c}^2.\end{array}$$

(39)

With $\eta =0.1$, $P_{\mathrm{jet}}\sim {\dot{M}}_{\mathrm{crit}}\times L_{\mathrm{Edd}}\sim {10}^{42-44}\mathrm{erg}{\mathrm{s}}^{-1}$, with the range of miniquasar mass $M_{\mathrm{mq}}={10}^{3-5}{\mathrm{M}}_{\odot }$. Potential Doppler effects due to the bulk motion of the jet may further enhance the observed jet power (Equation (7)).

As a demonstration case, we consider a EMRB system with the following parameters: $a=1500R_g$, $e=0.7$, $M_{\mathrm{SMBH}}={10}^8{\mathrm{M}}_{\odot }$ (corresponding to the magenta points in Figure 2 and Figure 4), $M_{\mathrm{mq}}=500M_{\odot }$ and $t_{\mathrm{visc}}=$ two weeks (see the magenta point in Figure A1). These parameters are summarized in Table 2. The orbital motion, Equation (17), is computed and presented in the panel (a) of Figure 6. The orbital phase-dependent accretion rate (Equation (36)) is shown in panel (b). The assumption is made that the miniquasar jet is initiated and launched near the pericenter, $t\approx 0$. This choice of such $t_{\mathrm{visc}}$ is consistent with the jet-launching conditions provided in Equation (38) (see also Figure A1). In the panel (b) of Figure 6, the gray regions within the panel demonstrate instances where the jet-launching condition, as specified by Equation (38), is not satisfied. Consequently, jet activity is expected to stop. As illustrated, the resulting jet lifetime is shorter than the orbital period.

In the computations, for demonstrative purpose, the jet direction is presumed to remain constant within the inertial frame, thus ensuring that the Doppler boosting effect due to the jet’s bulk motion remains invariant with orbital time, and the jet power is simply proportional to Equation (39), provided that the jet is launched. In a more realistic and sophisticated scenario, the jet’s orientation may change with respect to the orbital phase, resulting in an orbital-dependent Doppler factor, as described by Equation (8). The variation in the direction of the jet would also be correlated with the angle and column density at the points where the jet impacts and penetrates the disk.

3.4. Multi-Frequency Light Curves as Diagnostic for Jet Composition

Next, we model the time-dependent emission features during the collision of the miniquasar jet with the SMBH accretion flow. The multi-wavelength light curves are related to both the total jet power and the detailed emissions related to the leptonic and hadronic components within the jet.

By adopting the emission-related parameters summarized in Table 2, we examine two representative jet compositions: a pair-dominated jet (${\xi }_{{\mathrm{e}}^{+}}=1$, ${\xi }_{\mathrm{p}}=0$) and a baryonic jet (${\xi }_{{\mathrm{e}}^{+}}=0$, ${\xi }_{\mathrm{p}}=1$) in panels (c) and (d) in Figure 6. It is assumed that 50% of the lepton energy is converted into emissions, $\mathcal{Q}=0.5$. Additionally, we adopt $f_{\mathrm{IC}}=0.8\mathcal{Q}$ (so $U_{\mathrm{ph}}/U_{\mathrm{B}}=4$ to align with the expectation that there will be substantial IC emissions due to the effective amplification of the energy of the external photon from the SMBH accretion flow by the relativistic photons present within the jet. The corresponding model light curves are calculated as described in Section 3.2 and Section 3.3.

Although the results quantitatively depend on the chosen values of the parameters, qualitative trends reveal meaningful insights into how the emissions from pair-dominated and baryonic jets are modulated by orbital dynamics. The UV and gamma-ray power from the jet–disk interaction, displayed in panels (c) and (d) of Figure 6, respectively, are shown in units of the maximum jet power corresponding to the initial accretion rate at $t=0$. In addition, the jet power evolution is proportional to the accretion, as described in Equation (36). These profiles are obtained through observing the way the intrisic jet power is carried by different particle species (Section 2.1), how the carried power could be converted to radiation power, and the associated conversion efficiency at different frequencies (Section 3.2.1 and Section 3.2.2). For the pair-dominated jet, the gamma rays are contributed by the IC process via leptons. In comparison, the gamma rays are contributed by neutral pion decay and the IC process for both primary and secondary leptons. For both types of jet, the UV emissions comprise the synchrotron emissions from leptons. Note that the gray regions shown in the panels of Figure 6 indicate the system is sub-Eddinton and the jet is quenched. Consequently, in panels (c) and (d) of Figure 6, it is expected that the calculated emission profiles located within the gray-shaded area are expected to diminish to zero as a result of jet quenching. This observed discontinuity in the multi-frequency light curve profiles of electromagnetic radiation from EMRBs might therefore offer empirical evidence for the activity of miniquasar jets.

The pair-dominated jet has a relatively stronger UV power, and the baryonic jet instead has a relatively larger gamma-ray power. For the adopted ratio $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle ={10}^{-2}$, the power ratios are $P_{\gamma -\mathrm{ray}}/P_{\mathrm{UV}}=4$ and $P_{\gamma -\mathrm{ray}}/P_{\mathrm{UV}}\sim 20$ for the pair-dominated jet and the baryonic jet, as indicated in Figure 5 and discussed in Section 3.2.3. The power of multi-wavelength emissions is transformed from the power contained within the leptonic and hadronic components of the jet, and a reduction in $\langle {\gamma }_{\mathrm{p}}\rangle /\langle {\gamma }_{\mathrm{e}}\rangle$ indicates that more power is carried by protons, resulting in increased gamma-ray emissions by efficient pp collision. The presence of a hadronic component significantly enhances the potential of the EMRB system to appear as a bright transient gamma-ray source.

4. Summary and Final Remarks

Motivated by the recognition that quasi-steady-state conditions in black hole jets may obscure the diagnostic temporal variability that is essential for constraining jet composition, in this work, we investigate the emission properties of EMRBs, wherein a miniquasar orbits an SMBH. In the configurations considered, the pericenter and apocenter of the miniquasar orbit reside within and beyond the outer boundary of the geometrically thin accretion disk of SMBH, respectively (see also Section 3.1). This orbital structure induces a sequence of distinctive dynamical phenomena, including periodic penetration of the SMBH disk, episodic accretion onto the miniquasar, and subsequent jet-launching events. Furthermore, the jet emitted by the miniquasar may interact directly with the accretion environment of the SMBH, producing collision-driven high-energy signatures.

Compared to the relativistic jets in AGN, any baryonic component within the miniquasar jet is expected to achieve significantly lower maximum energies. As a consequence, hadronic interactions arising from jet–disk collisions are dominated by proton–proton collisions rather than proton–photon processes. It is demonstrated (Section 3.4) that, during the jet–disk interaction, the resulting multi-wavelength light curves encode characteristic temporal and spectral imprints of both leptonic and hadronic emission components. By integrating variability timescales, jet power evolution, and multi-wavelength radiation features, it is suggested that these systems offer a novel laboratory for constraining the composition of black hole jets.

The key results of the work are summarized as follows:

• The period of the EMRBs of interest here is on the order of years (see also Figure 2). Recurrent encounters between the miniquasar jet and the SMBH accretion flow lead to periodic accretion episodes and jet activity from the miniquasar, characterized by a two-stage accretion process described in Section 3.3.
• In the first stage, when the miniquasar traverses the thin accretion disk of the SMBH, it captures material through Bondi accretion. In the second stage, the captured material is then gradually funneled into the miniquasar, usually leading to super-Eddington accretion. The jet is assumed to be launching when the super-Eddinton accretion onto the miniquasar is satisfied, Equation (38). Typical intrinsic jet power is about ${10}^{42-44}\mathrm{erg}/{\mathrm{s}}^{-1}$ (see the estimation described after Equation (39)).
  The duration of both the accretion phase and the associated jet activity is determined primarily by the viscous timescale of the inflowing material. As the captured material is gradually depleted, both the accretion rate and the resulting jet power decrease over time before the jet activity is quenched.
• Once the miniquasar jet is launched and interacts with the SMBH’s accretion disk, the resulting emission signatures serve as a valuable diagnostic for probing the jet composition. The multi-frequency radiation power from the transient jet can be estimated based on the jet power carried by different particle species and their respective energy-conversion efficiencies to radiation (see also Section 2.2, Section 2.3, Section 3.2.1 and Section 3.2.2).
• The leptonic components within the jet are the source of optical/UV emissions through synchrotron radiation and approximate MeV/GeV gamma rays via IC scattering. In contrast, the proton–proton interactions from the hadronic constituents can yield MeV/GeV gamma-rays through either neutral pion decay or the IC process involving secondary leptons generated by charged pion decay. Additionally, synchrotron radiation from these secondary leptons may also contribute to optical/UV emissions. An analysis of the ratios of different frequency emissions is provided in Section 3.2.3.
• Among the diverse multi-frequency features associated with the EMRBs under consideration, we focus primarily on emissions from miniquasar jets and their interaction with the SMBH accretion flow. In Section 3.4, we model the orbital dynamics and corresponding light curves of UV and gamma-ray power resulting from jet–disk collisions for a demonstration case. With the efficient pp interaction, Equation (25), and the radiation conversion rate, Equation (26), during the jet–disk interaction, the ratio of gamma-ray-to-UV emission serves as a valuable diagnostic to identify the presence of hadronic components within the jet. This ratio increases with the fraction of jet power carried by hadrons.

Other emission components, such as radiation from the SMBH’s thin accretion disk, may further shape the observed spectral energy distribution, particularly in the optical and ultraviolet bands. However, the time variation of the UV light curve and the profile of the spectra may help distengle the sources of UV emissions. Moreover, recurrent interactions between the miniquasar jet and the SMBH accretion flow may induce variations in disk emissions, providing an additional avenue for identifying EMRBs. The transient accretion flow around the miniquasar itself could also produce detectable X-ray emission.

In this work, we estimate the emission properties without considering the full energy distribution of each particle species. We also explore how the multi-frequency power evolves over time as the accretion rate declines, neglecting for now the influence of relative motion between the system and the observer. More detailed spectral modeling that accounts for these effects will be presented in future work. In addition, to further advance our understanding of the complex physical processes involved, it is essential to pursue comprehensive modeling that properly accounts for the transient accretion episodes and the corresponding jet-launching mechanisms.

Complementing the electromagnetic signatures, neutrino production via pion decay will offer a useful means to verify the presence of hadronic interaction, and hence provide information regarding the hadronic content in relativistic jets of accreting black holes. A rough estimate of the resulting neutrino power can be obtained using (cf. Equation (22))

$$P_{\mathrm{p}\to \nu }\approx f_{\mathrm{pp}}\left({\displaystyle \frac{2}{3}}{f}_{{\pi }^{\pm }\to \nu }\right)P_{\mathrm{p}}.$$

(40)

With $f_{{\pi }^{\pm }\to \nu }\sim 10\%$ and a modest intrinsic jet power $P_{\mathrm{jet}}\sim {10}^{42-44}\mathrm{erg}/{\mathrm{s}}^{-1}$, the intrinsic neutrino power due to the interaction between the miniquasar baryonic jet and the SMBH disk can produce $P_{\mathrm{p}\to \nu }\sim {10}^{40-42}\mathrm{erg}/\mathrm{s}$. For an EMRB system at a distance of ∼100 Mpc, the corresponding neutrino flux would be ∼${10}^{-14}$–${10}^{-12}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^2$, potentially accessible to neutrino detectors such as IceCube [67]. We note that neutrino fluxes associated with the miniquasar jet are expected to have lower energy (∼GeV-TeV) compared to neutrinos produced via the SMBH jet, e.g., TXS 0506+056 [40,41].

The detection of high-energy (≥10 GeV–PeV) astrophysical neutrinos is primarily pursued by large-volume observatories such as IceCube [68], KM3NeT/ARCA [69,70], Baikal-GVD [71], and future facilities including IceCube-Gen2 [72] and P-ONE [73,74], which are designed to capture hadronic processes in energetic transients. Meanwhile, GeV-sensitive detectors such as Deep Underground Neutrino Experiment (DUNE) [75], the future Hyper-Komiokande (Hyper-K) [76], and the Oscillation Research with Cosmics in the Abyss (ORCA) in KM3NeT [77] can offer valuable complementary energy coverage that may help constrain low-energy extensions of these neutrino spectra in exceptional nearby or high-fluence events. Together with next-generation electromagnetic facilities, including the Thirty Meter Telescope (TMT) [78] and Extremely Large Telescope (ELT) [79,80] in the optical and infrared wavebands, Advanced Telescope for High Energy Astrophysics (ATHENA) [81,82] and Imaging and Spectroscopy Mission (XRISM) [83] in the X-ray bands, and the gamma-ray telescopes succeeding Fermi Gamma-ray Space Telescope [84], and the facilities capable of detecting neutrinos at GeV energies, these observatories will form an increasingly synergistic multi-messenger network for probing jet composition and orbital-phase-dependent emissions in EMRB systems.