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Magnetic Flux Transport in Advection Dominated Accretion Flow Towards the Formation of Magnetically Arrested Disk

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Hot accretion flows can assemble magnetically arrested inner disks.

desk verdict The coupled ADAF-MAD solution is a genuine step forward, but the claimed 100x jet-power enhancement is an artifact of inconsistent normalizations between Sections 4.1 and 4.2. read the letter →

arxiv 2411.18258 v1 pith:JZXKB6YY submitted 2024-11-27 astro-ph.HE

Jia-Wen Li (Yunnan Univ.) , Xinwu Cao (Zhejiang Univ.) This is my paper · ORCID
classification astro-ph.HE
keywords magneticallyarresteddiskadvectiondominatedaccretionflowmagneticfluxtransportblackholejetpowerBlandford–ZnajekmechanismPrandtlnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that an inner magnetically arrested disk (MAD) can form in an advection-dominated accretion flow (ADAF) by inward magnetic flux transport alone, without invoking a turbulent dynamo. The authors solve the ADAF structure equations together with the induction equation for a large-scale poloidal field, iterating until the gas dynamics and the field configuration are mutually consistent. With an external vertical field of strength $\beta_{\rm out} = 40$ at $R_{\rm out} = 1000 R_{\rm s}$, the solution develops a MAD region spanning roughly $5$–$45$ Schwarzschild radii in which the flow is sub-Keplerian ($\Omega \sim 0.4$–$0.5\,\Omega_{\rm K}$), magnetically dominated ($\beta \lesssim 1$), and radially slowed by about an order of magnitude compared with a normal ADAF. They also derive a rough threshold: for $\beta_{\rm out} \gtrsim 100$ at $R_{\rm out}$, inward advection cannot accumulate enough flux to reach the MAD state. If these claims are right, low-luminosity AGN such as FR I radio galaxies can naturally harbor inner MADs, whose enhanced Blandford–Znajek jet power is about two orders of magnitude above the normal-ADAF estimate.

What carries the argument

The machinery is the coupled problem of disk dynamics and large-scale poloidal field transport. The field is described by the stream function $R\psi(R,z)$ satisfying the steady axisymmetric induction equation, with the inflow speed $v_R(R)$ dragging field inward against Ohmic diffusion; the ratio of viscosity to diffusivity is fixed at $P_m = 3$, the value the paper argues makes flux advection efficient in a flow with $H/R \sim 0.2$–$0.5$. The disk equations include the magnetic torque $T_m$ and radial magnetic force $g_m$ exerted by the field, so the gas and field must be solved iteratively. A solution is accepted as a MAD when the radial derivative of $v_R$ drops to $\lesssim 0.01$ of the derivative of the Keplerian velocity, and the surface toroidal field is fixed by the approximation $B^s_\phi = -0.1 B^s_R$. The same iterative structure yields the scaling relation between $R_{\rm m}$ and $\beta_{\rm out}$ used for the jet-power estimate.

What would settle it

Measure the effective magnetic Prandtl number in a shearing-box or global simulation of ADAF-like turbulence: if it comes out below roughly $R/H \sim 2$–$5$ under the conditions assumed here, then with $\beta_{\rm out}=40$ the flux would diffuse outward faster than it is dragged inward and the predicted $\beta \lesssim 1$ inner region at several tens of Schwarzschild radii would not form.

Watch

Extended reading notes

Core claim

The central discovery, on the authors' own terms, is that flux advection in a hot, thick accretion flow is efficient enough to assemble a magnetically arrested inner disk self-consistently. Solving the coupled radial-momentum, angular-momentum, energy, continuity, and induction equations, they find a steady solution in which an inner MAD (about $5R_{\rm s}$–$45R_{\rm s}$ for mass-loss index $s=0.3$, and $6R_{\rm s}$–$20R_{\rm s}$ for $s=0.1$) joins smoothly to an outer ADAF. The accompanying estimate relates the MAD radius $R_{\rm m}$ to the external plasma $\beta_{\rm out}$: stronger external fields give larger arrested regions, an entirely MAD disk would need $\beta_{\rm out} \sim 1$–$2$, and $\beta_{\rm out} \gtrsim 100$ prevents MAD formation through advection. Evaluating the Blandford–Znajek jet power with the computed inner field strength (approximately $B \propto R^{-1.5}$), the paper finds that a black hole of mass $10^8 M_\odot$ surrounded by an ADAF with an inner MAD produces jets roughly two orders of magnitude more powerful than the same hole surrounded by a normal ADAF, matching the excess jet power seen in low-Eddington FR I galaxies.

Load-bearing premise

The load-bearing premise is that the magnetic Prandtl number is as high as $P_m = 3$ (needed because efficient flux advection requires $P_m \gtrsim R/H \sim 2$–$5$); if real ADAF turbulence has a lower viscosity-to-diffusivity ratio, inward flux transport would be too weak to build the inner MAD even with a strong external field.

Editorial extensions

If this is right

  • A low-luminosity AGN whose accretion flow is an ADAF with an external field $\beta_{\rm out} \lesssim 100$ at $R_{\rm out} = 1000 R_{\rm s}$ should develop an inner MAD rather than remaining a normal ADAF throughout.
  • In the arrested region, the flow rotates at roughly $0.4$–$0.5$ times the Keplerian rate and has $\beta \lesssim 1$, so observational signatures of sub-Keplerian, magnetically dominated inner disks should accompany these systems.
  • Jet powers from a spinning black hole in the MAD-ADAF case exceed the normal-ADAF case by about two orders of magnitude for the same mass, spin, and outer field, matching the excess jet power reported in FR I galaxies.
  • Stronger external fields push the MAD boundary outward; an entire disk in a MAD state would require $\beta_{\rm out} \sim 1$–$2$, while $\beta_{\rm out} \gtrsim 100$ would leave the disk in a normal or standard state.
  • Outflows widen the arrested region: increasing the mass-loss power-law index from $s=0.1$ to $s=0.3$ grows the inner MAD, so outflow-driven angular momentum loss assists flux accumulation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable consequence not developed in the paper: because the threshold is set at $\beta_{\rm out} \sim 100$ at $1000 R_{\rm s}$, estimating the external field at that radius in low-luminosity AGN would separate sources that can host an inner MAD from those that cannot.
  • The fixed $P_m = 3$ assumption means a targeted simulation measuring the effective viscosity-to-diffusivity ratio in ADAF-like turbulence would sharpen or overturn the formation claim; $P_m < 2$ would suppress the predicted flux pile-up.
  • If MAD phases are common enough to explain FR I jet powers, the enhanced spin-energy extraction implies those black holes should spin down faster than their normal-ADAF counterparts over cosmic time, a population-level trend the steady model does not follow.
  • The steady axisymmetric approximation neglects episodic flux eruptions seen in MAD simulations; a time-dependent extension would predict jet power variability and intermittent MAD states even when the mean $\beta_{\rm out}$ satisfies the formation threshold.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper presents a steady, axisymmetric model of an advection-dominated accretion flow (ADAF) around a Schwarzschild black hole, coupled self-consistently to the inward transport of a large-scale poloidal magnetic field. The disk structure equations—continuity with a power-law mass-loss rate (Eq. 1), angular momentum balance including the magnetic torque and outflow angular-momentum loss (Eq. 4), radial force balance with the magnetic curvature force (Eq. 6), and an advective energy equation (Eq. 8)—are solved iteratively together with the integral form of the induction equation (Eqs. 10–14), using prescribed parameters α = 0.1, s = 0.1 or 0.3, Pm = 3, B_sφ = −0.1B_sR, and an outer boundary β_out = 40 at R_out = 1000 R_s. The converged solutions develop an inner region (≈5–45 R_s for s = 0.3, ≈6–20 R_s for s = 0.1) identified as a MAD through the criterion dv_R/dR ≲ 0.01 dv_K/dR (Eq. 16), with β ≲ 1, Ω ≈ (0.4–0.5)Ω_K, and radial velocity reduced by about an order of magnitude relative to a normal ADAF. From power-law fits to the computed profiles the authors derive an approximate relation between the MAD radius R_m and β_out (Eq. 17), concluding that inward flux advection cannot form a MAD for β_out ≳ 100. Finally, they estimate Blandford–Znajek jet powers and report that an ADAF with an inner MAD produces about two orders of magnitude more jet power than a normal ADAF, which they suggest may explain powerful jets in low-Eddington-ratio FR I galaxies.

Significance. The paper's core contribution is a semi-analytic demonstration that inward flux advection in an ADAF can, for a sufficiently strong external field, self-consistently produce an inner region with the characteristic signatures of a MAD, together with quantitative predictions (MAD radius and its scaling with β_out, Ω ≈ (0.4–0.5)Ω_K, β ≲ 1, reduced radial velocity) that can be confronted with global simulations. Credit is due for the explicit iteration between the disk dynamics and the integral induction equation—the field is solved rather than prescribed—and for the quantitative comparisons with recent simulations (Dhang et al. 2023; Aktar et al. 2024; Ressler et al. 2020, 2023; Cho et al. 2024) and the candid acknowledgement of the model's limitations (transonic region, time-dependence, magnetic buoyancy, reconnection) in Section 5. If the jet-power comparison in Section 4.2 is recomputed with a consistent boundary normalization and Eq.

major comments (4)
  1. [§4.2, Eq. (18), Figs. 9–10] The claimed factor-of-~100 jet-power enhancement is not a prediction of the model because the two cases are normalized inconsistently. The MAD-case field is fixed by assuming B = 1 mG at R = 10^5 R_s and extrapolated inward with a single power law B ∝ R^{−1.5}, whereas the MAD structure solved in Section 4.1 used β_out = 40 at R_out = 1000 R_s. A β_out = 40 boundary at 1000 R_s, with the self-similar ADAF pressure used in the model, corresponds to B_out ≈ 0.1 G, which extrapolated to 10^5 R_s gives ≈ 0.1 mG—an order of magnitude below the adopted 1 mG, hence a factor ≈ 10^2 in P_jet ∝ φ_BH². The single power law also contradicts the model's own B_z profile, which flattens in the MAD region (Fig. 4), so the extrapolated horizon field is further overestimated. Since the normal-ADAF jet power is normalized at R_ms through Eq. (A1) with the Eddington ratio, the two curves in Fig. 9 correspond to different physical setups, and the abstract's 'two orders of magnitude' claim and the comparison with FR I galaxies should be recomputed with a common boundary normalization or explicitly qualified as conditional on the assumed 1 mG external field.
  2. [Eq. (17), Fig. 8, §4.1] The sign convention of the exponent ξ is inconsistent. The text states B_z ∝ R^ξ with ξ ≈ −(1.45–1.55) (field increasing inward), but the Fig. 8 caption says 'we adopt ξ = 1.5' while keeping ε = −1.4, τ = −0.42; inserting ξ = +1.5 into Eq. (17) changes the exponent 2ξ − ε − 2τ from ≈ −0.76 to ≈ +5.2 and reverses the R_m–β_out trend, so the plotted curve and the threshold statements are not reproducible as written. Since Eq. (17) and Fig. 8 are the basis for the β_out ≳ 100 'no MAD' threshold and the β_out ≈ 1–2 full-MAD estimate, the sign convention must be fixed and the derivation of Eq. (17) presented; the text jumps from 'substituting the MAD criteria into equation (6)' directly to Eq. (17) with no intermediate steps. The paper should also state that the power-law indices are read off the model's own profiles and that the β_out threshold is an extrapolation beyond the single computed value β_out = 40.
  3. [Eq. (16), §5, Fig. 2] The MAD criterion as written is inconsistent with the properties claimed for the MAD region. With the sign convention v_R < 0 implied by Ṁ = −2πRΣv_R, the Keplerian derivative dv_K/dR is negative, so 0.01 dv_K/dR on the right-hand side of Eq. (16) is negative; Section 5, however, states that in the MAD region dv_R/dR ≈ 0, which cannot satisfy dv_R/dR ≲ 0.01 dv_K/dR. The criterion presumably requires absolute values (e.g., |dv_R/dR| ≲ 0.01|dv_K/dR|), and the sign convention for v_R and v_K should be stated explicitly. Because this criterion is what defines the reported MAD boundaries (≈5–45 R_s for s = 0.3 and ≈6–20 R_s for s = 0.1), the statement of Eq. (16) and the associated panel of Fig. 2 must be corrected.
  4. [§2 (after Eq. 10), §3] The central formation claim depends on the efficiency of inward flux advection, and the paper's own text after Eq. (10) states that efficient flux transport requires Pm ≳ R/H, which for H/R ≈ 0.2–0.5 means Pm ≳ 2–5; the adopted Pm = 3 is therefore marginal, yet no calculation with other values of Pm (e.g., 1, 5, or 10) is presented. The same holds for the other hand-set parameters (χ = B_sφ/B_sR = −0.1, f_adv, α, s). In addition, Section 3 asserts that the n = 200, k = 40 grid 'can achieve a good performance in accuracy' without any convergence test, and no convergence criterion is given for the iteration between Steps 2 and 3. A sensitivity study, at minimum over Pm, and a grid-doubling/iteration-convergence check are required to establish that the inner-MAD solution is robust rather than an artifact of the adopted parameter values.
minor comments (4)
  1. [Eq. (8), §2] The value of the advection factor f_adv is never specified anywhere in the manuscript; since it enters the energy equation and therefore the sound speed and disk thickness, the adopted value should be stated.
  2. [Eq. (A1), §4.2] With β defined as P_g/P_m, the factor (1−β)/(1+β) under the square root in Eq. (A1) is negative for β > 1, so the value and convention of β used to produce the normal-ADAF jet-power curves (dashed lines in Figs. 9 and 10) must be specified for the estimate to be reproducible.
  3. [Figs. 6–7 captions] The captions of Figs. 6 and 7 are self-referential ('The same as figure 6 but...', 'The same as figure 7 but...'); they should refer to Figs. 1 and 5, respectively.
  4. [§1; Fig. 2 caption] There are several typographical slips, e.g., 'recently, down by Ressler et al. (2023)' should read 'done by', and in the Fig. 2 caption 'the derivation of the radial velocity' should read 'the derivative of the radial velocity'.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild self-referential scaling in Eq. (17); central MAD-formation claim is a forward solution.

  1. fitted input called prediction [Section 4.1, Eq. (17) and Fig. 8 caption.]
    "Based on the above calculations, we can have following approximations, i.e., Bz ∝ Rξ, ρ ∝ Rϵ and cs ∝ Rτ, thus substituting the MAD criteria, (i.e., equation 16) into equation (6), we can finally have a relation of, βout ≲ ... (17). From figure 8 we can simply infer that ... it is unlikely to form an inner MAD disk within an ADAF when the external magnetic field is weak enough with βout ≳ 100"

    The power-law exponents ξ, ε, τ in Eq. (17) are taken from the model's own numerical solutions (Fig. 8 caption gives ξ ∼ −(1.45−1.55), ε ∼ −(1.3−1.5), τ ∼ −0.42), and the relation is obtained by substituting these scalings into the radial momentum equation. The threshold βout ≳ 100 for 'no MAD' is then read off this same fitted curve, so it is an extrapolation of the model's output rather than an independent prediction. The two computed cases (s=0.1, s=0.3) already assume βout=40 and produce MADs; the βout threshold is essentially the value at which the fitted scaling would fail, not a result derived from new physics.

full rationale

The core calculation—self-consistently iterating the ADAF structure with the induction equation and boundary condition βout=40—is a forward solution, not a fit to the claimed MAD size or jet power. The emergence of an inner MAD region at ~5–45 Rs and its sub-Keplerian rotation follow from the equations, so the central formation claim is not circular. The one self-referential element is Eq. (17): its power-law exponents (ξ, ε, τ) are read off the paper's own numerical solutions, and the βout≳100 threshold is an extrapolation of that fitted scaling rather than an independent prediction. This is a mild internal-consistency/calibration issue, but it does not make the central result equivalent to its inputs. Note also that the §4.2 jet-power comparison uses a 1 mG field at 10^5 Rs for the MAD case, while §4.1 uses βout=40 at 1000 Rs; the two normalizations are not tied, so the 'two orders of magnitude' statement rests on an assumption external to the model. That is a correctness/consistency concern, not a circular reduction.

Assumptions & free parameters 8 free parameters · 9 assumptions · 0 invented entities

The model rests on a large number of hand-set parameters and simplifications, most notably Pm = 3, the MAD criterion (Eq. 16), and the power-law exponents fitted to the model output. No new physical entities are introduced.

free parameters (8)
  • alpha (viscosity parameter) = 0.1
    Standard alpha viscosity parameter, fixed by hand in all calculations (Section 2, Eq. 4).
  • s (mass-loss power-law index) = 0.1 and 0.3
    Chosen values for the outflow strength in ˙Macc ∝ R^s (Eq. 1); results depend on s.
  • Pm (magnetic Prandtl number) = 3
    Fixed in all calculations; critical for flux advection efficiency. The paper notes Pm ~ 1-5 is typical and Pm ≥ R/H is needed (Section 2, after Eq. 10).
  • Bs_phi / Bs_R (toroidal to radial surface field ratio) = -0.1
    Ad hoc ratio of toroidal to radial global field at the disk surface, adopted in all calculations (Section 2, after Eq. 15).
  • beta_out (outer boundary field strength) = 40
    Chosen as a moderately strong external magnetic field at R_out = 1000 R_s (Section 4.1, figure 1 caption).
  • fadv (energy advection factor) = not explicitly stated; described as constant
    Adopted as a constant to account for energy loss by radiation and outflow; typical values 0.5-0.9 are cited but the exact adopted value is not given (Section 2, after Eq. 9).
  • MAD criterion threshold = 0.01 in Eq. (16)
    Hand-set criterion dv_R/dR ≤ 0.01 dv_K/dR used to define the MAD region; directly determines the reported MAD radius.
  • Power-law exponents for Bz, rho, cs = xi = 1.5, epsilon = -1.4, tau = -0.42
    Fitted to the numerical solutions and then used in Eq. (17) to derive the Rm-beta_out relation and the beta_out ~ 100 threshold (Section 4.1, figure 8 caption).
assumptions (9)
  • domain assumption The accretion flow is steady, axisymmetric, and described by height-integrated ADAF equations with a pseudo-Newtonian potential.
    Used in Section 2, Eqs. (1)-(9); neglects time dependence, general relativity, and vertical structure.
  • domain assumption The external large-scale magnetic field is homogeneous and vertical at the outer boundary R_out = 1000 R_s.
    Section 2, after Eq. (12); the flux advection picture requires this imposed field.
  • domain assumption The magnetic Prandtl number Pm = 3 is constant and large enough to make flux advection efficient.
    Section 2, after Eq. (10); the paper notes Pm ≥ R/H is needed, and H/R ~ 0.2-0.5 makes Pm = 3 borderline.
  • ad hoc to paper The radial velocity v_R is vertically uniform (v_R(R,z) = v_R(R,0)).
    Section 2, before Eq. (13); adopted for simplicity, not derived from the vertical momentum equation.
  • ad hoc to paper The MAD state is defined by dv_R/dR ≤ 0.01 dv_K/dR.
    Section 2, Eq. (16); a hand-chosen 'less stringent' criterion that sets the MAD radius.
  • ad hoc to paper The toroidal-to-radial surface field ratio Bs_phi / Bs_R = -0.1.
    Section 2, after Eq. (15); based on an order-of-magnitude estimate v_R/v_phi ~ 0.1-0.01.
  • domain assumption The mass accretion rate follows ˙Macc ∝ R^s with constant s (Blandford & Begelman 1999).
    Eq. (1); adopted from prior work, with s treated as constant in this paper.
  • domain assumption The flow is around a Schwarzschild black hole (Paczynski-Wiita potential), while the jet power estimate assumes a spinning black hole.
    Eq. (2) for the disk model; Section 4.2 and Appendix A use spin a* for Rms and jet power; the mismatch is unaddressed.
  • ad hoc to paper The outflowing gas has the same angular velocity as the accretion flow at each radius.
    Section 2, after Eq. (4); assumed to avoid modeling the transition region between inflow and outflow.

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Pith. "Pith review of Magnetic Flux Transport in Advection Dominated Accretion Flow Towards the Formation of Magnetically Arrested Disk." pith.science (2026). https://pith.science/paper/JZXKB6YY

@misc{pith2026241118258,
  author       = {Pith},
  title        = {Pith review of: Magnetic Flux Transport in Advection Dominated Accretion Flow Towards the Formation of Magnetically Arrested Disk},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZXKB6YY}},
  note         = {Machine review of arXiv:2411.18258}
}
abstract

The magnetically arrested disks (MADs) have attracted much attention in recent years. The formation of MADs are usually attributed to the accumulation of a sufficient amount of dynamically significant poloidal magnetic flux. In this work, the magnetic flux transport within an advection dominated accretion flow and the formation of a MAD are investigated. The structure and dynamics of an inner MAD connected with an outer ADAF are derived by solving a set of differential equations with suitable boundary conditions. We find that an inner MAD disk is eventually formed at a region about several ten Schwarzschild radius outside the horizon. Due to the presence of strong large-scale magnetic field, the radial velocity of the accretion flow is significantly decreased. The angular velocity of the MAD region is highly subkeplerian with $\Omega \sim (0.4-0.5)\Omega_{\rm K}$ and the corresponding ratio of gas to magnetic pressure is about $\beta \lesssim 1$. Also, we find that MAD is unlikely to be formed through the inward flux advection process when the external magnetic field strength weak enough with $\beta_{\rm out}\gtrsim 100$ around $R_{\rm out}\sim 1000R_{\rm s}$. Based on the rough estimate, we find that the jet power of a black hole, with mass $M_{\rm BH}$ and spin $a_*$, surrounded by an ADAF with inner MAD region is about two order of magnitude larger than that of a black hole surrounded by a normal ADAF. This may account for the powerful jets observed in some Fanaroff Riley type I galaxies with a very low Eddington ratio.

Figures

Figures reproduced from arXiv: 2411.18258 by the authors.

Figure 1
Figure 1. Dynamical structures of the accretion disk for s = 0.3. The variations of ADAF variables with radius R are shown in four panels. Panel (a): The radial velocity (solid) and the sound speed (dashed) of the accretion flow vary with radius, the blue lines are calculated for normal ADAF without a large-scale magnetic field, and the red lines are calculated for an inner MAD disk connected with an outer ADAF. Panel (b): Th… view at source ↗
Figure 2
Figure 2. These lines are calculated for the criteria of the MAD state adopted in equation (16) for the case of s = 0.3. The solid line is for the derivation of the radial velocity (i.e., the solid red line in panel (a) of figure 1) along the radial direction, while the dotted line is for 0.01 times the derivation of the Keplerian rotating velocity [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Radial variation of the specific angular momentum and the Keplerian rotating rate of the disk for the case of s = 0.3. The solid and dotted lines show the variables calculated for the MAD-case and normal ADAF, respectively. The dash-dotted line is calculated for the Keplerian specific angular momentum jK [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Variations in the vertical component of the poloidal magnetic field at disk midplane with radius. The solid line is calculated for s = 0.3 while the dashed line for s = 0.1 region, the gas density of the MAD-case (see the solid line in panel (c) of figure 1) is about o…
Figure 5
Figure 5. Figure 5: The same as figure 1 but the power-law index s = 0.1 is adopted in the calculations [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: The same as figure 6 but the power-law index s = 0.1 is adopted in the calculations. with a mass of MBH and a dimensionless spin of a∗, the maximum jet power accelerated via the Blandford–Znajek mechanism (Blandford & Znajek 1977) for the black hole can be estimated as…
Figure 7
Figure 7. Figure 7: The same as figure 7 but the power-law index s = 0.1 is adopted in the calculations [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Variations of the inner MAD radius Rm with βout, in our calculations ξ ∼ −(1.45 − 1.55), ϵ ∼ −(1.3 − 1.5), τ ∼ −0.42, and Ω ∼ (0.4 − 0.5)ΩK. Thus in above estimation we adopt ξ = 1.5, ϵ = −1.4, τ = −0.42, Ω ∼ 0.45ΩK, and the disk aspect ratio H˜ = 0.5 are adopted, in t…
Figure 9
Figure 9. Figure 9: Variations of the jet power Pjet with the dimensionless black hole spin a∗ (see equation 18 for the details). The dashed lines are calculated for the jet power of a black hole surrounded by a normal ADAF, while the red and blue lines corresponding to the different Eddi…
Figure 10
Figure 10. Figure 10: The same as figure 9, but for the variations of the jet power Pjet with black hole mass MBH. The black hole spin a∗ = 0.95 is adopted as an upper limit. approximation for the force equilibrium within the MAD region, i.e., d dR vR ∼ 0, see the details in the paragraph …

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