Relativistic thick accretion disks: Morphology and evolutionary parameters

Daniela Pugliese and Giovanni Montani

Phys. Rev. D 91, 083011 – Published 27 April 2015

Abstract

We explore thick accretion disks around rotating attractors. We detail the configurations analyzing the fluid angular momentum and finally provide a characterization of the disk morphology and different possible topologies. Investigating the properties of orbiting disks, a classification of attractors, possibly identifiable in terms of their spin-mass ratio, is introduced; then an attempt to characterize dynamically a series of different disk topologies is discussed, showing that some of the toroidal configuration features are determined by the ratio of the angular momentum of the orbiting matter and the spin-mass ratio of the attractor. Then we focus on “multistructured” disks constituted by two or more rings of matter orbiting the same attractor, and we prove that some structures are constrained in the dimension of rings, spacing, location, and an upper limit of ring number is provided. Finally, assuming a polytropic equation of state, we study some specific cases.

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Received 5 December 2014

DOI:https://doi.org/10.1103/PhysRevD.91.083011

Authors & Affiliations

Daniela Pugliese^1,2 and Giovanni Montani^3,4

¹School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
²Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo náměstí 13, CZ-74601 Opava, Czech Republic
³ENEA, Unità Tecnica Fusione, ENEA C. R. Frascati, via Enrico Fermi 45, 00044 Frascati, Roma, Italy
⁴Physics Department, “Sapienza” University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy

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Vol. 91, Iss. 8 — 15 April 2015

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Figure 1
Angular momenta $ℓ_{B}^{\pm}$ as function of radius $r / M$ and the spin-mass ratio $a / M$ of the spacetime. The right plot is a zoom in the ergoregion $r \in] r_{ε}^{+}, r_{+} [$ . In the shaded regions, the effective potential function $V_{eff}^{2}$ is not defined. The white surface is the plane $ℓ_{B}^{\pm} = 0$ ; $ℓ_{B}^{+} < ℓ_{B}^{-}$ is the gray surface. The horizon $r_{+}$ and the static limit are also plotted (black surfaces).
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Figure 2
Left panel: Radii of the set $R \equiv {r_{d}, r_{ε}^{+}, r_{lsco}^{\pm}, r_{b}^{\pm}, r_{γ}^{\pm}}$ as a function of black hole spin-mass ratio $a / M$ , namely, the last circular orbit for timelike particles $r_{γ}^{\pm}$ . The last bounded orbit is $r_{b}^{\pm}$ , and the last stable circular orbit is $r_{lsco}^{\pm}$ , where $(\pm)$ is for counterrotating or corotating orbits with respect to the attractor on the equatorial plane $θ = π / 2$ . The outer horizon $r_{+} / M$ , the static limit $r_{ε}^{+} = 2 M$ , and $r_{d}$ are also plotted; the regions of positive fluid angular momentum $ℓ_{f}^{\pm}$ for counterrotating and corotating fluid are shaded. The gray region is $] r_{ε}^{+}, r_{+} [$ ; the black region $r < r_{+}$ . Vertical lines are the spin in $A \equiv {a_{\circ}, a_{•}, a_{♭}, a_{†}, a_{□}, a_{⋄}, \overset{ˇ}{a}, a_{♮}}$ where $a_{\circ} \equiv (8 \sqrt{2} - 11) M$ , it is $a_{•} \equiv (23 - 16 \sqrt{2}) M$ , $a_{♭} / M \equiv \sqrt{7 / 3} / 3$ , $a_{†} / M \equiv 1 / 2 (3 \sqrt{17} - 11) M$ , $a_{□} \equiv 1 / \sqrt{2} M$ , $a_{⋄} \equiv 2 (\sqrt{2} - 1) M$ , $\overset{ˇ}{a} \equiv 0.865157 M$ , $a_{♮} \equiv 2 \sqrt{2} / 3 M$ . Nine classes of black hole spacetimes BHI–IX are outlined. Right plot: Fluid angular momentum $ℓ_{f}^{\pm}$ of the orbits in units of mass $M$ of $R$ as a function of $a / M$ .
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Figure 3
Left panel: Spacetime spin $a = 8 / 10 M \in BHVI$ , fluid angular momentum in units of mass $M$ , $ℓ_{f}^{+}$ (black curve), $ℓ_{d} = 2.74035 M$ (gray curve) as functions of $r > r_{+}$ , the horizon $r_{+} = 1.6 M$ , and the static limit $r_{ε}^{+} = 2 M$ are plotted (dashed lines). The inset plot is the effective potential $V_{eff}^{2} (ℓ_{d}, r_{d})$ function of $a / M$ . The energy function $V_{eff}^{2} (ℓ_{d}, r_{d})$ is not well defined in $a = a_{♭} \approx 0.51 M$ , and it is negative in $a < a_{♭}$ . Right panel: Spacetime spin $a = 8 / 10 M \in BHVI$ and fluid angular momentum $ℓ_{f}^{-}$ for corotating matter as a function of $r / M$ . Dashed lines are the last circular orbit $r_{γ}^{-}$ , the static limit $r_{ε}^{+}$ , the last bounded orbit $r_{b}^{-}$ , the radius $r_{d}$ , and the last stable circular orbit $r_{lsco}^{-}$ . The curve $ℓ_{f}^{-} > 0$ spacetime spin $a = 8 / 10 M \in BHVI$ has a minimum point in $r_{lsco}^{-}$ .
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Figure 4
Left top panel: Plot of the effective potential $V_{eff} (ℓ_{f}^{\pm}, r_{lsco}^{\pm})$ as a function of $a / M$ . At $a = M$ , the $r_{lsco}^{-}$ reaches the horizon. Right top and bottom panels: Angular momentum in units of mass $M$ , $ℓ_{f}^{-}$ $(ℓ_{f}^{+})$ at ${r_{lsco}^{-}, r_{b}^{-}, r_{γ}^{-}}$ , $(r_{lsco}^{+}, r_{b}^{+}, r_{γ}^{+})$ for different spacetimes. It is $ℓ_{f}^{\mp} (r_{γ}^{\mp}) ≷ ℓ_{f}^{\mp} (r_{b}^{-}) ≷ ℓ_{f}^{\mp} (r_{lsco}^{\mp})$ . In terms of the fluid angular momentum, only $C_{x}$ configurations are at $K < 1$ and $ℓ \in] ℓ_{b}^{+}, ℓ_{lsco}^{+} [$ and $ℓ \in] ℓ_{lsco}^{-}, ℓ_{b}^{-} [$ . It is $C$ disks are for $K \in] K_{lsco}^{\pm}, 1 [$ , $ℓ < ℓ_{lsco}^{+} < 0 \cup ℓ > ℓ_{lsco}^{-} > 0$ , and $r > r_{lsco}^{\pm}$ . It is $O_{x}$ disks at $K > 1$ $ℓ \in] ℓ_{γ}^{+}, ℓ_{b}^{+} [\cup] ℓ_{b}^{-}, ℓ_{γ}^{-} [$ , see text.
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Figure 5
Left panel: Logarithm plot of the effective potential $V_{eff} (ℓ)$ as a function of $r / M$ . Right plot: The Boyer surface in the plane $(x, y)$ for a spacetime spin-mass ratio $a / M = 0.97 \in BHIX$ and fluid angular momentum $ℓ = 2.279$ in units of mass $M$ . The black regions are $r < r_{+}$ where $r_{+}$ is the black hole outer horizon, the gray region is $] r_{+}, r_{ε}^{+}]$ where $r_{ε}^{+}$ is the static limit, and the light gray region is $] r_{ε}^{+}, r_{Min}]$ . The critical points of $V_{eff} (ℓ)$ are signed in the plots: $r_{Min}$ is a minimum point of $V_{eff} (ℓ)$ and $r_{Max}$ a maximum point. The surfaces and the points at $K = V_{eff} (ℓ) = constant$ are black lines; the point $y_{1}$ (left plot) corresponds to the outer boundary of the disk section (right plot), $y_{3}$ to the inner one, and $y_{2}$ to the outer boundary of the innermost surface; $r_{Min}$ sets the disk center.
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Figure 6
Left panels: Logarithm plot of the effective potential $V_{eff} (ℓ)$ as a function of $r / M$ . The angular momentum $ℓ$ is in units of mass $M$ . Right panels: The Boyer surface in the plane $(x, y)$ (sequences $B_{K} \equiv B_{p} / Σ_{ℓ}$ of P-D tori loops). The black regions are $r < r_{+}$ where $r_{+}$ is the outer black hole outer horizon, the gray region is $] r_{+}, r_{ε}^{+}]$ where $r_{ε}^{+}$ is the static limit, and the light gray region is $] r_{ε}^{+}, r_{Min}]$ . The critical points of $V_{eff} (ℓ)$ are signed in the plots: $r_{Min}$ is a minimum point of $V_{eff} (ℓ)$ and $r_{Max}$ a maximum point. The surfaces and the points at $K = V_{eff} (ℓ) = constant$ are black lines. The point $y_{1}$ (left plot) corresponds to the outer boundary of the disk section, $y_{3}$ to the inner one, and $y_{2}$ to the outer boundary of the innermost surface; $r_{Min}$ sets the disk center. The first line is for spacetime at $a = 0.5 M \in BHIII$ , the second line is for $a = 0.7 M \in BHV$ , and the third and fourth lines $a = 0.97 M \in BHIX$ . Each contour in the right plots is at $K = constant$ . It is $ℓ = ℓ_{f}^{+} (r_{Max})$ , where $r_{Max}$ is the maximum point of the effective potential $K_{Max}$ . The first and third rows of plots are at $K_{Max} < 1$ . The fourth line is $K_{Max} = 1$ ; the second line is $K_{Max} > 1$ . The radii $r : V_{eff} (a, ℓ) = K$ within the critical points are shown in the plot of the effective potential.
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Figure 7
Left panel: Effective potential $V_{eff} (ℓ)$ versus the orbital radius $r / M$ at $a = 0.97 M$ (BHIX spacetime) and orbital fluid angular momentum $ℓ = 2.25$ in units of mass $M$ . Right panel: Boyer surfaces at fixed $K$ (sequences $B_{K} \equiv B_{p} / Σ_{ℓ}$ of P-D tori loops). The black region is $r < r_{+}$ ( $r_{+}$ being the outer horizon); the ergoregion (dark gray) is $r \in] r_{+}, r_{ε}^{+} [$ where $r_{ε}^{+}$ is the static limit. The critical points of the effective potential are both in the ergoregion; the maximum located in $r_{Max} = 1.66346 M$ and minimum $r_{Min} = 1.82079 M$ are signed by thick black lines. Closed configuration are centered in $r_{Min}$ ; the crossing points are at $r_{Max}$ . In the right panel are configurations entirely contained in the ergoregion. The light gray region is the $[r_{Max}, r_{Min}]$ ; the contours of the Boyer surfaces associated to the $K_{Max} \equiv V_{eff} (r_{Max}) = 0.786937$ and $K_{Min} \equiv V_{eff} (r_{Min}) = 0.786588$ (crossed closed $C_{x}$ ) are in black thick lines. Inside panel: The surface at $K = 0.7865 \in] K_{m n i n}, K_{Max} [$ is closed and contained in the ergoregion. The surfaces boundaries $y_{1} = 1.92614 M$ , $y_{3}^{Max} = 1.66346 M$ (configuration of maximum $K$ ) and $y_{2}^{Min} = 1.60392 M$ (configuration of minimum $K$ ) are signed.
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Figure 8
Configurations at $K < 1$ , $ℓ > a$ . Region $K \in] 0, K_{lsco}^{-}]$ . The spacetime spin is $a = 0.25 M \in BHI$ ; the outer horizon is at $r_{+} = 1.96825 M$ . The static limit $r_{ε}^{+} = 2 M$ , where $K^{2} = 0.7$ (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ ), is $ℓ_{K}^{i} = {0.233022, 0.270991}$ , and $ℓ_{f} (r_{+}) = 15.746$ , $ℓ_{ε}^{+} = 13.4216$ , $ℓ_{f} (r_{d}) = 8.22857$ , $ℓ_{γ}^{-} = 4.67535$ , $ℓ_{b}^{-} = 3.73205$ , $ℓ_{lsco}^{-} = 3.43993$ , in units of mass $M$ .
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Figure 9
Configurations at $K < 1$ and $ℓ > a$ . Region $K \in] K_{lsco}^{-}, K_{lsco}^{+} [$ . The spacetime spin is $a = 0.25 M \in BHI$ . The outer horizon is at $r_{+} = 1.96825 M$ ; the static limit $r_{ε}^{+} = 2 M$ (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ ), where $K^{2} = 0.9$ , is $ℓ_{K}^{i} = {0.235225, 0.268791, 3.53586, 3.69391}$ and $ℓ_{f} (r_{+}) = 15.746$ , $ℓ_{ε}^{+} = 13.4216$ , $ℓ_{f} (r_{d}) = 8.22857$ , $ℓ_{γ}^{-} = 4.67535$ , $ℓ_{b}^{-} = 3.73205$ , $ℓ_{lsco}^{-} = 3.43993$ . The fluid angular momentum is in units of mass $M$ .
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Figure 10
Configurations at $K < 1$ , $ℓ > a$ in region III $K \in [K_{lsco}^{+}, 1 [$ (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ ); see Sec. 4b1. The spacetime spin is $a = 0.25 M \in BHI$ . The outer horizon is at $r_{+} = 1.96825 M$ ; the static limit $r_{ε}^{+} = 2 M$ is $K^{2} = 0.93$ and $ℓ_{K}^{i} \in {- 4.29142, - 4.01885, 0.235492, 0.268524, 3.60729, 4.19897}$ , and $ℓ_{f} (r_{+})^{\pm} = 15.746$ , $ℓ_{ε}^{+} = 13.4216$ , $ℓ_{f} (r_{d}) = 8.22857$ , $ℓ_{γ}^{-} = 4.67535$ , $ℓ_{lsco}^{-} = 3.43993$ , $ℓ_{b}^{-} = 3.73205$ , and $ℓ_{γ}^{+} = - 5.68011$ , $ℓ_{b}^{+} = - 4.23607$ $ℓ_{lsco}^{+} = - 3.88089$ .
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Figure 11
Configurations at $K < 1$ , $ℓ < a$ in region I $K \in] 0, K_{lsco}^{+} [$ (see Sec. 4b2). The spacetime spin is $a = 0.25 M \in BHI$ (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ ), where $K = 0.81$ . The outer horizon is at $r_{+} = 1.96825 M$ ; the static limit $r_{ε}^{+} = 2 M$ , $ℓ_{K}^{i} = {0.234227, 0.269787}$ , in units of mass $M$ .
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Figure 12
Configurations at $K < 1$ , $ℓ < a$ in region III $K \in] K_{lsco}^{+}, 1 [$ (see Sec. 4b2). The spacetime spin is $a = 0.25 M \in BHI$ . The outer horizon is at $r_{+} = 1.96825 M$ ; the static limit $r_{ε}^{+} = 2 M$ (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ ), where $K^{2} = 0.91$ , is $ℓ_{K}^{i} = (- 3.96369, - 3.92872, 0.235316, 0.2687, 3.56152, 3.82688)$ . The angular momentum is in units of mass $M$ .
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Figure 13
Configurations of the fluid angular momentum $ℓ = ℓ (a) < 0$ (left panel) and $ℓ = ℓ (a) > 0$ (right panel). Existence regions of the zeros $y_{i}$ of $V_{eff} (p) \equiv V_{eff} (ℓ) - K$ for fixed ranges of angular momentum and the potential-parameter $K_{i} = K_{i} (a, ℓ) \in] 0, 1 [$ . In the dark-gray-shaded regions, closed $C$ Boyer disks exist.
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Figure 14
Closed Boyer configurations for fixed ranges of the fluid angular momentum $ℓ = ℓ (a) > 0$ in units of mass $M$ and the parameter $K_{i} = K_{i} (a, ℓ) \in] 0, 1 [$ for spacetime $a = 9 / 10 M \in BHVIII$ and $a = 7 / 10 M \in BHV$ .
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Figure 15
Closed Boyer surfaces (sequences $B_{K} \equiv B_{p} / Σ_{ℓ}$ ) for fixed ranges of the fluid angular momentum $ℓ = ℓ (a) < 0$ in units of mass $M$ and the potential parameter $K_{i} = K_{i} (a, ℓ) \in] 0, 1 [$ in different spacetimes. It is $a = 9 / 10 M \in BHVIII$ and $ℓ_{γ}^{+} = - 6.83232$ and $(K_{i})^{2} \in {0.977871, 0.000828153}$ . Second row: $ℓ_{f}^{+} (r_{b}^{+}) = - 4.75681$ and $(K_{i})^{2} \in {0.00146791, 0.951018}$ . Third row: $a / M = 5 / 10 \in BHIII$ , $ℓ_{f}^{+} (r_{b}^{+}) = - 4.44949$ and $ℓ_{f}^{+} (r_{lsco}^{+}) = - 4.06784$ $(K_{i})^{2} \in {0.000172721, 0.932708, 0.984353}$ . Bottom row: $a / M = 5 / 10 \in BHIII$ , $(K_{i})^{2} \in {0.00102636, 0.900794}$ .
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Figure 16
The Schwarzschild case: Sequence $B_{K}$ in $ℓ / M \in] 3 \sqrt{3 / 2}, 4 [$ . It is $ℓ = 3.9 M$ in units of mass $M$ and $K_{i}^{2} = {0.95529, 0.910634} i \in {a, b}$ . The angular momentum is in units of mass $M$ .
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Figure 17
The Schwarzschild case sequence $B_{K}$ . Left panel: $ℓ / M \in] 0, 3 \sqrt{3 / 2}]$ with $(K \in] 0, 1 [, y_{1})$ . Center and right panels: $ℓ / M \in [4, 3 \sqrt{3}]$ ; $ℓ = 4.5 M$ and $K_{i}^{2} = {1.47765, 0.939964} i \in {a, b}$ . The angular momentum is in units of mass $M$ .
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Figure 18
Configurations $ℓ = \pm a$ and different spin-mass ratio of the attractors. The angular momentum is in units of mass $M$ .
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Figure 19
Nonrotating attractor: Sequences $B_{ℓ}$ with fixed $K \in] \sqrt{8 / 9}, 1 [$ . The angular momentum is in units of mass $M$ ; $K^{2} = 0.9$ and $ℓ_{K}^{\pm} \in {3.72678, 3.77124}$ .
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Figure 20
Nonrotating attractor, sequences $B_{ℓ}$ for $K = 1$ and $K > 1$ , where $K^{2} = 1.5$ and $ℓ_{K}^{+} = 4.51276 M$ .
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Figure 21
Effective potential $V_{eff} (ℓ)$ (left panel) and contour plot of the Boyer surfaces (sequences $B_{ℓ} \equiv B_{p} / Σ_{K}$ of “loops” of $C$ , $C_{x}$ and $B$ configurations) as a function of $r / M$ at fixed spin $a = 0.25 M \in BHI$ and $K = 0.9539$ and different fluid angular momenta. Each contour is at $ℓ = constant$ in units of mass $M$ . The black region is $r < r_{+}$ ( $r_{+}$ is the outer horizon); the gray region is $r \in] r_{+}, r_{ε}^{+}]$ ( $r_{ε}^{+}$ is the static limit).
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Figure 22
Fluids orbiting in a BHIV spacetime with $a = 0.7 M$ . Left upper panel: The fluid angular momentum $ℓ_{f}^{+}$ for counterrotating (gray curve) and corotating $ℓ_{f}^{-}$ (black curve) matter as functions of the orbital radius $r / M$ . The black region is $r < r_{+}$ , the gray region is $r \in] r_{+}, r_{ε}^{+} [$ , the static limit is $r_{ε}^{+}$ , and the outer horizon is the $r_{+}$ . Points on the curves set the critical points of the angular momentum located at the last stable circular orbit $r_{lsco}^{+}$ for counterrotating matter and $r_{lsco}^{-}$ for corotating matter. The marginally bounded orbits $r_{b}^{\pm}$ are also plotted. The lines at constant $K$ parameters are plotted with dotted lines. Right panel: The effective potential $V_{eff} (ℓ_{f}^{+})$ for counterrotating (gray curve) and corotating $V_{eff} (ℓ_{f}^{-})$ (black curve) matter as functions of the orbital radius $r / M$ ; the values $V_{eff} (ℓ_{f}^{\pm})$ are the $K$ -parameter values of the Boyer surfaces, and the values $ℓ = constant$ are signed with dotted lines. Center panels: Plot of $V_{eff} (ℓ_{f}^{-})$ versus $r / M$ . The surfaces at $K = 0.90$ are for $ℓ_{f}^{-} = (2.899184, 2.905348)$ gray and black curves, respectively. At $K = 0.90$ are the maximum points $r_{Max}^{\pm}$ and the minimum $r_{Min}^{\pm}$ for the two curves, respectively. Double (separated and closed) Boyer configurations are allowed in the gray region where $y_{3}^{(o)}$ and $y_{1}^{(i)}$ must be located: the boundaries of the region are $r_{Min}$ for the outer configuration and the minimum point of the inner one (signed by a point on the gray curve), i.e., $[y_{1}^{(i)}, y_{3}^{(o)}] \subset [r_{Min}^{(i)}, r_{Min}^{(o)}]$ . The left panel shows a zoom of the gray region $[r_{Min}^{(i)}, r_{Min}^{(o)}]$ as the location of $(y_{1}^{(i)}, y_{3}^{(o)})$ (dotted-dashed lines). The bottom panel shows the couple of corresponding Boyer surfaces, in analogue couples of configurations at $ℓ = ℓ_{f}^{+}$ are located at $r > 4 M$ . The angular momentum is in units of mass $M$ . These are the maximum numbers $(n = 4)$ of $C$ configurations in this spacetime.
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Figure 23
Effective potential $V_{eff} (ℓ)$ as a function of $r / M$ for a black hole spacetime at $a = 0.5 M \in BHIII$ , for different values of the fluid angular momentum $ℓ_{f}^{-}$ (upper left panel), $ℓ_{f}^{+}$ (upper right panel), and $ℓ_{f}^{\pm}$ (bottom panel), in the range $ℓ \in] \mp ℓ_{lsco}^{\pm}, \mp ℓ_{b}^{\pm}]$ for corotating $(-)$ and counterrotating fluids $(+)$ , where $ℓ_{lsco}^{\pm} \equiv ℓ (r_{lsco}^{\pm})$ and $ℓ_{b}^{\pm} \equiv ℓ^{\pm} (r_{b})$ , where $r_{lsco}^{\pm}$ is the radius of the last stable circular orbits, and $r_{b}^{\pm}$ is the radius of the last circular bounded orbit. The potential at $ℓ_{γ}^{\pm} \equiv ℓ_{(r_{γ}^{\pm})}$ has been plotted as well where $r_{γ}^{\pm}$ is the last circular orbit. Thick lines mark the critical points of the effective potentials. The angular momentum is in units of mass $M$ .
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Figure 24
Plots of the function $λ \equiv y_{1} - y_{3}$ (left upper panel), surface of maximum torus diameter, and distance from the static limit surface $δ_{ε}^{+} \equiv y_{3} - r_{ε}^{+}$ (right upper panel). The distance of the torus inner surface from the structure inner surface $\hat{δ} \equiv y_{3} - y_{2}$ (bottom panel), the distance of the structure outer surface from the horizon $δ_{ε_{1}}^{+} (a; p) \equiv y_{1} - 2 M$ , and $δ_{ε_{2}}^{+} (a; p) \equiv y_{2} - 2 M$ where $y_{2}$ is the solution closest to the singularity, as functions of the parameter $K^{2}$ , at different spacetime spin-mass ratios $a / M$ : $a = 0.99 M \in BHIX$ (gray curves), $a = 0.59 M \in BHIV$ (dotted curves), $a = 0.19 M \in BHI$ (black curves), and fluid angular momentum $ℓ = - 4.5$ (left set of curves) and $ℓ = 4.5$ (right set of curves). Inset plot shows each function of the parameter $p \equiv (K, ℓ)$ at fixed $K$ , as a function $ℓ$ ; the black curve is for $a = 1 0^{- 5} M \in BHI$ , and the gray curve is for $a = 0.991 M$ . It is clear the symmetry between the $ℓ$ counterrotating configurations with respect to the transformation $ℓ \to - ℓ$ and $ℓ_{f}^{-} \to ℓ_{f}^{+}$ . The angular momentum is in units of mass $M$ .
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Figure 25
Plot of $r_{q I} / M$ and $r_{q II} / M$ as a function of $a / M$ . The black region is $r < r_{+}$ , the static limit $r_{ε}^{+}$ , the last circular orbit $r_{γ}^{-}$ , the last bounded orbit $r_{b}^{-}$ , and the last stable circular orbit radius $r_{lsco}^{-}$ (test particle with $L = L_{-}$ ) are also plotted.
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