Outermost Stable Circular Orbit (OSCO)

• Outermost Stable Circular Orbit (OSCO) is the maximum radius where test particles maintain stable circular motion under combined gravitational, centrifugal, and additional forces.
• Its location is determined by solving for critical points of the effective potential (V'(r) = 0) and ensuring stability via a positive second derivative (V''(r) > 0).
• OSCO analysis in systems like ring galaxies, magnetized Kerr black holes, and multi-black hole spacetimes provides insights into accretion disk boundaries and observable dynamical features.

The outermost stable circular orbit (OSCO) denotes the maximum radius at which stable circular motion can occur for test particles in a broad class of gravitational systems. Unlike the innermost stable circular orbit (ISCO), which sets the inner edge of the stable orbit band, the OSCO serves as the outer stability boundary due to the interplay of gravitational, centrifugal, and—in some cases—additional forces or geometrical effects. Across various settings including axially symmetric spacetimes, systems with multiple @@@@1@@@@ sources, higher-dimensional geometries, and those influenced by cosmological or external fields, determination of the OSCO involves detailed analysis of the effective potential governing geodesic or Newtonian motion.

1. Mathematical Formulation and General Stability Criteria

The position and existence of the OSCO are intrinsically linked to the effective potential $V(r)$ (or $V(\rho)$ in axially symmetric coordinates), which describes the radial dynamics of particle orbits. Circular orbits occur at the critical points where the derivative of the potential vanishes:

$V'(r_c) = 0$

or in axially symmetric Weyl coordinates:

$V'(\rho_c) = 0$

Stable circular orbits require the second derivative at the orbit radius to be positive:

$V''(r_c) > 0$

Marginal stability—or the boundary between stability and instability—occurs when $V''(r_c) = 0$. By solving for where this condition is met, one finds the OSCO radius.

In static axially symmetric spacetimes, explicit conditions are provided for equatorial orbits. For timelike geodesics in Weyl's metric,

$$\ell^2 = \frac{\rho^3 \psi_{,\rho} e^{-2\psi}}{1-2\rho \psi_{,\rho}}$$

$$E^2 = \frac{e^{2\psi} (1-\rho \psi_{,\rho})}{1-2\rho \psi_{,\rho}}$$

with marginal stability (OSCO) derived from setting the derivative of the angular momentum with respect to the radius to zero:

$\frac{d\ell^2}{d\rho} = 0$

or equivalently using the second derivative of the potential. These equations generalize across cylindrical, oblate spheroidal, and prolate spheroidal coordinate systems with appropriate forms for the metric function $\psi$ (González et al., 2011).

2. OSCOs in Systems with Multiple Sources and Nontrivial Geometry

In systems comprised of a central mass and a massive ring (e.g., Hoag-like ring galaxies), the gravitational dynamics depend on two conflicting influences: the inward pull of the core and the outward pull of the ring. The effective potential, such as

$$U_{\rm eff} = U_0 + U_{\rm mc} + \frac{I^2}{2R^2 \rho^2}$$

admits a local minimum (corresponding to a stable circular orbit) only over a finite radial range. The OSCO emerges at the radius where the potential extremum merges (i.e., the minimum and maximum coalesce), denoted mathematically as

$\frac{\partial W(\rho)}{\partial \rho} = 0$

with $W(\rho)$ involving elliptic integrals depending on the core-to-ring mass ratio and the radius (Bannikova, 2018).

The OSCO in such contexts is not only a theoretical boundary: it defines the gap between stable orbits and those that may be dynamically forbidden, contributing to observed features in mass distributions such as luminosity gaps in ring galaxies.

3. Influence of Cosmological and External Fields

The introduction of a cosmological constant or external force fundamentally alters the structure of stable orbits. In static spherically symmetric spacetimes with a cosmological constant $\Lambda$ (Kottler metric), the effective potential becomes quartic:

$e^{2\psi(r)} = 1 - \frac{2m}{r} - A r^2 / 3$

with the condition for marginally stable orbits (ISCO and OSCO) reducing to finding roots of quartic or higher order polynomials:

$4Ar^4 - 15mAr^3 - 3mr + 18m^2 = 0$

The OSCO marks the outer boundary in the presence of a repulsive cosmological term, with its location set by the positive real roots of such equations (Nasereldin et al., 2019).

In the context of massive gravity, the effective potential modifies further:

$$A(r) = 1 - \frac{2M}{r} - \frac{\Lambda}{3} r^2 + \gamma r + \eta$$

and the stability conditions yield a fifth-order polynomial whose roots are computed numerically to locate both ISCO and OSCO. Notably, the linear $\gamma$ term—arising from the mass of the graviton—plays a decisive role in setting the OSCO location, sometimes dominating over $\Lambda$ (Rincon et al., 2021).

4. Dimensional Dependence and Multi-Black Hole Systems

The nature and existence of the OSCO are sensitive to the dimensionality of space. For instance, in Newtonian ring potentials, 3D space allows stable circular orbits from infinity down to the ISCO. In 5D, the potential structure changes so that the stable region is sandwiched between an ISCO and an OSCO (e.g., ISCO at $(\sqrt{6}/2)R$ and OSCO at $2R$ for axisymmetric ring sources) (Igata, 2020). In 6D and above, all stable circular orbits disappear.

In multi-black-hole spacetimes such as the 5D Majumdar-Papapetrou dihole, stable circular orbits exist only within finite regions, bounded internally by the ISCO and externally by the OSCO. The location of OSCO is determined by the Hessian determinant in the effective potential, $h_0 = 0$, and its existence depends intricately on the separation of black holes and the spacetime dimension (Igata et al., 2020).

5. OSCOs in Modified, Regular, and Caged Black Hole Geometries

Models of regular black holes with Minkowskian or de Sitter cores (including horizonless compact massive objects) present double-valued extremal stable circular orbit (ESCO) curves for massive particles, with the upper branch as the ISCO and the lower branch as the OSCO. The presence and location of an OSCO differ quantitatively depending on the core model, affecting black hole shadows and disk truncation radii:

• For Minkowskian-core black holes, the OSCO generally lies at a lower radius compared to dS-core objects (Berry et al., 2020, Zeng et al., 2022).
• Observationally accessible properties, like the shadow radius or the edges of accretion disks, could differentiate these core types.

In caged black holes (higher-dimensional black holes with compactified extra dimensions), the OSCO analogue appears in the regime dominated by effective four-dimensional gravity at large scale, with the stability region stretching from the ISCO outwards. Compactification effects and extra-dimensional scale set the location and extent of the OSCO (Igata et al., 2021, Tomizawa et al., 2021).

6. OSCOs in Magnetized Spacetimes

In Kerr black holes immersed in an external magnetic field (magnetized Kerr solutions), the spacetime structure approaches a Melvin universe asymptotically, resulting in a magnetic barrier. Stable circular orbits are then confined between the ISCO and an OSCO. Analytically, the photon circular orbit condition:

$3B^2 r^3 - 5MB^2 r^2 - 4 r + 12 M = 0$

has two positive roots for subcritical $B$; the larger root denotes the OSCO. For $B \to 0$ the OSCO recedes to infinity, recovering the standard Kerr scenario. With increasing $B$, the OSCO moves inward, ultimately merging with the ISCO at $B_{\rm cr}$; above this, no circular orbits are possible (Iyer et al., 15 Oct 2025).

These modifications substantially alter orbital frequencies, epicyclic precession rates, and the range of quasi-periodic oscillations (QPOs) observable in accretion disks, serving as a diagnostic of astrophysical magnetization.

7. Physical Significance and Observational Implications

The OSCO provides a natural diagnostic for gravitational system boundaries:

• In ring galaxies, it determines gaps in radial luminosity profiles due to forbidden regions for stable orbits (Bannikova, 2018).
• In regular or caged black holes, the OSCO impacts the region of possible stable disk material or lensing structures, potentially observable with current astrophysical instrumentation (Berry et al., 2020, Igata et al., 2021).
• Magnetized black holes with finite OSCO radii and modified QPOs offer a direct avenue for constraining large-scale magnetic fields from timing data (Iyer et al., 15 Oct 2025).
• The dependence of OSCO on internal structure (spin-induced quadrupole moments) and cosmological parameters (e.g., $\Lambda$, graviton mass) emphasizes its utility in probing extensions of GR and possible modifications to compact object phenomenology (Zhang et al., 2022, Rincon et al., 2021).

The coexistence of ISCO and OSCO marks a qualitative transition: while the ISCO sets the inner truncation for accretion flows, the OSCO enforces outer boundaries that arise in certain classes of spacetimes or under additional physical ingredients (e.g., multiple sources, cosmological constant, external fields, higher dimensions, or modified gravity). Observational determination of the OSCO—through disk edges, lensing features, QPO range, or direct imaging—serves as a probe of underlying gravitational dynamics and the composition or geometry of the compact object system.