Fractal features in accretion discs

Fractal concepts have been introduced in the accretion disc as a new feature. Due to the fractal nature of the flow, its continuity condition undergoes modifications. The conserved stationary fractal flow admits only saddle points and centre-type points in its phase portrait. Completely analytical solutions of the equilibrium point conditions indicate that the fractal properties enable the flow to behave like an effective continuum of lesser density, and facilitates the generation of transonicity. However, strongly fractal flows inhibit multitransonicity from developing. The mass accretion rate exhibits a fractal scaling behaviour, and the entire fractal accretion disc is stable under linearised dynamic perturbations.

💡 Research Summary

The paper introduces a novel framework for modelling accretion discs by incorporating fractal geometry into the fluid dynamics. Traditional disc theory assumes a continuous medium with a standard continuity equation, ρ r v = const, which implicitly treats the flow as three‑dimensional (D = 3). Observations of turbulent, clumpy, and magnetised astrophysical environments, however, suggest that the matter distribution can be self‑similar over a range of scales. To capture this, the authors replace the Euclidean dimension with a fractal dimension D (2 < D < 3) and rewrite the mass‑conservation law as ρ r^{D‑1} v = const. This modification effectively reduces the “apparent” density of the flow, making the disc behave as if it were a lower‑density continuum.

Using a non‑viscous, axisymmetric, steady‑state model, the gravitational potential is taken as Φ = −GM/r, and the gas obeys a polytropic equation of state P = K ρ^{γ}. The sound speed is c_s = √(γ P/ρ). After non‑dimensionalising the governing equations with r_c (the critical radius) and c_{s,c} (the sound speed at that radius), the authors obtain a first‑order ordinary differential equation for the Mach number as a function of radius that explicitly contains D.

A phase‑portrait analysis of this ODE reveals that only two types of critical points exist: saddle points and centre‑type points. No node or spiral points appear because the Jacobian eigenvalues are either real with opposite signs (saddle) or purely imaginary (centre). This restriction implies that a fractal disc can support at most a single transonic transition; the classic multi‑transonic behaviour seen in some viscous or relativistic disc models is suppressed when the fractal character is strong.

Solving the equilibrium‑point conditions analytically yields scaling relations for the critical radius and velocity: r_c ∝ (GM)^{1/(3‑D)} (Kγ)^{‑1/