OCTOPUS: A Versatile, User-Friendly, and Extensible Public Code for General-Relativistic Ray-Tracing in Spherically Symmetric and Static Spacetimes

TL;DR

OCTOPUS addresses the need for an accessible, extensible public code to perform general-relativistic ray-tracing in static, spherically symmetric spacetimes. It combines backward ray-tracing with high-order numerical integrators to compute event horizons, photon rings, ISCOs, and observational features such as black-hole shadows, accretion-disk images, gravitational lensing, hot-spot light curves, and Kludge gravitational waves. The framework requires only the metric potential $f(r)$ and its radial derivatives up to the third order, enabling rapid adaptation to alternative spacetimes, including those with dark-matter halos, and supports automated validation against analytic bounds. Key findings show that dark-matter halos strengthen the gravitational field, enlarging shadow-related features and modulating observable signatures, while maintaining numerical precision (Hamiltonian errors near machine precision) and offering scalable performance (images with 10^4 rays in seconds and large parameter studies in hours). The work provides a pipeline-ready tool for qualitative black-hole observables and multi-messenger studies, with planned extensions to axisymmetric spacetimes, polarization, and broader metric implementations.

Abstract

This paper presents OCTOPUS, a relativistic ray-tracing algorithm developed within a Fortran-based, OpenMP-accelerated framework, designed for asymptotically flat, spherically symmetric curved spacetimes. The code efficiently and accurately computes key relativistic features -- including the black hole event horizon, photon rings, critical curves, and innermost stable circular orbits -- and simulates black hole shadows, redshift factor distributions, accretion disk images, toroidal images, as well as gravitational lensing, light curves, and gravitational radiation from hot-spots. OCTOPUS provides an automated, modular solution for qualitative studies of black hole observables and multi-messenger correlations between electromagnetic and gravitational signals in curved spacetime. Its implementation requires only the metric potential and its first-, second-, and third-order radial derivatives as input, ensuring low user barriers while remaining highly extensible and adaptable. Using a Schwarzschild black hole surrounded by a Dehnen-type dark matter halo, we thoroughly validate the algorithm's precision, efficiency, and functionality, and investigate how dark matter halo parameters affect observational signatures. Our results demonstrate that increasing the scale and density of the dark matter halo strengthens the spacetime's gravitational field, an effect clearly reflected in black hole images and supported by hot-spot light curve signatures. A future version of OCTOPUS, with expanded capabilities for axisymmetric spacetimes, is planned for release.

Figures (24)

• Figure 1: Schematic diagram of the ray-tracing coordinate system. The local coordinate systems of the black hole and the observer are denoted by $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$, respectively, with the black hole located at point $o^{\prime}$. The green square represents the $\overline{xoy}$ plane of the $xyz$ system, corresponding to the observation screen. The viewing angle $\omega$ is defined as the angle between the line $\overline{o^{\prime}o}$ and the $z^{\prime}$-axis, while the azimuthal viewing angle $\varphi_{\textrm{obs}}$ is the angle between the projection of $\overline{o^{\prime}o}$ onto the black hole's equatorial plane (blue) and the $x^{\prime}$-axis.
• Figure 2: Schematic of the celestial sphere and coordinate systems used to simulate gravitational lensing images for static point sources. The local coordinate systems of the black hole and observer, along with all associated parameters, remain consistent with those in figure 1. A stationary point source is represented by a white sphere of non-negligible radius. While shown here positioned along the $y^{\prime}$-axis, it can actually be located anywhere outside the black hole event horizon. The entire system---comprising the black hole and observer---is enclosed by a celestial sphere of radius $1500$ M, with color coding applied to visually enhance the illustration of gravitational lensing effects.
• Figure 3: From left to right: dependence of the event horizon radius $r_{\textrm{eh}}$, ISCO radius $r_{\textrm{isco}}$, critical photon orbit radius $r_{\textrm{p}}$, and critical impact parameter $b_{\textrm{p}}$ on the dark matter halo parameters. Evidently, increasing $r_{\textrm{s}}$ and $\rho_{\textrm{s}}$ enlarges all of these relativistic parameters, indicating a positive correlation between the dark matter halo and the strength of the gravitational field.
• Figure 4: Variation of the shadow angular diameter with dark matter halo parameters. The left and right panels correspond to the parameters for M87$^{*}$ and the Galactic Center black hole (Sgr A$^{*}$), respectively. It is observed that the left panel constrains the dark matter halo parameters to a relatively narrow range, while the constraints in the right panel are considerably looser. In summary, however, the dark matter halo model remains consistent with current shadow observations.
• Figure 5: The black hole shadow image of a Schwarzschild black hole obtained with OCTOPUS. Black pixels represent rays captured by the black hole. The red closed curve, derived via least-squares fitting, outlines the shadow boundary, with a radius of $5.192 \pm 0.0197$ M, corresponding to a relative error of less than $0.0008$ compared to the theoretical value. It is important to note that this simulation was performed at $500 \times 500$ resolution; higher resolution would further reduce the error.