Distinguishing Black Holes and Neutron Stars via Optical Images Illuminated by Thick Accretion Disks

Abstract: This paper investigates the optical images of neutron stars within the framework of the radiatively inefficient accretion flow model, taking into account a polytropic equation of state. After obtaining the numerical solutions of the neutron star, we solved numerically the geodesic equations together with the radiative transfer equation. We mainly examine the effects of the polytropic index $N$ and the observer inclination angle $θ_o$ on the image morphology. The obtained images are also compared with the shadow of a Schwarzschild black hole. It is shown that, under the assumption that photon trajectories are terminated at the neutron star surface, the image exhibits a bright higher order structure surrounding an inner dark region. As $N$ increases, the size of the higher-order image gradually expands. As $θ_o$ increases, the obscuration of the neutron star silhouette by radiation originating outside the equatorial plane becomes more pronounced. Compared with the black hole shadow obtained under the same parameter configuration, the neutron star exhibits a larger higher order image and a more extended obscured inner dark region, whereas the higher order image of the black hole is more readily distinguishable. These results indicate significant differences in the optical appearance of neutron stars and black holes, and thus provide a theoretical basis for distinguishing between them through high resolution imaging.

• The paper demonstrates that neutron star images exhibit an expanded higher order photon ring due to gravitational lensing and a softer equation-of-state compared to black holes.
• It employs TOV-based modeling, numerical ray-tracing, and complete radiative transfer in a RIAF framework to generate high-resolution optical images.
• Results indicate that observer inclination and anisotropic radiation critically influence image morphology, providing clear observational discriminants.

Distinguishing Black Holes and Neutron Stars via Optical Images Illuminated by Thick Accretion Disks

Abstract and Motivation

The paper "Distinguishing Black Holes and Neutron Stars via Optical Images Illuminated by Thick Accretion Disks" (2604.00117) provides a rigorous theoretical treatment of optical imaging for neutron stars embedded in radiatively inefficient accretion flows (RIAF), employing a polytropic equation of state for the neutron star interior. The primary objective is to quantify and characterize observable differences in image morphologies between neutron stars and Schwarzschild black holes, especially under high-resolution imaging conditions akin to those realized by the Event Horizon Telescope (EHT).

Methodology

Stellar Structure and Ray-Tracing

Neutron star equilibrium is modeled through the Tolman-Oppenheimer-Volkoff (TOV) equations, with the polytropic equation of state parameterized by the polytropic index $N$. The metric is smoothly fitted across the stellar surface to enable numerical ray-tracing. Photon geodesics in the neutron star's spacetime—alongside radiative transfer equations—are solved, imposing a termination condition at the neutron star surface owing to its optically thick nature. This contrasts with black holes, where photon trajectories are truncated at the event horizon.

Accretion Flow and Radiation

The accretion environment follows the RIAF paradigm, reflecting the empirical geometry inferred for supermassive black holes. Electron number density, temperature, and magnetic field are modeled as radially stratified, with both isotropic and anisotropic synchrotron emission mechanisms considered. Full radiative transfer is implemented, including relativistic invariant intensity, emissivity, and absorptivity, accounting for Doppler shifts and gravitational redshifts.

Imaging Setup

A zero angular momentum observer (ZAMO) is used at various inclination angles $\theta_o$. The imaging pipeline projects celestial coordinates onto a discrete pixel grid for intensity mapping.

Numerical Results

Parameter Dependence and Morphology

The study reveals robust morphological structures across neutron star optical images:

• Higher Order Image Expansion: Increasing polytropic index $N$ (i.e., softer equation of state) results in an enlarged ring of higher order photon trajectories, reflecting greater gravitational lensing due to larger stellar radii and mass distributions.
• Inclination Angle Effects: For small $\theta_o$, the higher order image remains symmetric; as $\theta_o$ increases, obscuration by non-equatorial accretion flow becomes prominent, with inner dark regions receding and sometimes fragmenting.
• Radiation Anisotropy: Anisotropic emission further enhances intensity in the ring, particularly at high $\theta_o$, and significantly diminishes the prominence of inner dark regions.

Comparison with Black Hole Shadows

When comparing neutron star images to Schwarzschild black hole shadows under identical RIAF parameters:

• Ring Size: The neutron star’s higher order ring is consistently larger.
• Inner Darkness: The neutron star exhibits a more extended inner region of reduced intensity; in the black hole, this region is directly associated with the event horizon.
• Image Distinguishability: The higher order ring in the black hole shadow is more readily isolatable, especially at high inclination angles.
• Origin of Darkness: The neutron star’s inner darkness arises from optical depth at the stellar surface (computational truncation), whereas the black hole’s is physical, caused by the event horizon.

Implications and Future Directions

Observational Discrimination

The reported differences—most notably the expansion and morphology of the higher order ring, and the obscured inner dark region—establish critical criteria for distinguishing neutron stars from black holes in compact binary systems and galactic nuclei using VLBI and other high-resolution facilities. The sensitivity of image features to both the polytropic index and observer inclination underscores the necessity of accurate equation-of-state modeling for robust phenomenological inference.

Theoretical Extensions

The implementation of RIAF and ray-tracing methods for neutron stars invites further exploration in several directions:

• GRMHD Integration: The study acknowledges limitations of RIAF and points toward the integration of full general relativistic magnetohydrodynamics (GRMHD) for improved modeling of accretion dynamics, including angular momentum transfer and magnetic reconnection.
• Polarization Effects: Future work will need to incorporate polarized radiative transfer, especially given EHT's polarization mapping capabilities.
• Equation of State Variation: Extending to a wider variety of neutron star equations of state—beyond polytropes—could facilitate constraints on dense matter physics via imaging.

Numerical Robustness and Observational Relevance

The robustness of ray-tracing with truncation at the stellar surface provides a practical computational framework for all optically thick compact objects. The congruence of theoretical images with EHT data for M87* indicates that such models are operationally relevant for interpreting real astrophysical systems.

Conclusion

This paper systematically analyzes the optical images of neutron stars in thick accretion environments, contrasting their morphological features against those of Schwarzschild black holes. The expansion of the higher order image ring and extended obscured inner region in neutron stars, as modulated by polytropic index and inclination, supply theoretically justified, observationally actionable discriminants for compact object identification. The methodology offers a framework for probing dense matter physics via astrophysical imaging and points toward the integration of more sophisticated accretion and radiation models for future studies.