Distinguishing Black Holes and Neutron Stars through Optical Images

TL;DR

The paper addresses distinguishing neutron stars from black holes through optical imaging by combining a polynomial EOS $p=K ho^{1+1/n_c}$ with TOV-derived interior solutions and backward ray tracing in static, spherically symmetric spacetimes. It analyzes images produced by spherical illumination and thin accretion disks, showing that increasing the polynomial index $n_c$ increases mass, radius, and compactness, which enlarges the Einstein ring and stellar outline, and shifts redshift patterns with observer inclination. Unlike Schwarzschild black holes, neutron stars exhibit a surface-brightness maximum and a distinct central shadow shaped by light truncation rather than an event horizon, with Doppler effects becoming prominent at high inclinations. These findings provide a theoretical basis for constraining the EOS and differentiating NSs from BHs using high-resolution imaging, and suggest avenues for incorporating more realistic disk models and additional EOS forms in future work.

Abstract

This paper employs the backward ray tracing method to study the optical images of neutron stars under the conditions of a spherical light source and a thin accretion disk, considering a polynomial equation of state given by $p = K ρ^{1 + 1/n_c}$. By numerically solving the TOV equations, we obtain the interior solutions of neutron stars for different densities. The results indicate that as the polynomial index $n_c$ increases, the mass, radius, and compactness of the neutron star all increase, which has a significant impact on its optical properties. Under the assumption that the light is truncated at the surface of the neutron star, we find that for a spherical light source, an increase in $n_c$ leads to an enlargement of the Einstein ring radius. For a thin accretion disk, the light intensity always reaches its maximum at the surface of the neutron star. The increase in $n_c$ also causes the outline of the neutron star to grow. When the observer inclination angle $θ_o$ changes, the neutron star's outline deforms from a circular shape to a D shape, with the left side being significantly brighter than the right side. In addition, this paper also investigates the distribution characteristics of the redshift factor. At lower observer inclination angles, gravitational redshift dominates, while at higher inclination angles, the Doppler effect induces noticeable blueshift. Compared to the Schwarzschild black hole, the optical appearance of the neutron star shows significant differences. The study provides a theoretical basis for distinguishing neutron stars from black holes using high-resolution imaging and for constraining the equation of state.

Figures (6)

• Figure 1: Metric components corresponding to different polynomial indices $n_c$. The left figure shows $-g_{tt}$, and the right figure shows $g_{rr}$. The red, green, blue, and orange curves correspond to $n_c = 1.1, 1.2, 1.3, 1.4$, respectively. The solid and dashed lines represent the fitted metric and numerical metric, while the dotted line represents the Schwarzschild metric for the mass $M$ (neutron star mass).
• Figure 2: Optical image of the neutron star under a spherical light source. The black region represents the neutron star, while the white ring represents the Einstein ring. The fixed parameters are observer inclination angle $\theta_o = 45^\circ$, observer distance $r_o = 200$, and field of view angle $\Phi_{\mathrm{fov}} = 20^\circ$.
• Figure 3: Optical images of the neutron star under a thin accretion disk. From top to bottom, the polynomial indices are $n_c = 1.1, 1.2, 1.3, 1.4$, and from left to right, the observer inclination angles are $\theta_o = 0^\circ, 17^\circ, 60^\circ, 80^\circ$. The fixed parameters are observer distance $r_o = 200$ and field of view angle $\Phi_{\mathrm{fov}} = 10^\circ$.
• Figure 4: Comparison of optical images between the neutron star and the Schwarzschild black hole. Figures \ref{['fig44a']} and \ref{['fig44c']} correspond to the neutron star with $n_c = 1.4$. Figures \ref{['fig44b']} and \ref{['fig44d']} correspond to the Schwarzschild black hole. For all images, the fixed parameters are mass $M = 0.948$, observer distance $r_o = 200$, and field of view angle $\Phi_{\mathrm{fov}} = 10^\circ$.
• Figure 5: Distribution of the redshift factor for the neutron star under a thin accretion disk. In the figure, red represents redshift and blue represents blueshift. The intensity of the color reflects the strength of the effect, with a linear relationship between the two. From top to bottom, the polynomial indices are $n_c = 1.1, 1.2, 1.3, 1.4$, and from left to right, the observer inclination angles are $\theta_o = 0^\circ, 17^\circ, 60^\circ, 80^\circ$. The fixed parameters are observer distance $r_o = 200$ and field of view angle $\Phi_{\mathrm{fov}} = 10^\circ$.