We study the influence of parameter $$\alpha $$α on the optical features of Schwarzschild-MOG black holes with different thin accretions in scalar-tensor-vector gravity. As $$\alpha $$α increases from 0, the radii of the event horizon, photon sphere, and observed shadow increase in comparison with the Schwarzschild black hole. We constrain the parameter $$\alpha $$α with the experimental data reported by the Event Horizon Telescope Collaboration for M87 $$^{*}$$∗ and Sagittarius A $$^{*}$$∗ . In the situation of spherical accretions, we unveil that the parameter $$\alpha $$α has a positive effect on the shadow size but a negative effect on the observed specific intensities. Considering that the Schwarzschild-MOG black hole is surrounded by an optical and geometrically thin accretion disk, we find that the total observed specific intensities are mainly contributed by the direct emissions, while the photon rings and lensed rings provide small contributions. It is also found that with the increase of $$\alpha $$α , the black hole shadow expands, the photon rings and lensed rings become larger and thicker. Besides, we emphasize that the boundary of the observed shadow cast by the aim black hole in the disk accretion scenario is determined by the direct emissions rather than the photon ring emissions. Consequently, we unveil that there is a linear relationship involving the critical impact parameter and the starting point of the direct emissions. This finding helps to use the experimental results of the Event Horizon Telescope to infer the critical impact parameter and to test General Relativity.

The present paper contains the investigations with respect to the optical appearance, redshift distribution, and observed energy flux of the thin accretion disk described by the Novikov-Thorne model in modified gravity (MOG) proposed by Moffat. The results show that the gravity coupling parameter $$\alpha $$α plays an indispensable role in the size of direct and secondary images of the accretion disk, while the image profile is mainly affected by the observational angles. In addition, an increase of the parameter $$\alpha $$α leads to that of the redshift factor, but a decrease in the radiation energy flux of the accretion disk. These results are due to the enhancement of the parameter $$\alpha $$α that strengthens the black hole’s gravitational effects. By comparing the accretion disk images of the Schwarzschild-MOG black hole, the Schwarzschild black hole, and the Reissner–Nordström black hole, we point out that the Schwarzschild-MOG black hole exhibits the largest inner shadow and the lowest luminosity. This finding provides a potential way to distinguish Schwarzschild-MOG black holes from their counterparts in General Relativity.

The prompt emission and afterglow phases of gamma-ray bursts (GRBs) have been extensively studied, yet the transition between these two phases remains inadequately characterized due to limited multiwavelength observational coverage. Among the recent growing samples of fast X-ray transients observed by Einstein Probe (EP), a subgroup of GRBs are captured with long-duration X-ray emission, potentially containing featured evolution from prompt emission to the afterglow phase. In this Letter, we present a detailed analysis of GRB 250404A/EP250404a, a bright, fast X-ray transient detected simultaneously by EP and the Fermi Gamma-ray Burst Monitor in X-rays and gamma rays. Its continuous X-ray emission reveals a long-duration tail, accompanied by distinct spectral evolution manifested by the spectral index α _X with an initial softening, followed by an evident hardening, eventually reaching a plateau at the value of ∼–2. Early optical and near-infrared observations enable broadband modeling with forward- and reverse-shock components, confirming that the X-ray hardening signals the emergence of the external-shock afterglow. From this spectral hardening, we infer that the prompt phase in soft X-rays lasted ∼300 s, which is more than 3 times longer than the gamma-ray T ₉₀. This well-tracked soft–hard–flat spectral pattern provides a clear indication of afterglow emergence from the fading prompt emission and offers a practical criterion for identifying a distinct population of GRBs among fast X-ray transients, even when the detection of the gamma-ray counterpart or obvious temporal break is absent.

A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 10⁸ yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.

The inner shadow of a black hole, as a projection of the event horizon, is regarded as a potential tool for testing gravitational theories and constraining system parameters. Whether this holds in the case of a tilted accretion disk warrants further investigation. In this paper, we employ a ray-tracing algorithm to simulate images of the Kerr black hole with a tilted thin accretion disk, with particular attention to the relationship between the inner shadow and system parameters. Our findings reveal that in the case of an equatorial accretion disk, the Kerr black hole exhibits a minimum inner shadow size of S _min = 13.075 M², where M denotes the black hole mass. This minimum is achieved when the viewing angle is 0° and the spin parameter approaches 1. However, with a non-zero disk tilt, the inner shadow exhibits novel configurations — taking on petal, crescent, or eyebrow shapes — significantly smaller than S _min across various parameter spaces. This indicates that the inner shadow is highly sensitive to the accretion environment, suggesting that caution is needed when using it as a diagnostic tool for black holes. Notably, an observed inner shadow smaller than S _min would either indicate the presence of a tilted accretion disk or support the viability of modified gravity. Moreover, in certain parameter spaces, we identify the emergence of a dual-shadow structure, which could also serve as a probe for the tilted accretion disk.

Gamma-ray bursts (GRBs) are luminous stellar explosions characterized by the ejection of relativistic jets. This work proposes a novel paradigm to study these GRB jets. By analyzing the timing information of prompt pulses and X-ray flares, in conjunction with the multiwavelength afterglow observations, we identify three distinct jets in the extraordinary GRB 060729, with initial bulk Lorentz factors ranging from approximately 20 to 80, smaller than typical values of >100. These three jets undergo two successive collisions, producing the observed pair of X-ray flares. Following these interactions, the system evolves into a fast, narrow jet and a slower, hollow jet that continues to propagate in the circumburst medium, evidenced by the notable twin bumps observed in the X-ray and optical afterglow of GRB 060729. Our findings demonstrate that the timing of the early emission enables us to measure the velocities of the GRB jets. The proposed paradigm enhances our understanding of jet dynamics and shock interactions and serves as a powerful tool for probing the physics of the central engine with the expanded sample in the current golden era of GRB research.