Second-order multipole response of two nearby incoherent sources in wave-optical imaging of a Schwarzschild black hole
Abstract
We study wave-optical imaging of two nearby, mutually incoherent point sources by a Schwarzschild black hole. Using the spherical-harmonic addition theorem, we construct partial-wave fields for arbitrary source directions and form images through a finite-aperture Fourier transform. To isolate the binary structure, we compare the binary image with that of a single source of equal total brightness placed at the brightness centroid. Expanding about the centroid removes the first-order term exactly, so the leading structural correction is controlled by the second central moment, μ2=4a2q/(1+q)2. Numerical calculations confirm the expected dependence of the residual on source half-separation and brightness ratio. We then study its wavelength dependence. At long wavelengths, the imaging kernel varies little across the source separation, making the binary nearly indistinguishable from the centroid reference. As the wavelength decreases, finer fringes enhance the residual, which grows by about a factor of three over 3≤ Mω≤12 (2.09≥λ/M≥0.52), even though the pair remains unresolved in the Rayleigh sense. Meanwhile, the second-order approximation gradually breaks down: its relative error first reaches unity for 17<Mω<18 (0.349<λ/M<0.370), indicating the need for higher-order source moments. These results demonstrate that the distinguishability of a nearby binary is intrinsically chromatic and identify the wavelength range in which the binary is both detectable through its residual and accurately described at second order.
Related
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- Second-order scattering response of a Schwarzschild black hole
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BibTeX
@misc{cao2026secondorder,
title = {{Second-order multipole response of two nearby incoherent sources in wave-optical imaging of a Schwarzschild black hole}},
author = {Wenfu Cao and Hongsheng Zhang and Zong-Kuan Guo and Rong-Gen Cai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39110},
eprint = {2609.39110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39110},
}
RIS
TY - UNPB
TI - Second-order multipole response of two nearby incoherent sources in wave-optical imaging of a Schwarzschild black hole
AU - Wenfu Cao
AU - Hongsheng Zhang
AU - Zong-Kuan Guo
AU - Rong-Gen Cai
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39110
UR - https://arxiv.org/abs/2609.39110
AB - We study wave-optical imaging of two nearby, mutually incoherent point sources by a Schwarzschild black hole. Using the spherical-harmonic addition theorem, we construct partial-wave fields for arbitrary source directions and form images through a finite-aperture Fourier transform. To isolate the binary structure, we compare the binary image with that of a single source of equal total brightness placed at the brightness centroid. Expanding about the centroid removes the first-order term exactly, so the leading structural correction is controlled by the second central moment, μ_2=4a^2q/(1+q)^2. Numerical calculations confirm the expected dependence of the residual on source half-separation and brightness ratio. We then study its wavelength dependence. At long wavelengths, the imaging kernel varies little across the source separation, making the binary nearly indistinguishable from the centroid reference. As the wavelength decreases, finer fringes enhance the residual, which grows by about a factor of three over 3≤ Mω≤12 (2.09≥λ/M≥0.52), even though the pair remains unresolved in the Rayleigh sense. Meanwhile, the second-order approximation gradually breaks down: its relative error first reaches unity for 17<Mω<18 (0.349<λ/M<0.370), indicating the need for higher-order source moments. These results demonstrate that the distinguishability of a nearby binary is intrinsically chromatic and identify the wavelength range in which the binary is both detectable through its residual and accurately described at second order.
ER -