Gelles and colleagues presented a prescription for adaptive ray tracing, which enables efficient computation of extremely high resolution images of black holes and revealed it’s subrings.
When surrounded by emitting material, black holes imprint distinctive properties of their spacetimes on the image seen by a distant observer. Black hole images can then offer valuable insights into the astrophysical processes that govern the accretion and outflow, the physical processes that produce heating and dissipation in the nearby plasma, and the geometrical lensing of light. Over the past few decades, images of black holes have evolved from being studied primarily for their rich theoretical features to being directly accessible via very long baseline interferometry. With progressively sharper images of black holes expected as these observations continue to improve, increasingly accurate simulated images of black holes are imperative to guide analysis and interpretation.
One limitation of image accuracy is related to finite image sampling at discrete points on the screen. Namely, the intensity at each point on an image is computed by ray tracing the path of the corresponding null geodesic and computing the radiative transfer along the trajectory. The computational expense of forming an image then increases with the number of rays at which this intensity function is sampled. A crucial question is how to efficiently distribute a finite sample of rays across an image to reach a prescribed image fidelity.
Black hole ray-tracing programs typically distribute rays on an evenly spaced grid. In this approach, regions of the image with sharp, bright features are sampled with the same density of rays as the faint regions of only diffuse structure. Black hole images are expected to have regions of both categories. Near the black hole, the accretion flow is turbulent and bright, requiring high resolution to adequately resolve. Far from the black hole, tightly collimated outflows or “jets” produce narrow regions with significant flux. The strong lensing of Kerr black holes is manifest in the “photon ring,” a bright ring with self-similar substructure that emerges in the limit of no absorption and scattering. Apart from these distinctive parts of the image, black hole images often have the bulk of their flux density concentrated in a small fraction of the image.
Now, Gelles and colleagues in their recently published paper in arxiv, presented a recursive algorithm for adaptive ray-tracing, with natural applications to current and future high-resolution black hole imaging efforts. Whereas most conventional ray-tracing programs spread rays evenly across a uniform grid, their method preferentially samples rays in regions of an image with small-scale structure. When applied to both GRMHD and semianalytic models, they found that their algorithm reduces the time required to generate images by an order of magnitude or more.
They then used this code to generate images with resolutions of 1025 × 1025, 4097 × 4097, and 32769 × 32769 pixels. These images directly visualize the fine structure present in both the accretion flow and the photon ring, revealing the n = 1, 2, and 3 subrings (Figure 1).
Our goal is to maximize the interpolation fraction (IF) while minimizing the flux error (FE) and mean squared error (MSE), as this achieves both efficiency and accuracy. We ran our adaptive scheme on a single GRMHD snapshot (the same snapshot/parameters used in Figure 2) with a wide variety of error tolerances, and then evaluate the IF/FE/MSE for each.
— told Gelles, lead author of the study.
They note additionally from Table 1 that as resolution increases, IF increases as well. This behavior is expected because the error estimates decrease at smaller separations, allowing more pixels to satisfy the error tolerances required for interpolation. Indeed, Figure 3 shows that the density of rays in the diffuse parts of the image quickly saturates – upon increasing the spatial resolution from 1025 × 1025 to 4097 × 4097, the sampled ray density only increases in the photon ring region.
Unlike the IF, however, the MSE and FE do not depend strongly on resolution. These quantities instead depend on the tolerances Rrel and Rabs. Thus, even though more pixels are interpolated at a resolution of 4097 × 4097 compared to a resolution of 1025 × 1025, the output images have comparable accuracy by these two metrics.
Finally, they explored the utility of time averaging in reducing stochastic noise in high-resolution images.
While their algorithm reduces the computational expense required to produce high-resolution images, significant limitations in the physical modeling remain. Although they verified that the magnetization (σ) cut negligibly alters the mean squared error (MSE) of the adaptively ray-traced image, a sharp cut on σ may introduce spurious high-frequency image power.
They are also fundamentally limited in resolution by the MHD cell size. For the MAD model, they used a GRMHD simulation on a spherical polar grid with a resolution of 384 × 192 × 192 in the radial, polar, and azimuthal directions, respectively (zones are compressed exponentially toward the event horizon and lightly toward the midplane). For the SANE model, they used a resolution of 288×128×128. The finite MHD cell size may introduce unphysical, high-power noise from sharp boundaries and will not reproduce subgrid turbulent power.
Future applications of adaptive ray-tracing could extend their results to selectively sample the time and frequency domains. For the purposes of this study, they used one frequency (230 GHz), but fine-scale frequency structure is expected from GRMHD simulations. Adaptive sampling in time would allow efficient generation of high-resolution movies from numerical simulations.
They have integrated this approach into ipole, but it should be compatible with any GRRT scheme that raytraces on a rectangular grid. And while they have used unpolarized transport to generate the images in this paper, the approach generalizes to polarized images as well by simply replacing the total intensity I(~x) with Stokes parameters Q(~x), U(~x), and V (~x). Highly lensed structure near the critical curve resolved with adaptive ray-tracing may show interesting, spin-dependent symmetries in the polarization.
Reference: Z. Gelles, B.S. Prather, D.C.M. Palumbo, M.D. Johnson, G.N. Wong, B. Georgiev, “The Role of Adaptive Ray Tracing in Analyzing Black Hole Structure”, ArXiv, pp. 1-17, 2021. https://arxiv.org/abs/2103.07417
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