Abstract
We revisit monochromatic and isotropic photon emissions from the zero-angular-momentum sources (ZAMSs) near a Kerr black hole. We investigate the escape probability of the photons that can reach to infinity and study the energy shifts of these escaping photons, which could be expressed as the functions of the source radius and the black hole spin. We study the cases for generic source radius and black hole spin, but we pay special attention to the near-horizon (near-)extremal Kerr ((near-)NHEK) cases. We reproduce the relevant numerical results using a more efficient method and get new analytical results for (near-)extremal cases. The main non-trivial results are: in the NHEK region of a (near-)extremal Kerr black hole, the escape probability for a ZAMS tends to frac{7}{24}approx 29.17%, independent of the NHEK radius; at the innermost of the photon shell (IPS) in the near-NHEK region, the escape probability for a ZAMS tends to 512-17+27πarctan17≈12.57%.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{5}{12} -\\frac{1}{\\sqrt{7}} + \\frac{2}{\\sqrt{7}\\pi }\\arctan \\frac{1}{\\sqrt{7}}\\approx 12.57\\% . \\end{aligned}$$\\end{document}
Highlights
Many black holes are thought to be illuminated by accretion disks surrounding them
The results show that the photon escape probability remains a relatively large nonzero value even though the zero-angular-momentum sources (ZAMSs) is in the deepest region of a near-horizon throat of a high spin Kerr black hole, as long as the ZAMS is outside the innermost photon shell (IPS)
Main conclusions for the photon emissions from the circular sources are that [20,21], the escape probabilities are always greater than 50% regardless of the black hole spin and orbital radius, and there are always sufficient energies carried by these escaping photons due to the fact that the Doppler boost can overcome the gravitational redshift in that case
Summary
Many black holes are thought to be illuminated by accretion disks surrounding them. A toy model of thin accretion disk are composed of photon emitters that orbit around a black hole on equatorial circular geodesics which terminate at the innermost stable circular orbit (ISCO). Photon emissions from ZAMSs near a Kerr(-Newmann) black hole were first studied in [19] last year, where the photon motions were exhaustively classified and the photon escape probabilities were produced numerically. We revisit the observability of ZAMSs near a Kerr black hole that was first studied in [19] by adapting the method introduced in [21] which crucially relies on the critical curve in the source sky. Taking the advantages of the NHEK and near-NHEK geometries, we analytically compute the photon escape probabilities and the redshift factors by applying a high-spin perturbation method [21].
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