<p>Astrophysical plasma environments and quantum-gravity-inspired spacetime regularization collectively modify photon dynamics near compact objects, yet their coupled impact on photon sphere structure remains unquantified. This study examines the photon sphere radius (<InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi mathvariant="normal">ph</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$r_{\mathrm{ph}}$</EquationSource> </InlineEquation>) for static Hayward regular black holes immersed in a cold, non-magnetized plasma with a power-law density profile (<InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msubsup> <mi>ω</mi> <mi>p</mi> <mn>2</mn> </msubsup> <mo>∝</mo> <msup> <mi>r</mi> <mrow> <mo>−</mo> <mi>k</mi> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$\omega _{p}^{2} \propto r^{-k}$</EquationSource> </InlineEquation>). The modified condition for circular photon orbits, incorporating the regularization parameter <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>l</mi> </math></EquationSource> <EquationSource Format="TEX">$l$</EquationSource> </InlineEquation> and plasma strength <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> <EquationSource Format="TEX">$K$</EquationSource> </InlineEquation>, was numerically solved across physical parameter ranges. Results demonstrate a consistent inward shift of <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi mathvariant="normal">ph</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$r_{\mathrm{ph}}$</EquationSource> </InlineEquation> relative to the Schwarzschild vacuum case, with individual increases in <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>l</mi> </math></EquationSource> <EquationSource Format="TEX">$l$</EquationSource> </InlineEquation> (to <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mn>1.0</mn> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$1.0M$</EquationSource> </InlineEquation>) or <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>K</mi> </math></EquationSource> <EquationSource Format="TEX">$K$</EquationSource> </InlineEquation> (to 0.5) reducing <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi mathvariant="normal">ph</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$r_{\mathrm{ph}}$</EquationSource> </InlineEquation> by up to <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <mn>12.7</mn> <mi mathvariant="normal">%</mi> </math></EquationSource> <EquationSource Format="TEX">$12.7\%$</EquationSource> </InlineEquation> and over <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math> <mn>18</mn> <mi mathvariant="normal">%</mi> </math></EquationSource> <EquationSource Format="TEX">$18\%$</EquationSource> </InlineEquation>, respectively. Crucially, combined effects exhibit nonlinear synergy: the contraction exceeds additive expectations, peaking at <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math> <mi>l</mi> <mo>≈</mo> <mn>0.8</mn> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$l \approx 0.8M$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="MATHML"><math> <mi>K</mi> <mo>≈</mo> <mn>0.3</mn> </math></EquationSource> <EquationSource Format="TEX">$K \approx 0.3$</EquationSource> </InlineEquation> with a residual shift <InlineEquation ID="IEq14"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> <msub> <mi>r</mi> <mi mathvariant="normal">ph</mi> </msub> <mo stretchy="false">/</mo> <mi>M</mi> <mo>≈</mo> <mo>−</mo> <mn>0.32</mn> </math></EquationSource> <EquationSource Format="TEX">$\Delta r_{\mathrm{ph}}/M \approx -0.32$</EquationSource> </InlineEquation> and a total shift <InlineEquation ID="IEq15"> <EquationSource Format="MATHML"><math> <mi>δ</mi> <msubsup> <mi>r</mi> <mi mathvariant="normal">ph</mi> <mi mathvariant="normal">total</mi> </msubsup> <mo stretchy="false">/</mo> <mi>M</mi> <mo>≈</mo> <mo>−</mo> <mn>0.32</mn> </math></EquationSource> <EquationSource Format="TEX">$\delta r_{\mathrm{ph}}^{\mathrm{total}}/M \approx -0.32$</EquationSource> </InlineEquation> for typical accretion parameters (<InlineEquation ID="IEq16"> <EquationSource Format="MATHML"><math> <mi>k</mi> <mo>=</mo> <mn>1.5</mn> </math></EquationSource> <EquationSource Format="TEX">$k = 1.5$</EquationSource> </InlineEquation>). This synergy, arising from the combined amplification by plasma gradients and quantum-corrected curvature, enhances sensitivity to plasma strength by <InlineEquation ID="IEq17"> <EquationSource Format="MATHML"><math> <mn>60</mn> <mi mathvariant="normal">%</mi> </math></EquationSource> <EquationSource Format="TEX">$60\%$</EquationSource> </InlineEquation> compared to singular geometries. The corresponding 4–<InlineEquation ID="IEq18"> <EquationSource Format="MATHML"><math> <mn>12</mn> <mi mathvariant="normal">%</mi> </math></EquationSource> <EquationSource Format="TEX">$12\%$</EquationSource> </InlineEquation> reduction in black hole shadow diameter underscores the significance of these interactions for interpreting next-generation interferometric observations of strong-gravity lensing features.</p>

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Photon sphere structure in Hayward regular black holes immersed in power-law plasma media

  • Ali Hasnain

摘要

Astrophysical plasma environments and quantum-gravity-inspired spacetime regularization collectively modify photon dynamics near compact objects, yet their coupled impact on photon sphere structure remains unquantified. This study examines the photon sphere radius ( r ph $r_{\mathrm{ph}}$ ) for static Hayward regular black holes immersed in a cold, non-magnetized plasma with a power-law density profile ( ω p 2 r k $\omega _{p}^{2} \propto r^{-k}$ ). The modified condition for circular photon orbits, incorporating the regularization parameter l $l$ and plasma strength K $K$ , was numerically solved across physical parameter ranges. Results demonstrate a consistent inward shift of r ph $r_{\mathrm{ph}}$ relative to the Schwarzschild vacuum case, with individual increases in l $l$ (to 1.0 M $1.0M$ ) or K $K$ (to 0.5) reducing r ph $r_{\mathrm{ph}}$ by up to 12.7 % $12.7\%$ and over 18 % $18\%$ , respectively. Crucially, combined effects exhibit nonlinear synergy: the contraction exceeds additive expectations, peaking at l 0.8 M $l \approx 0.8M$ and K 0.3 $K \approx 0.3$ with a residual shift Δ r ph / M 0.32 $\Delta r_{\mathrm{ph}}/M \approx -0.32$ and a total shift δ r ph total / M 0.32 $\delta r_{\mathrm{ph}}^{\mathrm{total}}/M \approx -0.32$ for typical accretion parameters ( k = 1.5 $k = 1.5$ ). This synergy, arising from the combined amplification by plasma gradients and quantum-corrected curvature, enhances sensitivity to plasma strength by 60 % $60\%$ compared to singular geometries. The corresponding 4– 12 % $12\%$ reduction in black hole shadow diameter underscores the significance of these interactions for interpreting next-generation interferometric observations of strong-gravity lensing features.