The characteristics of circular motion and quasiperiodic oscillations around accelerating black hole

Abstract This study explores the motion of massive test particles and associated quasi-periodic oscillations (QPOs) around an accelerating black hole. The acceleration factor A suppresses the radial effective potential $$V_{\text {eff}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> </mml:math> , thereby lowering the energy $${\mathcal {E}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> and angular momentum $${\mathcal {L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> required for stable circular orbits. Stability demands $$\partial _r^2 V_{\text {eff}} \ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:msub> <mml:mi>V</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , setting an upper bound AM $$\le 0.0161$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>0.0161</mml:mn> </mml:mrow> </mml:math> . As A increases, the innermost stable circular orbit (ISCO) radius grows, while $${\mathcal {L}}{\text {ISCO}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mtext>ISCO</mml:mtext> </mml:mrow> </mml:math> and $${\mathcal {E}}{\text {ISCO}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mtext>ISCO</mml:mtext> </mml:mrow> </mml:math> decrease. Radiative efficiency $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> rises with A , peaking at $$6.9\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>6.9</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> . Fundamental frequencies show that A accelerates the decay of the Keplerian $$\Omega _{\phi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> and vertical $$\Omega _{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:math> frequencies, while suppressing the radial frequency E. The divergence between $$\Omega _{\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>θ</mml:mi> </mml:msub> </mml:math> and $$\Omega _{\phi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> increases with A , differing from spherical black hole behavior. Using the RP, ER3, ER4, and WD QPO models, the WD model predicts the highest frequencies. ER4’s resonant radius remains fixed across frequency ratios, unlike ER3. Although A suppresses twin-peak QPO frequencies, it enhances the nodal precession frequency $$\nu _{\text {nod}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ν</mml:mi> <mml:mtext>nod</mml:mtext> </mml:msub> </mml:math> . Fitting observational data from GRO J1655−40 and XTE J1859+226 and applying the TOV limit, the ER4 model uniquely fits GRO J1655−40 with $$(10^3A, M, r/M) \approx (4.31, 3.43 M_\odot , 8.08)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>/</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≈</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4.31</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3.43</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>8.08</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . For XTE J1859+226, three models yield $$10^3A \approx 1.4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>A</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>1.4</mml:mn> </mml:mrow> </mml:math> , excluding ER3, suggesting stronger acceleration in GRO J1655−40.