Horizon fluxes of binary black holes in eccentric orbits

Abstract We compute the rate of change of mass and angular momentum of a black hole, namely tidal heating, in an eccentric orbit. The change is caused due to the tidal field of the orbiting companion. We compute the result for both the spinning and non-spinning black holes in the leading order of the mean motion, namely $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> . We demonstrate that the rates get enhanced significantly for nonzero eccentricity. Since eccentricity in a binary evolves with time we also express the results in terms of an initial eccentricity and azimuthal frequency $$\xi _{\phi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> . In the process, we developed a prescription that can be used to compute all physical quantities in a series expansion of initial eccentricity, $$e_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> . These results are computed taking account of the spin of the binary components. The prescription can be used to compute very high-order corrections of initial eccentricity. We use it to find the contribution to eccentricity up to $${\mathcal {O}}(e_0^5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the spinning binary. Using the computed expression of eccentricity, we derived the rate of change of mass and angular momentum of a black hole, both rotating and non-rotating, in terms of initial eccentricity and azimuthal frequency up to $${\mathcal {O}}(e_0^6)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> <mml:mn>6</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . With the computed fluxes we also compute for the first time the leading order dephasing in both cases analytically up to $${\mathcal {O}}(e_0^6)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>0</mml:mn> <mml:mn>6</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and study its impact. We argue that for high signal-to-noise ratio sources, these contributions require inclusion in the waveform modeling.