The Common Envelope Evolution Outcome—A Case Study on Hot Subdwarf B Stars
Hongwei Ge, Christopher A. Tout, Xuefei Chen, Matthias U. Kruckow, Hailiang Chen, Dengkai Jiang, Zhenwei Li, Zheng-Wei Liu, Zhanwen Han
Abstract
Abstract Common envelope evolution (CEE) physics plays a fundamental role in the formation of binary systems, such as merging stellar gravitational wave sources, pulsar binaries, and Type Ia supernovae. A precisely constrained CEE has become more important in the age of large surveys and gravitational wave detectors. We use an adiabatic mass-loss model to explore how the total energy of the donor changes as a function of the remnant mass. This provides a more self-consistent way to calculate the binding energy of the donor. For comparison, we also calculate the binding energy through integrating the total energy from the core to the surface. The outcome of CEE is constrained by total energy conservation at the point at which both components’ radii shrink back within their Roche lobes. We apply our results to 142 hot subdwarf binaries. For shorter orbital period hot subdwarf B stars (sdBs), the binding energy is highly consistent. For longer orbital period sdBs in our samples, the binding energy can differ by up to a factor of 2. The common envelope (CE) efficiency parameter β CE becomes smaller than α CE for the final orbital period <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>orb</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi>days</mml:mi> <mml:mo>></mml:mo> <mml:mo>−</mml:mo> <mml:mn>0.5</mml:mn> </mml:math> . We also find the mass ratios <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:mi>q</mml:mi> </mml:math> and CE efficiency parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>CE</mml:mi> </mml:mrow> </mml:msub> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>log</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>CE</mml:mi> </mml:mrow> </mml:msub> </mml:math> linearly correlate in sdBs, similarly to the findings of De Marco et al. for post-AGB binaries.