On the Common Envelope Efficiency

TL;DR

This paper uses the luminosity versus displacement correlation of high mass X-ray binaries to constrain the common envelope efficiency parameter, finding a preference for higher values around 0.8-1.0, with uncertainties due to core boundary definitions.

Contribution

It introduces a method to constrain the CE efficiency using observational data and updates the models with a variable lambda and new CE criteria for Hertzsprung gap donors.

Findings

High CE efficiency (0.8-1.0) is favored by the data.

Lower CE efficiency (<0.4) cannot reproduce observations.

Uncertainty in core boundary definitions affects lambda and alpha_CE estimates.

Abstract

In this work, we try to use the apparent luminosity versus displacement (i.e., $L_{\rm X}$ vs. $R$) correlation of high mass X-ray binaries (HMXBs) to constrain the common envelope (CE) efficiency $\alpha_{\rm CE}$, which is a key parameter affecting the evolution of the binary orbit during the CE phase. The major updates that crucial for the CE evolution include a variable $\lambda$ parameter and a new CE criterion for Hertzsprung gap donor stars, both of which are recently developed. We find that, within the framework of the standard energy formula for CE and core definition at mass $X=10$\%, a high value of $\alpha_{\rm CE}$, i.e., around 0.8-1.0, is more preferable, while $\alpha_{\rm CE}< \sim 0.4$ likely can not reconstruct the observed $L_{\rm X}$ vs. $R$ distribution. However due to an ambiguous definition for the core boundary in the literature, the used $\lambda$ here still carries almost two order of magnitude uncertainty, which may translate directly to the expected value of $\alpha_{\rm CE}$. We present the detailed components of current HMXBs and their spatial offsets from star clusters, which may be further testified by future observations of HMXB populations in nearby star-forming galaxies.