On the Common Envelope Efficiency

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.Comment: 14 pages, 10 figures, 7 tables, accepted for publication in MNRA

A Note on the Maximum Genus of Graphs with Diameter 4

Let G be a simple graph with diameter four, if G does not contain complete subgraph K3 of order three