Subdwarf B (sdB) stars (and related sdO/sdOB stars) are believed to be helium‐core‐burning objects with very thin hydrogen‐rich envelopes. In recent years it has become increasingly clear from observational surveys that a large fraction of these objects are members of binary systems. To understand their formation better, we present the results of a detailed investigation of the three main binary evolution channels that can lead to the formation of sdB stars: the common‐envelope (CE) ejection channel, the stable Roche lobe overflow (RLOF) channel, and the double helium white dwarfs (WDs) merger channel. The CE ejection channel leads to the formation of sdB stars in short‐period binaries with typical orbital periods between 0.1 and 10 d, very thin hydrogen‐rich envelopes and a mass distribution sharply peaked around ∼0.46 M⊙. On the other hand, under the assumption that all mass transferred is soon lost, the stable RLOF channel produces sdB stars with similar masses but long orbital periods (400–1500 d) and with rather thick hydrogen‐rich envelopes. The merger channel gives rise to single sdB stars whose hydrogen‐rich envelopes are extremely thin but which have a fairly wide distribution of masses (0.4−0.65 M⊙). We obtained the conditions for the formation of sdB stars from each of these channels using detailed stellar and binary evolution calculations where we modelled the detailed evolution of sdB stars and carried out simplified binary population synthesis simulations. The observed period distribution of sdB stars in compact binaries strongly constrains the CE ejection parameters. The best fits to the observations are obtained for very efficient CE ejection where the envelope ionization energy is included, consistent with previous results. We also present the distribution of sdB stars in the Teff−log g diagram, the Hertzsprung–Russell diagram and the distribution of mass functions.

We have carried out a detailed binary population synthesis (BPS) study of the formation of subdwarf B (sdB) stars and related objects (sdO, sdOB stars) using the latest version of the BPS code developed by Han and co-workers. We systematically investigate the importance of the five main evolutionary channels in which the sdB stars form after one or two common-envelope (CE) phases, one or two phases of stable Roche lobe overflow (RLOF) or as the result of the merger of two helium white dwarfs (WDs). Our best BPS model can satisfactorily explain the main observational characteristics of sdB stars, in particular their distributions in the orbital period-minimum companion mass (log P-M comp ) diagram and in the effective temperaturesurface gravity (T eff -log g) diagram, their distributions of orbital period, log (gθ 4 ) (θ = 5040 K /T eff ) and mass function, their binary fraction and the fraction of sdB binaries with WD companions, their birth rates and their space density. We obtain a Galactic formation rate for sdB stars of 0.014-0.063 yr −1 with a best estimate of ∼0.05 yr −1 and a total number in the Galaxy of 2.4-9.5 × 10 6 with a best estimate of ∼6 × 10 6 ; half of these may be missing in observational surveys owing to selection effects. The intrinsic binary fraction is 76-89 per cent, although the observed frequency may be substantially lower owing to the selection effects. The first CE ejection channel, the first stable RLOF channel and the merger channel are intrinsically the most important channels, although observational selection effects tend to increase the relative importance of the second CE ejection and merger channels. We also predict a distribution of masses for sdB stars that is wider than is commonly assumed and that some sdB stars have companions of spectral type as early as B. The percentage of A-type stars with sdB companions can in principle be used to constrain some of the important parameters in the binary evolution model. We conclude that (i) the first RLOF phase needs to be more stable than is commonly assumed, either because the critical mass ratio q crit for dynamical mass transfer is higher or because of tidally enhanced stellar wind mass loss; (ii) mass transfer in the first stable RLOF phase is non-conservative, and the mass lost from the system takes away a specific angular momentum similar to that of the system; and (iii) common-envelope ejection is very efficient.

This work aims to present our current best physical understanding of common-envelope evolution (CEE). We highlight areas of consensus and disagree- ment, and stress ideas which should point the way forward for progress in this important but long-standing and largely unconquered problem. Unusually for CEE-related work, we mostly try to avoid relying on results from population synthesis or observations, in order to avoid potentially being misled by previous misunderstandings. As far as possible we debate all the relevant issues starting from physics alone, all the way from the evolution of the binary system immediately before CEE begins to the processes which might occur just after the ejection of the envelope. In particular, we include extensive discussion about the energy sources and sinks operating in CEE, and hence examine the foundations of the standard energy formalism. Special attention is also given to comparing the results of hydrodynamic simulations from different groups and to discussing the potential effect of initial conditions on the differences in the outcomes. We compare current numerical techniques for the problem of CEE and also whether more appropriate tools could and should be produced (including new formulations of computational hydrodynamics, and attempts to include 3D processes within 1D codes). Finally we explore new ways to link CEE with observations. We compare previous simulations of CEE to the recent outburst from V1309 Sco, and discuss to what extent post-common-envelope binaries and nebulae can provide information, e.g. from binary eccentricities, which is not currently being fully exploited.

We have carried out a detailed study of one of the most favoured evolutionary channels for the production of Type Ia supernova (SN Ia) progenitors, the single-degenerate channel (CO + MS), where a carbon/oxygen (CO) white dwarf (WD) accretes matter from an unevolved or slightly evolved non-degenerate star until it reaches the Chandrasekhar mass limit. Employing Eggleton's stellar evolution code and adopting the prescription of Hachisu et al. for the accretion efficiency, we performed binary stellar evolution calculations for about 2300 close WD binary systems and mapped out the initial parameters in the orbital period-secondary mass (P-M 2 ) plane (for a range of WD masses) which lead to a successful Type Ia supernova. We obtained accurate, analytical fitting formulae to describe this parameter range which can be used for binary population synthesis (BPS) studies. The contours in the P-M 2 plane differ from those obtained by Hachisu et al. for low-mass CO WDs, which are more common than massive CO WDs. We confirm that WDs with a mass as low as 0.67 M can accrete efficiently and reach the Chandrasekhar limit. We have implemented these results in a BPS study to obtain the birth rates for SNe Ia and the evolution of birth rates with time of SNe Ia for both a constant star formation rate and a single starburst. The birth rates are lower than (but comparable to) those inferred observationally.