Interacting binary stars

An excellent account of stellar structure and evolution is given in the classic text R. Kippenhahn and A. Weigert, Stellar Structure and Evolution, Springer-Verlag 1990. 

A more recent also very useful text giving applications to observational material for clusters and galaxies and a list of useful websites is 

Salaris and S. Cassisi, Evolution of Stars and Stellar Populations, John Wiley, Chichester, 2005. 

These are usefully supplemented (especially for the evolution of close binaries) by the following review article  

Single and Binary Star Evolution , Ap. J. Suppl, 76, 55 (1991). 

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Sahade, J. and Wood, F.B. 1978, Interacting Binary Stars (Pergamon Press), p. 

The structure factor Ss r(q) mirrors repulsive interaction between the stars in good and 6 solvents. Star polymers in dilute solution can be assimilated to soft colloidal particles. Interaction between star polymers in dilute solutions has a character of soft repulsion that arises at separation d 2Rs,ar between the star centers. The binary repulsive potential between the stars was evaluated by Witten and... 

The starred interaction parameters have their usual significance. It was noted that, in this instance, x 3 may represent a better picture of polymer-polymer interaction than X23 since binary mixtures of poly(e-caprolactone) and poly(vinyl chloride) are known to be stable and a small or even negative interaction parameter would be ex-... 

This very simple description applies to those stars which evolve as single stars or as members of a wide binary system which do not interact. It is increasingly clear that a large fraction of stars are born in binary or multiple systems in which two stars exchange material at some point during their evolution. The possibilities of what can happen thereafter are too numerous to be able to cover here, but some of the more bizarre possibilities will be considered later. 

Solution. We use our program (Dimian and Groenendijk, 1994). Bubble point pressures, equilibrium X-factors and densities are presented in Table 6.2. Values calculated using interaction coefficients are marked by star. The differences between properties with and without binary interaction parameters are considerable. 

The first theories that implemented a proper balance of intramolecular interactions and conformational elasticity of the branches were developed by Daoud and Cotton [21] and by Zhulina and Birshtein [22-24]. These theories use scaling concepts (the blob model), originally developed by de Gennes and Alexander to describe the structure of semidilute polymer solutions [64] and planar polymer brushes [65, 66]. Here, the monomer-monomer interactions were incorporated on the level of binary or ternary contacts (corresponding to good and theta-solvent conditions, respectively), and both dilute and semidilute solutions of star polymers were considered. Depending on the solvent quality and the intrinsic stiffness of the arms, the branches of a star could be locally swollen, or exhibit Gaussian statistics [22-24]. 

For a neutral star, 7 = 2 or 3 in a good or a theta-solvent, respectively. The same exponent, 7= 2, is found for a PE star in a concentrated salt solution, see (60). The latter is not surprising, because at a high salt concentration, the electrostatic repulsion between charged monomers is partially screened and the monomer-monomer interactions are described via binary short-range repulsion. 

An increase in concentration Oion of added salt ions, leads to the penetration of salt ions into the star interior and a decrease in the differential osmotic pressure. When the concentration of added ions sufficiently exceeds the average concentration of counterions in the osmotic star, the polyion is found in the so-called salt-dominated regime. Here, the differential osmotic pressure of ions is equivalent to that created by binary monomer-monomer interactions with an effective second viral coefficient Ueff = a /24>ion. As a result, one recovers the same scaling dependence for the size of a PE star as that found for neutral star polymer under good solvent conditions, (4), with replacement u —> Uetr ... 

We, therefore, find that the PE star size, R, in the salt-dominated regime decreases as upon an increase in salt concentration. Note that the same scaling dependence is obtained when the electrostatic repulsions between the charged monomers are accounted for through the screened Coulomb binary interaction potential, u r)/k T = l r exp -r/rj)). A further increase in Oion leads to an additional decrease in When Uett becomes on the order of the bare virial coefficient u, the star size R approaches that of a neutral star, (4). Therefore, the size 7 of a quenched PE star demonstrates a plateau at low salt concentrations (in the osmotic regime), decreases in the salt-dominated regime as 7 und ap-... 

How would binary interaction model expect to fare for star polymers ... 

Fig. 35 Phase diagram of the PS/PVME/THP system at 20°C. Circles measured cloud points, closed stars calculated critical points, open stars calculated tie lines. The values of the specific ternary interaction parameters (61) are indicated on the edges, the binary interaction parameters are the same as in Fig. 34 [74]...

Close binaries interact via tides, and the stars are deformable. Therefore, they are able to show changes on relatively short time scales in the structure of the stellar envelope and to provide important clues to the origin of tidal coupling. 

During the final few minutes before coalescence, the gravitational wave from a neutron star binary sweeps up in amplitude and frequency ( chirps ) through the HF band. When the binary system reaches a frequency of 1 kHz, the orbit will become unstable either due to the tidal interaction between the two stars or because of a dynamical instability of orbital motion in general relativity. At this stage, the details of the merger may depend on the internal properties and spins of the two neutron stars. 

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