A Simple Prescription for Envelope Binding EnergyANDREW LOVERIDGE, MARC VAN DER SLUYS, VICKY KALOGERA

1. IntroductionBetween thirty and fifty percent of all stars in the night sky belong to binary, or double star, systems. Under the right conditions, binary systems can enter stages of evolution that do not occur for single stars. One particularly interesting example is known as a common envelope (CE) phase, during which the hydrogen envelope of the primary star engulfs the secondary star. The outcome of a CE is determined using a quantity called the binding energy of the envelope of the primary, which requires detailed knowledge of the internal structure of the primary star. Population-synthesis models, which compute the evolution of large numbers of binary stars, generally lack this detailed information. In this study, we use stellar-structure models to calculate the envelope binding energy for stars of varying age and mass, and determine the best fit to these data. The result will be a simple prescription for this important parameter, requiring only basic macroscopic input values like the stellar mass and radius, which are available in the large-scale synthesis models.

2. Common Envelope Phase

A star expands and contracts during its lifetime. This expansion of the primary can lead to mass transfer from the primary to the secondary (Figure 1a-c), and, for certain orbital periods, this mass transfer is hydrodynamically unstable and leads to the formation of a common envelope engulfing the entire binary orbit. Although a detailed model is difficult to compute, a simple "cartoon" model can serve to illustrate the processes that play a role.

If the mass transfer is unstable, the envelope of the primary star will expand rapidly, so that it soon engulfs the secondary star (Fig. 1d-e) and a common envelope (CE) is formed. Inside the CE, the core of the primary and the entire secondary continue to orbit, ploughing through the gas (Fig. 1f). Friction between these bodies and the surrounding gas will heat the gas. This energy is supplied by a shrinkage of the binary orbit; the more the orbit shrinks, the more energy is released. The heating of the envelope will cause it to expand. The shrinking of the orbit will end only when the envelope is expelled, hence, the final orbital separation of the binary depends on how much energy is needed to expel the envelope: the 'binding energy' of the envelope.

3. Detailed Stellar Models In this study we use the TWIN binary evolution code to compute stellar models for a range of masses. The program begins with a young star and computes it's entire evolution, in each step calculating stellar properties like temperature, luminosity, radius, and most importantly for this study, the envelope binding energy. For this investigation, we computed a grid of models with masses between 0.8 and 20 times that of the Sun.

Figure 2 shows the surface temperature (T_eff) and the luminosity (L) for the model stars. Each track represents the evolution of one of the stars in our grid. The colored parts of each track correspond to the evolutionary phases where the primary can cause a CE, and are discussed in more detail in the next section.

4.. Regions of Interest and Variable Choice Each evolutionary track in Figure 2 is divided into three or four parts. The grey parts of the tracks are evolutionary phases during which the primary star in a binary cannot initiate a CE, and are therefore of no interest to this study. The phases in which a CE can occur are the red giant branch (RGB, drawn in red) and the asymptotic giant branch (AGB, drawn in blue). During both stages, the star expands rapidly and any resulting mass transfer will be unstable, which are the conditions needed for a CE. Hence, we will need to provide a prescription for the envelope binding energy in terms of basic stellar parameters for both of these phases. We found that the envelope binding energy varies regularly and sensitively with the stellar radius. Thus, the radius provides a good, basic stellar property to use for our fits.

5. Fitting We are in the process of generating fits for

the data, in order to describe the envelope binding energy as a function of the stellar radius. Figure 3 shows the binding energy (U_bind) as a function of the radius (R) for the stellar model of 20 solar masses on the RGB as an example . The blue solid line shows the polynomial of the 5th degree that fits the data best. The fits are generated using the software package Mathematica, using the standard mean square difference minimization to determine the coefficients.

For x=log(Radius/Radius of the Sun) and Y=log(Ubind/ergs):

Y= 9.74466×1017-2.09408×1018 x + 1.92873×1018 x2 - 9.06088×1017 x3 + 2.14035×1017 x4 - 2.02066×1016 x5

R2= 0.995336

6. Future Work This project is yet to be finished and a fair amount of future work will still need to becompleted. The following will require attention in particular:

-A final decision on the appropriate criteria for data selection (that is, the separation of the data into four regions of interest for fitting) will need to be made.

-Since the data represents a two dimensional surface rather than a curve when the mass is not held constant, a multivariable fit will ultimately need to be computed. The current curve fits are only a first step.

-An appropriate set of basis functions will need to be decided upon for the fit. Most probably a polynomial function of degree n will be chosen, where n will be picked after inspection of data on goodness of fit versus n.

-Some way of incorporating varying initial composition into the fit will need to be decided upon and implemented. New grids will need to be computed to obtain data on the relationship between the binding energy and composition.