Transcript Slide 1

A Simple Prescription for Envelope Binding Energy
ANDREW LOVERIDGE, MARC VAN DER SLUYS, VICKY KALOGERA
1. Introduction
Between 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. Populationsynthesis models, which compute the evolution of
large numbers of binary stars, generally lack this
detailed information. In this study, we use stellarstructure 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.
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
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.
6. Future Work
This project is yet to be finished and a fair
amount of future work will still need to be
completed. 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.