A Simple Prescription for Envelope Binding Energy

1
A Simple Prescription for Envelope Binding Energy
Andrew Loveridge, Marc van der Sluys, Vicky
Kalogera
5. Fitting
3. Detailed Stellar Models
1. Introduction
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.
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.
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.
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.
For xlog(Radius/Radius of the Sun) and
Ylog(Ubind/ergs) Y 9.744661017-2.094081018
x 1.928731018 x2 - 9.060881017 x3
2.140351017 x4 - 2.020661016 x5 R2 0.995336
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.
4.. Regions of Interest and Variable Choice
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.
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.