Project for ASTR7741
Fall Semester, 2007
Instead of the "old" project described, below, I am thinking that it would be interesting to learn how to add an equilibrium isothermal (or polytropic gas) atmosphere to the surface of both distorted stars in a double white dwarf (DWD) binary star system. My idea is to assume that the mass contained in the atmosphere is sufficiently small that its influence on the underlying gravitational field can be neglected. Hence, the distorted effective potential could be constructed using the self-consistent-field techniques that already exist (in the literature as well as in my research group). Then a numerical scheme could be developed to attach an equilibrium atmosphere to both stars while assuming the effective potential is fixed. It would be particularly interesting to add an atmosphere to the donor star that would marginally fill the Roche lobe surrounding the donor.
It probably would be best to start by doing this in a spherically symmetric system, focusing on a determination of the boundary conditions that must be set at the base of the atmosphere. Then the same basic ideas should be easily extendable to nonspherical configurations.
Part I: Add a Thin, Polytropic Atmosphere to an n=1 Polytropic Star
First, examine the analytic solution defining an n=1 polytropic star whose mass is 1 Msolar and whose radius is 1 Rsolar. What is the value of the polytropic constant "K" for this star?
Now, suppose that you strip "epsilon" percent (for example, epsilon = 0.1%) of the star's mass from its outermost layers but assume that the rest of the star's structure remains unchanged. What will the new radius Rnew be? At the outer boundary (Rnew) of this star, what is the density rb, the pressure Pb, and the gravitational potential Fb ? Plot Rnew as a function of epsilon.
What is the formula that defines the gravitational potential outside the new surface of this truncated star? Assume that this potential does not change as you add on a new atmosphere to the star. Pick a polytropic index n' to characterize the equation of state of the atmosphere. Add an atmospheric layer that contains the same total mass as the original atmosphere (that is, m = epsilon times one solar mass) and is everywhere in hydrostatic balance; in particular, make sure that the gradient of the pressure matches the gradient of the gravitational field at the base of the atmosphere, and have the pressure go smoothly to zero at the outer edge of the atmosphere. It might prove best to work with the Bernoulli constant for the gas, as we do in the standard SCF technique.
What is the new radius of this star+atmosphere? What value of the polytropic constant K' was required in order to construct an atmosphere of the desired mass? (Maybe we need to set some other constraint besides the atmospheric mass. Perhaps we should have the outermost edge of this new star+atmosphere extend to a particular effective potential.)
Part II: Add a Thin, Polytropic Atmosphere to the "donor" star in one of Wes Even's binary star systems
The technique developed in "Part I" of this project should be straightforwardly extended to the case of a star that resides in the 3D potential well of a binary system.
OLD PROJECT OUTLINE
Equilibrium Structures for Rotating Polytropes and Polytropic Binary Stars
The procedures that you should follow to construct equilibrium models of rotating polytropes are described fully in my on-line textbook. Specifically, go to the following web page,
and scroll down to the "Summary" section that includes equations III.F.13, III.F.14, and III.F.15. This summary, on through the end of this web page, sets the stage for the more technical description given on the following two pages: The most difficult part of these procedures is the solution of the two- dimensional Poisson equation. I can hand you a Fortran subroutine that provides this solution, if you like; or you can try to see if Mathematica has an appropriate solver. If you develop your own solver, the first thing you should attempt is a solution for the gravitational potential of a uniform-density spheroid. This solution is known analytically.
Part II: Equilibrium, Polytropic Binary Stars
The procedures needed to build polytropic binaries is not technically more difficult than the procedures needed to build rotating, single stars. But it is more computationally demanding. I'm not sure that you have time during this semester course to complete this part of the project. But, let's talk about it.