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Polytropic Stars

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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.

The code public_poly.tbz computes the structure of stars that are in hydrostatic equilibrium and obey a polytropic equation of state \begin{equation} P = K \rho^{\gamma} = K \rho^{1 + 1/n} \ . \label{eq1} \tag{1} \end{equation} The solution to the resulting Lane-Emden equation \begin{equation} \dfrac{d^2y}{dx^2} + \dfrac{2}{x} \dfrac{dy}{dx} + y^n = 0 \hskip 0.5in y(x=0)=1 \hskip 0.5in \left . \dfrac{dy}{dx}\right |_{x=0} = 0 \ , \label{eq2} \tag{2} \end{equation} where $x$ is a dimensionless radius and $y$ is a dimensionless density, is writen out in dimensionless form and in physical units. Certain polytropic stars are related cold white dwarfs.

image Pathways in the PV-plane for a polytropic equation of state. Note γ ≡ 1 + 1/n = n for the golden ratio of n = Φ = 1.6180…
image Numerical solutions to the Lane-Emden equation. For n < 5, the solutions can be continued to negative y. Although not physical, such solutions exists mathematically, and are quite useful for determining the values of x and dy/dx when y is zero. It's an exercise for the user to compare the computed surface values with other's tabulated values.
image Difference between the analytical and numerical solutions for n = 0, 1, and 5 for various integration accuracies.
image The larger n, the more extended the object. Other physical properties can be gleaned from the output files.
image It was a good day. Chicago. 2nd floor LASR. One in an impeccable brown suit and the other in blue overalls, white t-shirt, and Sear's DieHard steel-toe black shoes.
 



Please cite the relevant references if you publish a piece of work that use these codes, pieces of these codes, or modified versions of them. Offer co-authorship as appropriate.