Polytropic Stars


Astronomy research
Software instruments
   Stellar equation of states
   EOS with ionization
   EOS for supernovae
   Chemical potentials
   Stellar atmospheres

   Voigt Function
   Jeans escape
   Polytropic stars
   Cold white dwarfs
   Hotter white dwarfs

   Cold neutron stars
   Stellar opacities
   Neutrino energy loss rates
   Ephemeris routines
   Fermi-Dirac functions

   Galactic chemical evolution
   Coating an ellipsoid
   Universal two-body problem

   Nuclear reaction networks
   Nuclear statistical equilibrium
   Laminar deflagrations
   CJ detonations
   ZND detonations

   Fitting to conic sections
   Unusual linear algebra
   Derivatives on uneven grids
   Pentadiagonal solver
   Quadratics, Cubics, Quartics

   Supernova light curves
   Exact Riemann solutions
   1D PPM hydrodynamics
   Verification problems
   Plane - cube Intersection

   The pendulum
   Circular and elliptical 3 body


   Zingale's software
   Brown's dStar
   GR1D code
   Iliadis' STARLIB database
   Herwig's NuGRID
   Meyer's NetNuc
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Education and Public Outreach

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^{1 + 1/n} \ , \label{eq1} \tag{1} \end{equation} where $x$ is a dimensionless radius and $y$ is a dimensionless density. The solution to the 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} is writen out in dimensionless form and in physical units. Certain polytropic stars are related cold white dwarfs.

image Various pathways in the PV-plane for a polytropic equation of state. Note that gamma = 1/n + 1 = n when n is equal to the golden ratio.
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 plotted from the generated output files.
image It was a good day. Chicago. 2nd floor LASR. One in an impeccable brown suit and the other in overalls, white t-shirt, and sear's die-hard 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.