*
Cococubed.com


Polytropic Stars

Home

Astronomy Research
   Radiative Opacity
   2024 Neutrino Emission from Stars
   2023 White Dwarfs & 12C(α,γ)16O
   2023 MESA VI
   2022 Earendel, A Highly Magnified Star
   2022 Black Hole Mass Spectrum
   2021 Skye Equation of State
   2021 White Dwarf Pulsations & 22Ne
   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
     Adiabatic white dwarfs

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

     Polyhedra volume
     Plane - cube intersection
     Coating an ellipsoid

     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
     Hydrodynamic test cases
     Galactic chemical evolution

     Universal two-body problem
     Circular and elliptical 3 body
     The pendulum
     Phyllotaxis

     MESA
     MESA-Web
     FLASH

     Zingale's software
     Brown's dStar
     GR1D code
     Iliadis' STARLIB database
     Herwig's NuGRID
     Meyer's NetNuc

AAS Journals
   2024 AAS YouTube
   2024 AAS Peer Review Workshops

2024 ASU Energy in Everyday Life
2024 MESA Classroom
Outreach and Education Materials

Other Stuff:
   Bicycle Adventures
   Illustrations
   Presentations



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 written 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 complex solutions exists and are 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.